Concurrency-Induced Transitions in Epidemic Dynamics on Temporal Networks

Tomokatsu Onaga12 James P Gleeson2 and Naoki Masuda31Department of Physics Kyoto University Kyoto 606-8502 Japan

2MACSI Department of Mathematics and Statistics University of Limerick Limerick V94 T9PX Ireland3Department of Engineering Mathematics University of Bristol Woodland Road Bristol BS8 1UB United Kingdom

(Received 16 February 2017 revised manuscript received 13 June 2017 published 6 September 2017)

Social contact networks underlying epidemic processes in humans and animals are highly dynamic Thespreading of infections on such temporal networks can differ dramatically from spreading on staticnetworks We theoretically investigate the effects of concurrency the number of neighbors that a node hasat a given time point on the epidemic threshold in the stochastic susceptible-infected-susceptible dynamicson temporal network models We show that network dynamics can suppress epidemics (ie yield a higherepidemic threshold) when the nodersquos concurrency is low but can also enhance epidemics when theconcurrency is high We analytically determine different phases of this concurrency-induced transition andconfirm our results with numerical simulations

DOI 101103PhysRevLett119108301

IntroductionmdashSocial contact networksmdashon whichinfectious diseases occur in humans and animals or viralinformation spreads online and offlinemdashare mostlydynamic Switching of partners and the (usually non-Markovian) activity of individuals for example shapenetwork dynamics on such temporal networks [1ndash3] Abetter understanding of epidemic dynamics on temporalnetworks is needed to help improve predictions of andinterventions in emergent infectious diseases to designvaccination strategies and to identify viral marketingopportunities This is particularly so because what weknow about epidemic processes on static networks [4ndash7]is only valid when the time scales of the network dynamicsand of the infectious processes are well separated In facttemporal properties of networks such as long-tailed dis-tributions of intercontact times temporal and cross-edgecorrelation in intercontact times and entries and exits ofnodes considerably alter how infections spread in a net-work [1ndash389]In the present study we focus on a relatively neglected

component of temporal networks ie the number ofconcurrent contacts that a node has Even if two temporalnetworks are the samewhen aggregated over a time horizonthey may be different as temporal networks due to differentlevels of concurrency Concurrency is a longstanding con-cept in epidemiology in particular in the context ofmonogamy or polygamy affecting sexually transmittedinfections [10ndash12] Modeling studies to date largely agreethat a level of high concurrency (eg polygamy as opposedto monogamy) enhances epidemic spreading in a popula-tion However this finding while intuitive lacks theoreticalunderpinning First some models assume that the meandegree or equivalently the average contact rate of nodesincreases as the concurrency increases [13ndash16] In thesecases the observed enhancement in epidemic spreading isan obvious outcomeof a higher density of edges rather than ahigh concurrency Second other models that vary the level

of concurrency while preserving the mean degree arenumerical [10111718] In the present study we use theanalytically tractable activity-driven model of temporalnetworks [19ndash23] to explicitly modulate the size of theconcurrently active network with the structure of theaggregate network fixed With this machinery we carefullytreat extinction effects derive an analytically tractablematrix equation using a probability generating functionfor dynamical networks and reveal nonmonotonic effects oflink concurrency on spreading dynamics We show that thedynamics of networks can either enhance or suppressinfection depending on the amount of concurrency thatindividual nodes have Note that analysis of epidemicprocesses driven by discrete pairwise contact events whichis a popular approach [1ndash3923ndash27] does not address theproblem of concurrency because we must be able to controlthe number of simultaneously active links possessed by anode in order to examine the role of concurrency withoutconfounding with other aspectsModelmdashWe consider the following continuous-time

susceptible-infected-susceptible (SIS) model on a dis-crete-time variant of activity-driven networks which is agenerative model of temporal networks [19ndash23] The num-ber of nodes is denoted by N Each node i eth1 le i le NTHORN isassigned an activity potential ai drawn from a probabilitydensity FethaTHORN eth0 lt a le 1THORN Activity potential ai is theprobability with which node i is activated in a window ofconstant duration τ If activated node i createsm undirectedlinks each of which connects to a randomly selected node

0 τ τ2 τ3 t

FIG 1 Schematic of an activity-driven network with m frac14 3

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8 SEPTEMBER 2017

0031-9007=17=119(10)=108301(5) 108301-1 copy 2017 American Physical Society

(Fig 1) If two nodes are activated and send edges to eachother we only create one edge between them However forlarge N and relatively small ai such events rarely occurAfter a fixed time τ all edges are discarded Then in the nexttime window each node is again activated with probabilityai independently of the activity in the previous timewindow and connects to randomly selected nodes by mundirected links We repeat this procedure Therefore thenetwork changes from one timewindow to another and is anexample of a switching network [28ndash31] A large τ impliesthat network dynamics are slow compared to epidemicdynamics In the limit of τ rarr 0 the network blinksinfinitesimally fast enabling the dynamical process to beapproximated on a time-averaged static network as in [30]For the SIS dynamics each node takes either the

susceptible or infected state At any time each susceptiblenode contracts infection at rate β per infected neighboringnode Each infected node recovers at rate μ irrespectively ofthe neighborsrsquo states Changing τ to cτ ethc gt 0THORN is equivalentto changing β and μ to β=c and μ=c respectively whileleaving τ unchanged Thereforewe setμ frac14 1without loss ofgeneralityAnalysismdashWe calculate the epidemic threshold as fol-

lows First we formulate SIS dynamics near the epidemicthreshold on a static star graph which is the building block ofthe activity-driven model while explicitly consideringextinction effects Second we convert the obtained set oflinear difference equations into a tractablemathematical formwith the use of a probability generating function of an activitydistribution Third the epidemic threshold is obtained froman implicit function For the sake of the analysis we assumethat star graphs generated by an activated nodewhichwe callthe hub are disjoint from each other Because a star graphwith hub node i overlaps with another star graph withprobability asympm

Pjneiajethmthorn 1THORN=N prop m2hai where haiequivR

daFethaTHORNa is the mean activity potential we imposem2hai ≪ 1 (However our method works better than theso-called individual-based approximation even whenm2hai frac14 05 as shown in the Supplemental Material[32]) We denote by ρetha tTHORN the probability that a nodewith activity a is infected at time t The fraction of infectednodes in the entire network at time t is given byhρethtTHORNiequiv R

daFethaTHORNρetha tTHORN Let c1 be the probability withwhich the hub in an isolated star graph is infected at timetthorn τ when the hub is the only infected node at time t and thenetwork has switched to a new configuration right at time tLet c2 be the probability with which the hub is infected attthorn τ when only a single leaf node is infected at t Theprobability that a hubwith activity potentiala is infected afterthe duration τ of the star graph denoted by ρ1 is given by

ρ1etha tthorn τTHORN frac14 c1ρetha tTHORN thorn c2mhρethtTHORNi eth1THORNIn deriving Eq (1) we considered the situation near theepidemic threshold such that at most one node is infected inthe star graph at time t [and hence ρetha tTHORN hρethtTHORNi ≪ 1] The

probability that a leaf with activity potential a that has a hubneighbor with activity potential a0 is infected after time τ isanalogously given by

ρ2etha a0 tthorn τTHORN frac14 c3ρetha tTHORN thorn c4ρetha0 tTHORNthorn c5ethm minus 1THORNhρethtTHORNi eth2THORN

where c3 c4 and c5 are the probabilities with which a leafnodewith activity potentiala is infected after duration τwhenonly that leaf node the hub and a different leaf node isinfected at time t respectively We derive formulas for cieth1 le i le 5THORN in the Supplemental Material [32] The proba-bility that an isolated nodewith activity potentiala is infectedafter time τ is given by eminusτρetha tTHORN By combining thesecontributions we obtain

ρetha tthorn τTHORN frac14 aρ1etha tthorn τTHORN thornZ

da0Fetha0THORNma0ρ2ethaa0 tthorn τTHORN

thorn eth1minus aminusmhaiTHORNeminusτρetha tTHORN eth3THORNTo analyze Eq (3) further we take a generating function

approach With this approach one trades a probabilitydistribution for a probability generating function whosederivatives provide us with useful information about thedistribution such as its moments Furthermore it oftenmakes analysis easier in particular linear analysis Bymultiplying Eq (3) by za and averaging over a we obtain

Θethz tthorn τTHORN frac14 c01Θeth1THORNethz tTHORN thorn c02Θeth1 tTHORNgeth1THORNethzTHORN thorn c03Θethz tTHORNthorn frac12c04Θeth1THORNeth1 tTHORN thorn c05Θeth1 tTHORNgethzTHORN eth4THORN

where c01equivc1minuseminusτ c02 equivmc2 c03 equiv eminusτ thornmhaiethc3 minus eminusτTHORNc04 equivmc4 c05 equivmethm minus 1THORNhaic5 gethzTHORNequiv R

daFethaTHORNza isthe probability generating function of a Θethz tTHORNequivRdaFethaTHORNρetha tTHORNza and throughout the paper the super-

script (n) represents the nth derivative with respect to ln zAlthough Eq (3) is an infinite dimensional system of lineardifference equations Eq (4) is a single difference equationof Θethz tTHORN and its derivative [35]We expand ρetha tTHORN as a Maclaurin series as follows

ρetha tTHORN frac14Xinfinnfrac141

wnethtTHORNanminus1 eth5THORN

Using this polynomial basis representation (the conver-gence is proven in the Supplemental Material [32]) we canconsider the differentiations in Eq (4) [ie Θeth1THORNethz tTHORN andgeth1THORNethzTHORN] as an exchange of bases and convert Eq (4) into atractable matrix form Let p0 be the fraction of initiallyinfected nodes which are selected uniformly at randomindependently of a We represent the initial condition aswetht frac14 0THORNequiv ethw1eth0THORN w2eth0THORNhellipTHORN⊤ frac14 ethp0 0 0hellipTHORN⊤Epidemic dynamics near the epidemic threshold obey lineardynamics given by

wethtthorn τTHORN frac14 TethτTHORNwethtTHORN eth6THORN

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108301-2

By substituting Θethz tTHORN frac14 Pinfinnfrac141 wnethtTHORNgethnminus1THORNethzTHORN and gethnminus1THORNeth1THORN frac14 hanminus1i in Eq (4) we obtain

T frac14

0BBBBBBBBB

c03 thorn haic04 thorn c05 ha2ic04 thorn haic05 ha3ic04 thorn ha2ic05 ha4ic04 thorn ha3ic05 ha5ic04 thorn ha4ic05 c01 thorn c02 haic02 thorn c03 ha2ic02 ha3ic02 ha4ic02

0 c01 c03 0 0 0 0 c01 c03 0 0 0 0 c01 c03

1CCCCCCCCCA eth7THORN

A positive prevalence hρethtTHORNi (ie a positive fraction ofinfected nodes in the equilibrium state) occurs only if thelargest eigenvalue of TethτTHORN exceeds 1 because in thissituation the probability of being infected grows in timeat least in the linear regime Therefore we get the followingimplicit function for the epidemic threshold denoted by βc

fethτ βcTHORNequiv eth1 minus rTHORNeth1 minus sTHORN minus eth1thorn qTHORNuSethqTHORN

minus qr minus qsthorn qrs minus q2u minus rs frac14 0 eth8THORNwhere SethqTHORNequivPinfin

nfrac140ethhanthorn2i=hainthorn2THORNqn frac14eth1=hai2THORNfhetha2THORN=frac121minus etha=haiTHORNqig qequiv haic01=eth1 minus c03THORN requiv haic02=eth1 minus c03THORNsequiv haic04=eth1 minus c03THORN and uequiv c05=eth1 minus c03THORN (see Supplemen-tal Material [32] for the derivation) Note that f is afunction of βethfrac14 βcTHORN through q r s and u which arefunctions of β In general we obtain βc by numericallysolving Eq (8) but some special cases can be determinedanalyticallyIn the limit τ rarr 0 Eq (8) gives βc frac14 frac12methhaithornffiffiffiffiffiffiffiffiffiha2i

pTHORNminus1 which coincides with the epidemic threshold

for the activity-driven model derived in the previous studies[1922] In fact this βc value is the epidemic threshold forthe aggregate (and hence static) network whose adjacencymatrix is given by A

ij asympmethai thorn ajTHORN=N [331] as demon-strated in Fig S1 [32]For general τ if all nodes have the same activity potential

a and if m frac14 1 we obtain βc as the solution of thefollowing implicit equation

2aefrac12ethβcminus1THORNτ=2cosh

κcτ

2

thorn 1thorn 3βc

κcsinh

minusκcτ2

minus eτ thorn 1 minus 2a frac14 0 eth9THORNwhere κc frac14

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ2c thorn 6βc thorn 1

p

The theoretical estimate of the epidemic threshold[Eq (8) we use Eq (9) in the case of m frac14 1] is shownby the solid lines in Figs 2(a) and 2(b) It is compared withnumerically calculated prevalence values for various τ andβ values shown in different colors Equations (8) and (9)describe the numerical results fairly well Whenm frac14 1 theepidemic threshold increases with τ and diverges at τ asymp 01[Fig 2(a)] Furthermore slower network dynamics (ie

larger values of τ) reduce the prevalence for all values of βIn contrast when m frac14 10 the epidemic thresholddecreases and then increases as τ increases [Fig 2(b)]The network dynamics (ie finite τ) impact epidemicdynamics in a qualitatively different manner dependingon m ie the number of concurrent neighbors that a hubhas Note that the estimate of βc by the individual-basedapproximation ([31] see Supplemental Material [32] forthe derivation) which may be justified when m ≫ 1 isconsistent with the numerical results and our theoreticalresults only at small τ [a dashed line in Fig 2(b)]

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(a)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(b)

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(c)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(d)

lt gtρ

lt gtρ

lt gtρ

lt gtρ

FIG 2 Epidemic threshold and the numerically-simulatedprevalence when m frac14 1 (a)(c) and m frac14 10 (b)(d) In (a)and (b) all nodes have the same activity potential value a Thesolid lines represent the analytical estimate of the epidemicthreshold [Eq (8) we plot Eq (9) instead in (a)] The dashedlines represent the epidemic threshold obtained from theindividual-based approximation (Supplemental Material [32])The color indicates the prevalence In (c) and (d) the activitypotential (ϵ le ai le 09 1 le i le N) obeys a power-law distri-bution with exponent 3 In (a)ndash(d) we set N frac14 2000 and adjustthe values of a and ϵ such that the mean degree is the same(hki frac14 01) in the four cases We simulate the stochastic SISdynamics using the quasistationary state method [36] as in[31] and calculate the prevalence averaged over 100 realiza-tions after discarding the first 15 000 time steps We set thestep size Δt frac14 0002 Qualitatively similar results are obtainedfor the variant of the activity-driven model with a reinforcementmechanism of link creation [37] (Fig S3 [32])

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108301-3

Qualitatively similar results are found when the activitypotential a is power-law distributed [Figs 2(c) and 2(d)]To illuminate the qualitatively different behaviors of the

epidemic threshold as τ increases we determine a phasediagram for the epidemic threshold We focus our analysison the case in which all nodes share the activity potentialvalue a noting that qualitatively similar results arealso found for power-law distributed activity potentials[Fig 3(b)] We calculate the two boundaries partitioningdifferent phases as follows First we observe that theepidemic threshold diverges at τ frac14 τ In the limit β rarr infininfection starting from a single infected node in a star graphimmediately spreads to the entire star graph leading toci rarr 1 eth1 le i le 5THORN By substituting ci rarr 1 in Eq (8) weobtain fethτ βc rarr infinTHORN frac14 0 where

τ frac14 ln1 minus eth1thornmTHORNa1 minus eth1thornmTHORN2a eth10THORN

When τ gt τ infection always dies out even if the infectionrate is infinitely large This is because in a finite networkinfection always dies out after a sufficiently long time dueto stochasticity [38ndash40] Second although βc eventuallydiverges as τ increases there may exist τc such that βc atτ lt τc is smaller than the βc value at τ frac14 0 Motivated bythe comparison between the behavior of βc at m frac14 1and m frac14 10 (Fig 2) we postulate that τc (gt 0) existsonly for m gt mc Then we obtain dβc=dτ frac14 0 atethτ mTHORN frac14 eth0 mcTHORN The derivative of Eq (8) gives partf=partτthornethpartf=partβcTHORNethdβc=dτTHORN frac14 0 Because dβc=dτ frac14 0 at ethτ mTHORN frac14eth0 mcTHORN we obtain partf=partτ frac14 0 which leads to

mc frac143

1 minus 4a eth11THORN

When m lt mc network dynamics (ie finite τ) alwaysreduce the prevalence for any τ [Figs 2(a) and 2(c)] Whenm gt mc a small τ raises the prevalence as compared to

τ frac14 0 (ie static network) but a larger τ reduces theprevalence [Figs 2(b) and 2(d)]The phase diagram based on Eqs (10) and (11) is shown

in Fig 3(a) The βc values numerically calculated bysolving Eq (8) are also shown in the figure It shouldbe noted that the parameter values are normalized such thatβc has the same value for all m at τ frac14 0 We find that thedynamics of the network may either increase or decreasethe prevalence depending on the number of connectionsthat a node can simultaneously have extending the resultsshown in Fig 2These results are not specific to the activity-drivenmodel

The phase diagram is qualitatively similar for randomlydistributed m (Fig S4 [32]) for different distributions ofactivity potentials (Fig S5 [32]) and for a different model inwhich an activated node induces a clique instead of a star(Fig S6 [32]) modeling a group conversation event as sometemporal network models do [41ndash43]DiscussionmdashOur analytical method shows that the

presence of network dynamics boosts the prevalence(and decreases the epidemic threshold βc) when theconcurrency m is large and suppresses the prevalence(and increases βc) when m is small for a range of valuesof the network dynamic time scale τ This result lendstheoretical support to previous claims that concurrencyboosts epidemic spreading [101113ndash1944] The resultmay sound unsurprising because a large m value impliesthat there exists a large connected component at any giventime However our finding is not trivial because a largecomponent consumes many edges such that other parts ofthe network at the same time or the network at other timeswould be more sparsely connected as compared to the caseof a smallm We confirmed that qualitatively similar resultsare found when the activity potentials were constructedfrom two empirical social contact networks (Fig S7 [32])Our results confirm that a monogamous sexual relationshipor a small group of people chatting face to face as opposedto polygamous relationships or large groups of conversa-tions hinders epidemic spreading where we compare likewith like by constraining the aggregate (static) network tobe the same in all cases For general temporal networksimmunization strategies that decrease concurrency (egdiscouraging polygamy) may be efficient Restricting thesize of the concurrent connected component (eg size of aconversation group) may also be a practical strategyAnother important contribution of the present study is

the observation that infection dies out for a sufficientlylarge τ regardless of the level of concurrency As shown inFigs 3 and S6 [32] the transition to the ldquodie outrdquo phaseoccurs at values of τ that correspond to network dynamicsand epidemic dynamics having comparable time scalesThis is a stochastic effect and cannot be captured byexisting approaches to epidemic processes on temporalnetworks that neglect stochastic dying out such as differ-ential equation systems for pair formulation-dissolutionmodels [1115ndash18] and individual-based approximations[314546] Our analysis methods explicitly consider such

τ

1

5mc

10

5

100

0 02 080604 1

β c

10

50

die out

enhanced

supp

ress

ed

(b)

τ

1

5

mc

10

0 02 080604 1

mm die out

enhanced

supp

ress

ed

(a)

FIG 3 Phase diagrams for the epidemic threshold βc whenthe activity potential is (a) equal to a for all nodes or (b) obeys apower-law distribution with exponent 3 (ϵ le ai le 09) We sethki frac14 01 at m frac14 1 and adjust the value of a and ϵ such that βctakes the same value for all m at τ frac14 0 In the ldquodie outrdquo phaseinfection eventually dies out for any finite β In the ldquosuppressedrdquophase βc is larger than the βc value at τ frac14 0 In the ldquoenhancedrdquophase βc is smaller than the βc value at τ frac14 0 The solid anddashed lines represent τ [Eq (10)] and τc respectively The colorbar indicates the βc values In the gray regions βc gt 100

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108301-4

stochastic effects and are therefore expected to be usefulbeyond the activity-driven model (or the clique-basedtemporal networks analyzed in the SupplementalMaterial [32]) and the SIS model

We thank Leo Speidel for discussion We thank theSocioPatterns collaboration (http wwwsociopatternsorg)for providing the data set T O acknowledges the supportprovided through JSPS Research Fellowship for YoungScientists J G acknowledges the support providedthrough Science Foundation Ireland (Grants No 15SPPE3125 and No 11PI1026) N M acknowledges thesupport provided through JST CREST and JSTERATO Kawarabayashi Large Graph Project

naokimasudabristolacuk[1] P Holme and J Saramaumlki Phys Rep 519 97 (2012)[2] P Holme Eur Phys J B 88 234 (2015)[3] N Masuda and R Lambiotte A Guide to Temporal

Networks (World Scientific Singapore 2016)[4] M J Keeling and K T D Eames J R Soc Interface 2 295

(2005)[5] A Barrat M Bartheacutelemy and A Vespignani Dynamical

Processes on Complex Networks (Cambridge UniversityPress Cambridge England 2008)

[6] R Pastor-Satorras C Castellano P Van Mieghem and AVespignani Rev Mod Phys 87 925 (2015)

[7] M A Porter and J P Gleeson Dynamical Systems onNetworks Frontiers in Applied Dynamical SystemsReviews and Tutorials (Springer International PublishingCham Switzerland 2016) Vol 4

[8] S Bansal J Read B Pourbohloul and L A Meyers JBiol Dyn 4 478 (2010)

[9] N Masuda and P Holme F1000Prime Rep 5 6 (2013)[10] M Morris and M Kretzschmar Soc Networks 17 299

(1995)[11] M Kretzschmar and M Morris Math Biosci 133 165

(1996)[12] J C Miller and A C Slim arXiv161104800[13] C H Watts and R M May Math Biosci 108 89 (1992)[14] I A Doherty S Shiboski J M Ellen A A Adimora and

N S Padian Sex Transm Dis 33 368 (2006)[15] K Gurski and K Hoffman Math Biosci 282 91 (2016)[16] K T D Eames and M J Keeling Math Biosci 189 115

(2004)[17] C Bauch and D A Rand Proc R Soc B 267 2019 (2000)[18] K Y Leung and M Kretzschmar AIDS 29 1097 (2015)[19] N Perra B Gonccedilalves R Pastor-Satorras and A

Vespignani Sci Rep 2 469 (2012)[20] M Starnini and R Pastor-Satorras Phys Rev E 87 062807

(2013)[21] B Ribeiro N Perra and A Baronchelli Sci Rep 3 3006

(2013)[22] S Liu N Perra M Karsai and A Vespignani Phys Rev

Lett 112 118702 (2014)

[23] L Zino A Rizzo and M Porfiri Phys Rev Lett 117228302 (2016)

[24] A Vazquez B Raacutecz A Lukaacutecs and A L Barabaacutesi PhysRev Lett 98 158702 (2007)

[25] M Karsai M Kivelauml R K Pan K Kaski J Kerteacutesz A LBarabaacutesi and J Saramaumlki Phys Rev E 83 025102(R)(2011)

[26] G Miritello E Moro and R Lara Phys Rev E 83 045102(R) (2011)

[27] J Stehleacute N Voirin A Barrat C Cattuto V Colizza LIsella C Reacutegis J F Pinton N Khanafer W Van denBroeck and P Vanhems BMC Med 9 87 (2011)

[28] D Liberzon Switching in Systems and Control Systemsand Control Foundations and Applications (BirkhaumluserBoston 2003)

[29] N Masuda K Klemm and V M Eguiacuteluz Phys Rev Lett111 188701 (2013)

[30] M Hasler V Belykh and I Belykh SIAM J Appl DynSyst 12 1007 (2013)

[31] L Speidel K Klemm V M Eguiacuteluz and N MasudaNew J Phys 18 073013 (2016)

[32] See Supplemental Material at httplinkapsorgsupplemental101103PhysRevLett119108301 which in-cludes Refs [3334] for details of mathematical derivationsand extensions of the model

[33] M Geacutenois C L Vestergaard J Fournet A Panisson IBonmarin and A Barrat Netw Sci 3 326 (2015)

[34] A Paranjape A R Benson and J Leskovec in Proceed-ings of the Tenth ACM International Conference on WebSearch and Data Mining WSDMrsquo17 (ACM New York2017) pp 601ndash610

[35] H Silk G Demirel M Homer and T Gross New J Phys16 093051 (2014)

[36] MM de Oliveira and R Dickman Phys Rev E 71 016129(2005)

[37] M Karsai N Perra and A Vespignani Sci Rep 4 4001(2014)

[38] M J Keeling and J V Ross J R Soc Interface 5 171(2008)

[39] P L Simon M Taylor and I Z Kiss J Math Biol 62 479(2011)

[40] J Hindes and I B Schwartz Phys Rev Lett 117 028302(2016)

[41] C Tantipathananandh T Berger-Wolf and D Kempe inProceedings of the Thirteenth ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining(ACM New York 2007) pp 717ndash726

[42] J Stehleacute A Barrat and G Bianconi Phys Rev E 81035101(R) (2010)

[43] K Zhao M Karsai and G Bianconi PLoS One 6 e28116(2011)

[44] J W Eaton T B Hallett and G P Garnett AIDS Behav15 687 (2011)

[45] E Valdano L Ferreri C Poletto and V Colizza Phys RevX 5 021005 (2015)

[46] L E C Rocha and N Masuda Sci Rep 6 31456(2016)

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108301-5

By substituting Θethz tTHORN frac14 Pinfinnfrac141 wnethtTHORNgethnminus1THORNethzTHORN and gethnminus1THORNeth1THORN frac14 hanminus1i in Eq (4) we obtain

T frac14

0BBBBBBBBB

c03 thorn haic04 thorn c05 ha2ic04 thorn haic05 ha3ic04 thorn ha2ic05 ha4ic04 thorn ha3ic05 ha5ic04 thorn ha4ic05 c01 thorn c02 haic02 thorn c03 ha2ic02 ha3ic02 ha4ic02

0 c01 c03 0 0 0 0 c01 c03 0 0 0 0 c01 c03

1CCCCCCCCCA eth7THORN

A positive prevalence hρethtTHORNi (ie a positive fraction ofinfected nodes in the equilibrium state) occurs only if thelargest eigenvalue of TethτTHORN exceeds 1 because in thissituation the probability of being infected grows in timeat least in the linear regime Therefore we get the followingimplicit function for the epidemic threshold denoted by βc

fethτ βcTHORNequiv eth1 minus rTHORNeth1 minus sTHORN minus eth1thorn qTHORNuSethqTHORN

minus qr minus qsthorn qrs minus q2u minus rs frac14 0 eth8THORNwhere SethqTHORNequivPinfin

nfrac140ethhanthorn2i=hainthorn2THORNqn frac14eth1=hai2THORNfhetha2THORN=frac121minus etha=haiTHORNqig qequiv haic01=eth1 minus c03THORN requiv haic02=eth1 minus c03THORNsequiv haic04=eth1 minus c03THORN and uequiv c05=eth1 minus c03THORN (see Supplemen-tal Material [32] for the derivation) Note that f is afunction of βethfrac14 βcTHORN through q r s and u which arefunctions of β In general we obtain βc by numericallysolving Eq (8) but some special cases can be determinedanalyticallyIn the limit τ rarr 0 Eq (8) gives βc frac14 frac12methhaithornffiffiffiffiffiffiffiffiffiha2i

pTHORNminus1 which coincides with the epidemic threshold

for the activity-driven model derived in the previous studies[1922] In fact this βc value is the epidemic threshold forthe aggregate (and hence static) network whose adjacencymatrix is given by A

ij asympmethai thorn ajTHORN=N [331] as demon-strated in Fig S1 [32]For general τ if all nodes have the same activity potential

a and if m frac14 1 we obtain βc as the solution of thefollowing implicit equation

2aefrac12ethβcminus1THORNτ=2cosh

κcτ

2

thorn 1thorn 3βc

κcsinh

minusκcτ2

minus eτ thorn 1 minus 2a frac14 0 eth9THORNwhere κc frac14

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ2c thorn 6βc thorn 1

p

The theoretical estimate of the epidemic threshold[Eq (8) we use Eq (9) in the case of m frac14 1] is shownby the solid lines in Figs 2(a) and 2(b) It is compared withnumerically calculated prevalence values for various τ andβ values shown in different colors Equations (8) and (9)describe the numerical results fairly well Whenm frac14 1 theepidemic threshold increases with τ and diverges at τ asymp 01[Fig 2(a)] Furthermore slower network dynamics (ie

larger values of τ) reduce the prevalence for all values of βIn contrast when m frac14 10 the epidemic thresholddecreases and then increases as τ increases [Fig 2(b)]The network dynamics (ie finite τ) impact epidemicdynamics in a qualitatively different manner dependingon m ie the number of concurrent neighbors that a hubhas Note that the estimate of βc by the individual-basedapproximation ([31] see Supplemental Material [32] forthe derivation) which may be justified when m ≫ 1 isconsistent with the numerical results and our theoreticalresults only at small τ [a dashed line in Fig 2(b)]

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(a)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(b)

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(c)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(d)

lt gtρ

lt gtρ

lt gtρ

lt gtρ

FIG 2 Epidemic threshold and the numerically-simulatedprevalence when m frac14 1 (a)(c) and m frac14 10 (b)(d) In (a)and (b) all nodes have the same activity potential value a Thesolid lines represent the analytical estimate of the epidemicthreshold [Eq (8) we plot Eq (9) instead in (a)] The dashedlines represent the epidemic threshold obtained from theindividual-based approximation (Supplemental Material [32])The color indicates the prevalence In (c) and (d) the activitypotential (ϵ le ai le 09 1 le i le N) obeys a power-law distri-bution with exponent 3 In (a)ndash(d) we set N frac14 2000 and adjustthe values of a and ϵ such that the mean degree is the same(hki frac14 01) in the four cases We simulate the stochastic SISdynamics using the quasistationary state method [36] as in[31] and calculate the prevalence averaged over 100 realiza-tions after discarding the first 15 000 time steps We set thestep size Δt frac14 0002 Qualitatively similar results are obtainedfor the variant of the activity-driven model with a reinforcementmechanism of link creation [37] (Fig S3 [32])

PRL 119 108301 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

8 SEPTEMBER 2017

108301-3

Qualitatively similar results are found when the activitypotential a is power-law distributed [Figs 2(c) and 2(d)]To illuminate the qualitatively different behaviors of the

epidemic threshold as τ increases we determine a phasediagram for the epidemic threshold We focus our analysison the case in which all nodes share the activity potentialvalue a noting that qualitatively similar results arealso found for power-law distributed activity potentials[Fig 3(b)] We calculate the two boundaries partitioningdifferent phases as follows First we observe that theepidemic threshold diverges at τ frac14 τ In the limit β rarr infininfection starting from a single infected node in a star graphimmediately spreads to the entire star graph leading toci rarr 1 eth1 le i le 5THORN By substituting ci rarr 1 in Eq (8) weobtain fethτ βc rarr infinTHORN frac14 0 where

τ frac14 ln1 minus eth1thornmTHORNa1 minus eth1thornmTHORN2a eth10THORN

When τ gt τ infection always dies out even if the infectionrate is infinitely large This is because in a finite networkinfection always dies out after a sufficiently long time dueto stochasticity [38ndash40] Second although βc eventuallydiverges as τ increases there may exist τc such that βc atτ lt τc is smaller than the βc value at τ frac14 0 Motivated bythe comparison between the behavior of βc at m frac14 1and m frac14 10 (Fig 2) we postulate that τc (gt 0) existsonly for m gt mc Then we obtain dβc=dτ frac14 0 atethτ mTHORN frac14 eth0 mcTHORN The derivative of Eq (8) gives partf=partτthornethpartf=partβcTHORNethdβc=dτTHORN frac14 0 Because dβc=dτ frac14 0 at ethτ mTHORN frac14eth0 mcTHORN we obtain partf=partτ frac14 0 which leads to

mc frac143

1 minus 4a eth11THORN

When m lt mc network dynamics (ie finite τ) alwaysreduce the prevalence for any τ [Figs 2(a) and 2(c)] Whenm gt mc a small τ raises the prevalence as compared to

τ frac14 0 (ie static network) but a larger τ reduces theprevalence [Figs 2(b) and 2(d)]The phase diagram based on Eqs (10) and (11) is shown

in Fig 3(a) The βc values numerically calculated bysolving Eq (8) are also shown in the figure It shouldbe noted that the parameter values are normalized such thatβc has the same value for all m at τ frac14 0 We find that thedynamics of the network may either increase or decreasethe prevalence depending on the number of connectionsthat a node can simultaneously have extending the resultsshown in Fig 2These results are not specific to the activity-drivenmodel

The phase diagram is qualitatively similar for randomlydistributed m (Fig S4 [32]) for different distributions ofactivity potentials (Fig S5 [32]) and for a different model inwhich an activated node induces a clique instead of a star(Fig S6 [32]) modeling a group conversation event as sometemporal network models do [41ndash43]DiscussionmdashOur analytical method shows that the

presence of network dynamics boosts the prevalence(and decreases the epidemic threshold βc) when theconcurrency m is large and suppresses the prevalence(and increases βc) when m is small for a range of valuesof the network dynamic time scale τ This result lendstheoretical support to previous claims that concurrencyboosts epidemic spreading [101113ndash1944] The resultmay sound unsurprising because a large m value impliesthat there exists a large connected component at any giventime However our finding is not trivial because a largecomponent consumes many edges such that other parts ofthe network at the same time or the network at other timeswould be more sparsely connected as compared to the caseof a smallm We confirmed that qualitatively similar resultsare found when the activity potentials were constructedfrom two empirical social contact networks (Fig S7 [32])Our results confirm that a monogamous sexual relationshipor a small group of people chatting face to face as opposedto polygamous relationships or large groups of conversa-tions hinders epidemic spreading where we compare likewith like by constraining the aggregate (static) network tobe the same in all cases For general temporal networksimmunization strategies that decrease concurrency (egdiscouraging polygamy) may be efficient Restricting thesize of the concurrent connected component (eg size of aconversation group) may also be a practical strategyAnother important contribution of the present study is

the observation that infection dies out for a sufficientlylarge τ regardless of the level of concurrency As shown inFigs 3 and S6 [32] the transition to the ldquodie outrdquo phaseoccurs at values of τ that correspond to network dynamicsand epidemic dynamics having comparable time scalesThis is a stochastic effect and cannot be captured byexisting approaches to epidemic processes on temporalnetworks that neglect stochastic dying out such as differ-ential equation systems for pair formulation-dissolutionmodels [1115ndash18] and individual-based approximations[314546] Our analysis methods explicitly consider such

τ

1

5mc

10

5

100

0 02 080604 1

β c

10

50

die out

enhanced

supp

ress

ed

(b)

τ

1

5

mc

10

0 02 080604 1

mm die out

enhanced

supp

ress

ed

(a)

FIG 3 Phase diagrams for the epidemic threshold βc whenthe activity potential is (a) equal to a for all nodes or (b) obeys apower-law distribution with exponent 3 (ϵ le ai le 09) We sethki frac14 01 at m frac14 1 and adjust the value of a and ϵ such that βctakes the same value for all m at τ frac14 0 In the ldquodie outrdquo phaseinfection eventually dies out for any finite β In the ldquosuppressedrdquophase βc is larger than the βc value at τ frac14 0 In the ldquoenhancedrdquophase βc is smaller than the βc value at τ frac14 0 The solid anddashed lines represent τ [Eq (10)] and τc respectively The colorbar indicates the βc values In the gray regions βc gt 100

PRL 119 108301 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

8 SEPTEMBER 2017

108301-4

stochastic effects and are therefore expected to be usefulbeyond the activity-driven model (or the clique-basedtemporal networks analyzed in the SupplementalMaterial [32]) and the SIS model

We thank Leo Speidel for discussion We thank theSocioPatterns collaboration (http wwwsociopatternsorg)for providing the data set T O acknowledges the supportprovided through JSPS Research Fellowship for YoungScientists J G acknowledges the support providedthrough Science Foundation Ireland (Grants No 15SPPE3125 and No 11PI1026) N M acknowledges thesupport provided through JST CREST and JSTERATO Kawarabayashi Large Graph Project

naokimasudabristolacuk[1] P Holme and J Saramaumlki Phys Rep 519 97 (2012)[2] P Holme Eur Phys J B 88 234 (2015)[3] N Masuda and R Lambiotte A Guide to Temporal

Networks (World Scientific Singapore 2016)[4] M J Keeling and K T D Eames J R Soc Interface 2 295

(2005)[5] A Barrat M Bartheacutelemy and A Vespignani Dynamical

Processes on Complex Networks (Cambridge UniversityPress Cambridge England 2008)

[6] R Pastor-Satorras C Castellano P Van Mieghem and AVespignani Rev Mod Phys 87 925 (2015)

[7] M A Porter and J P Gleeson Dynamical Systems onNetworks Frontiers in Applied Dynamical SystemsReviews and Tutorials (Springer International PublishingCham Switzerland 2016) Vol 4

[8] S Bansal J Read B Pourbohloul and L A Meyers JBiol Dyn 4 478 (2010)

[9] N Masuda and P Holme F1000Prime Rep 5 6 (2013)[10] M Morris and M Kretzschmar Soc Networks 17 299

(1995)[11] M Kretzschmar and M Morris Math Biosci 133 165

(1996)[12] J C Miller and A C Slim arXiv161104800[13] C H Watts and R M May Math Biosci 108 89 (1992)[14] I A Doherty S Shiboski J M Ellen A A Adimora and

N S Padian Sex Transm Dis 33 368 (2006)[15] K Gurski and K Hoffman Math Biosci 282 91 (2016)[16] K T D Eames and M J Keeling Math Biosci 189 115

(2004)[17] C Bauch and D A Rand Proc R Soc B 267 2019 (2000)[18] K Y Leung and M Kretzschmar AIDS 29 1097 (2015)[19] N Perra B Gonccedilalves R Pastor-Satorras and A

Vespignani Sci Rep 2 469 (2012)[20] M Starnini and R Pastor-Satorras Phys Rev E 87 062807

(2013)[21] B Ribeiro N Perra and A Baronchelli Sci Rep 3 3006

(2013)[22] S Liu N Perra M Karsai and A Vespignani Phys Rev

Lett 112 118702 (2014)

[23] L Zino A Rizzo and M Porfiri Phys Rev Lett 117228302 (2016)

[24] A Vazquez B Raacutecz A Lukaacutecs and A L Barabaacutesi PhysRev Lett 98 158702 (2007)

[25] M Karsai M Kivelauml R K Pan K Kaski J Kerteacutesz A LBarabaacutesi and J Saramaumlki Phys Rev E 83 025102(R)(2011)

[26] G Miritello E Moro and R Lara Phys Rev E 83 045102(R) (2011)

[27] J Stehleacute N Voirin A Barrat C Cattuto V Colizza LIsella C Reacutegis J F Pinton N Khanafer W Van denBroeck and P Vanhems BMC Med 9 87 (2011)

[28] D Liberzon Switching in Systems and Control Systemsand Control Foundations and Applications (BirkhaumluserBoston 2003)

[29] N Masuda K Klemm and V M Eguiacuteluz Phys Rev Lett111 188701 (2013)

[30] M Hasler V Belykh and I Belykh SIAM J Appl DynSyst 12 1007 (2013)

[31] L Speidel K Klemm V M Eguiacuteluz and N MasudaNew J Phys 18 073013 (2016)

[32] See Supplemental Material at httplinkapsorgsupplemental101103PhysRevLett119108301 which in-cludes Refs [3334] for details of mathematical derivationsand extensions of the model

[33] M Geacutenois C L Vestergaard J Fournet A Panisson IBonmarin and A Barrat Netw Sci 3 326 (2015)

[34] A Paranjape A R Benson and J Leskovec in Proceed-ings of the Tenth ACM International Conference on WebSearch and Data Mining WSDMrsquo17 (ACM New York2017) pp 601ndash610

[35] H Silk G Demirel M Homer and T Gross New J Phys16 093051 (2014)

[36] MM de Oliveira and R Dickman Phys Rev E 71 016129(2005)

[37] M Karsai N Perra and A Vespignani Sci Rep 4 4001(2014)

[38] M J Keeling and J V Ross J R Soc Interface 5 171(2008)

[39] P L Simon M Taylor and I Z Kiss J Math Biol 62 479(2011)

[40] J Hindes and I B Schwartz Phys Rev Lett 117 028302(2016)

[41] C Tantipathananandh T Berger-Wolf and D Kempe inProceedings of the Thirteenth ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining(ACM New York 2007) pp 717ndash726

[42] J Stehleacute A Barrat and G Bianconi Phys Rev E 81035101(R) (2010)

[43] K Zhao M Karsai and G Bianconi PLoS One 6 e28116(2011)

[44] J W Eaton T B Hallett and G P Garnett AIDS Behav15 687 (2011)

[45] E Valdano L Ferreri C Poletto and V Colizza Phys RevX 5 021005 (2015)

[46] L E C Rocha and N Masuda Sci Rep 6 31456(2016)

PRL 119 108301 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

8 SEPTEMBER 2017

108301-5

Qualitatively similar results are found when the activitypotential a is power-law distributed [Figs 2(c) and 2(d)]To illuminate the qualitatively different behaviors of the

epidemic threshold as τ increases we determine a phasediagram for the epidemic threshold We focus our analysison the case in which all nodes share the activity potentialvalue a noting that qualitatively similar results arealso found for power-law distributed activity potentials[Fig 3(b)] We calculate the two boundaries partitioningdifferent phases as follows First we observe that theepidemic threshold diverges at τ frac14 τ In the limit β rarr infininfection starting from a single infected node in a star graphimmediately spreads to the entire star graph leading toci rarr 1 eth1 le i le 5THORN By substituting ci rarr 1 in Eq (8) weobtain fethτ βc rarr infinTHORN frac14 0 where

τ frac14 ln1 minus eth1thornmTHORNa1 minus eth1thornmTHORN2a eth10THORN

When τ gt τ infection always dies out even if the infectionrate is infinitely large This is because in a finite networkinfection always dies out after a sufficiently long time dueto stochasticity [38ndash40] Second although βc eventuallydiverges as τ increases there may exist τc such that βc atτ lt τc is smaller than the βc value at τ frac14 0 Motivated bythe comparison between the behavior of βc at m frac14 1and m frac14 10 (Fig 2) we postulate that τc (gt 0) existsonly for m gt mc Then we obtain dβc=dτ frac14 0 atethτ mTHORN frac14 eth0 mcTHORN The derivative of Eq (8) gives partf=partτthornethpartf=partβcTHORNethdβc=dτTHORN frac14 0 Because dβc=dτ frac14 0 at ethτ mTHORN frac14eth0 mcTHORN we obtain partf=partτ frac14 0 which leads to

mc frac143

1 minus 4a eth11THORN

When m lt mc network dynamics (ie finite τ) alwaysreduce the prevalence for any τ [Figs 2(a) and 2(c)] Whenm gt mc a small τ raises the prevalence as compared to

τ frac14 0 (ie static network) but a larger τ reduces theprevalence [Figs 2(b) and 2(d)]The phase diagram based on Eqs (10) and (11) is shown

in Fig 3(a) The βc values numerically calculated bysolving Eq (8) are also shown in the figure It shouldbe noted that the parameter values are normalized such thatβc has the same value for all m at τ frac14 0 We find that thedynamics of the network may either increase or decreasethe prevalence depending on the number of connectionsthat a node can simultaneously have extending the resultsshown in Fig 2These results are not specific to the activity-drivenmodel

The phase diagram is qualitatively similar for randomlydistributed m (Fig S4 [32]) for different distributions ofactivity potentials (Fig S5 [32]) and for a different model inwhich an activated node induces a clique instead of a star(Fig S6 [32]) modeling a group conversation event as sometemporal network models do [41ndash43]DiscussionmdashOur analytical method shows that the

presence of network dynamics boosts the prevalence(and decreases the epidemic threshold βc) when theconcurrency m is large and suppresses the prevalence(and increases βc) when m is small for a range of valuesof the network dynamic time scale τ This result lendstheoretical support to previous claims that concurrencyboosts epidemic spreading [101113ndash1944] The resultmay sound unsurprising because a large m value impliesthat there exists a large connected component at any giventime However our finding is not trivial because a largecomponent consumes many edges such that other parts ofthe network at the same time or the network at other timeswould be more sparsely connected as compared to the caseof a smallm We confirmed that qualitatively similar resultsare found when the activity potentials were constructedfrom two empirical social contact networks (Fig S7 [32])Our results confirm that a monogamous sexual relationshipor a small group of people chatting face to face as opposedto polygamous relationships or large groups of conversa-tions hinders epidemic spreading where we compare likewith like by constraining the aggregate (static) network tobe the same in all cases For general temporal networksimmunization strategies that decrease concurrency (egdiscouraging polygamy) may be efficient Restricting thesize of the concurrent connected component (eg size of aconversation group) may also be a practical strategyAnother important contribution of the present study is

the observation that infection dies out for a sufficientlylarge τ regardless of the level of concurrency As shown inFigs 3 and S6 [32] the transition to the ldquodie outrdquo phaseoccurs at values of τ that correspond to network dynamicsand epidemic dynamics having comparable time scalesThis is a stochastic effect and cannot be captured byexisting approaches to epidemic processes on temporalnetworks that neglect stochastic dying out such as differ-ential equation systems for pair formulation-dissolutionmodels [1115ndash18] and individual-based approximations[314546] Our analysis methods explicitly consider such

τ

1

5mc

10

5

100

0 02 080604 1

β c

10

50

die out

enhanced

supp

ress

ed

(b)

τ

1

5

mc

10

0 02 080604 1

mm die out

enhanced

supp

ress

ed

(a)

FIG 3 Phase diagrams for the epidemic threshold βc whenthe activity potential is (a) equal to a for all nodes or (b) obeys apower-law distribution with exponent 3 (ϵ le ai le 09) We sethki frac14 01 at m frac14 1 and adjust the value of a and ϵ such that βctakes the same value for all m at τ frac14 0 In the ldquodie outrdquo phaseinfection eventually dies out for any finite β In the ldquosuppressedrdquophase βc is larger than the βc value at τ frac14 0 In the ldquoenhancedrdquophase βc is smaller than the βc value at τ frac14 0 The solid anddashed lines represent τ [Eq (10)] and τc respectively The colorbar indicates the βc values In the gray regions βc gt 100

PRL 119 108301 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

8 SEPTEMBER 2017

108301-4

stochastic effects and are therefore expected to be usefulbeyond the activity-driven model (or the clique-basedtemporal networks analyzed in the SupplementalMaterial [32]) and the SIS model

We thank Leo Speidel for discussion We thank theSocioPatterns collaboration (http wwwsociopatternsorg)for providing the data set T O acknowledges the supportprovided through JSPS Research Fellowship for YoungScientists J G acknowledges the support providedthrough Science Foundation Ireland (Grants No 15SPPE3125 and No 11PI1026) N M acknowledges thesupport provided through JST CREST and JSTERATO Kawarabayashi Large Graph Project

naokimasudabristolacuk[1] P Holme and J Saramaumlki Phys Rep 519 97 (2012)[2] P Holme Eur Phys J B 88 234 (2015)[3] N Masuda and R Lambiotte A Guide to Temporal

Networks (World Scientific Singapore 2016)[4] M J Keeling and K T D Eames J R Soc Interface 2 295

(2005)[5] A Barrat M Bartheacutelemy and A Vespignani Dynamical

Processes on Complex Networks (Cambridge UniversityPress Cambridge England 2008)

[6] R Pastor-Satorras C Castellano P Van Mieghem and AVespignani Rev Mod Phys 87 925 (2015)

[7] M A Porter and J P Gleeson Dynamical Systems onNetworks Frontiers in Applied Dynamical SystemsReviews and Tutorials (Springer International PublishingCham Switzerland 2016) Vol 4

[8] S Bansal J Read B Pourbohloul and L A Meyers JBiol Dyn 4 478 (2010)

[9] N Masuda and P Holme F1000Prime Rep 5 6 (2013)[10] M Morris and M Kretzschmar Soc Networks 17 299

(1995)[11] M Kretzschmar and M Morris Math Biosci 133 165

(1996)[12] J C Miller and A C Slim arXiv161104800[13] C H Watts and R M May Math Biosci 108 89 (1992)[14] I A Doherty S Shiboski J M Ellen A A Adimora and

N S Padian Sex Transm Dis 33 368 (2006)[15] K Gurski and K Hoffman Math Biosci 282 91 (2016)[16] K T D Eames and M J Keeling Math Biosci 189 115

(2004)[17] C Bauch and D A Rand Proc R Soc B 267 2019 (2000)[18] K Y Leung and M Kretzschmar AIDS 29 1097 (2015)[19] N Perra B Gonccedilalves R Pastor-Satorras and A

Vespignani Sci Rep 2 469 (2012)[20] M Starnini and R Pastor-Satorras Phys Rev E 87 062807

(2013)[21] B Ribeiro N Perra and A Baronchelli Sci Rep 3 3006

(2013)[22] S Liu N Perra M Karsai and A Vespignani Phys Rev

Lett 112 118702 (2014)

[23] L Zino A Rizzo and M Porfiri Phys Rev Lett 117228302 (2016)

[24] A Vazquez B Raacutecz A Lukaacutecs and A L Barabaacutesi PhysRev Lett 98 158702 (2007)

[25] M Karsai M Kivelauml R K Pan K Kaski J Kerteacutesz A LBarabaacutesi and J Saramaumlki Phys Rev E 83 025102(R)(2011)

[26] G Miritello E Moro and R Lara Phys Rev E 83 045102(R) (2011)

[27] J Stehleacute N Voirin A Barrat C Cattuto V Colizza LIsella C Reacutegis J F Pinton N Khanafer W Van denBroeck and P Vanhems BMC Med 9 87 (2011)

[28] D Liberzon Switching in Systems and Control Systemsand Control Foundations and Applications (BirkhaumluserBoston 2003)

[29] N Masuda K Klemm and V M Eguiacuteluz Phys Rev Lett111 188701 (2013)

[30] M Hasler V Belykh and I Belykh SIAM J Appl DynSyst 12 1007 (2013)

[31] L Speidel K Klemm V M Eguiacuteluz and N MasudaNew J Phys 18 073013 (2016)

[32] See Supplemental Material at httplinkapsorgsupplemental101103PhysRevLett119108301 which in-cludes Refs [3334] for details of mathematical derivationsand extensions of the model

[33] M Geacutenois C L Vestergaard J Fournet A Panisson IBonmarin and A Barrat Netw Sci 3 326 (2015)

[34] A Paranjape A R Benson and J Leskovec in Proceed-ings of the Tenth ACM International Conference on WebSearch and Data Mining WSDMrsquo17 (ACM New York2017) pp 601ndash610

[35] H Silk G Demirel M Homer and T Gross New J Phys16 093051 (2014)

[36] MM de Oliveira and R Dickman Phys Rev E 71 016129(2005)

[37] M Karsai N Perra and A Vespignani Sci Rep 4 4001(2014)

[38] M J Keeling and J V Ross J R Soc Interface 5 171(2008)

[39] P L Simon M Taylor and I Z Kiss J Math Biol 62 479(2011)

[40] J Hindes and I B Schwartz Phys Rev Lett 117 028302(2016)

[41] C Tantipathananandh T Berger-Wolf and D Kempe inProceedings of the Thirteenth ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining(ACM New York 2007) pp 717ndash726

[42] J Stehleacute A Barrat and G Bianconi Phys Rev E 81035101(R) (2010)

[43] K Zhao M Karsai and G Bianconi PLoS One 6 e28116(2011)

[44] J W Eaton T B Hallett and G P Garnett AIDS Behav15 687 (2011)

[45] E Valdano L Ferreri C Poletto and V Colizza Phys RevX 5 021005 (2015)

[46] L E C Rocha and N Masuda Sci Rep 6 31456(2016)

PRL 119 108301 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

8 SEPTEMBER 2017

108301-5

stochastic effects and are therefore expected to be usefulbeyond the activity-driven model (or the clique-basedtemporal networks analyzed in the SupplementalMaterial [32]) and the SIS model

We thank Leo Speidel for discussion We thank theSocioPatterns collaboration (http wwwsociopatternsorg)for providing the data set T O acknowledges the supportprovided through JSPS Research Fellowship for YoungScientists J G acknowledges the support providedthrough Science Foundation Ireland (Grants No 15SPPE3125 and No 11PI1026) N M acknowledges thesupport provided through JST CREST and JSTERATO Kawarabayashi Large Graph Project

naokimasudabristolacuk[1] P Holme and J Saramaumlki Phys Rep 519 97 (2012)[2] P Holme Eur Phys J B 88 234 (2015)[3] N Masuda and R Lambiotte A Guide to Temporal

Networks (World Scientific Singapore 2016)[4] M J Keeling and K T D Eames J R Soc Interface 2 295

(2005)[5] A Barrat M Bartheacutelemy and A Vespignani Dynamical

Processes on Complex Networks (Cambridge UniversityPress Cambridge England 2008)

[6] R Pastor-Satorras C Castellano P Van Mieghem and AVespignani Rev Mod Phys 87 925 (2015)

[7] M A Porter and J P Gleeson Dynamical Systems onNetworks Frontiers in Applied Dynamical SystemsReviews and Tutorials (Springer International PublishingCham Switzerland 2016) Vol 4

[8] S Bansal J Read B Pourbohloul and L A Meyers JBiol Dyn 4 478 (2010)

[9] N Masuda and P Holme F1000Prime Rep 5 6 (2013)[10] M Morris and M Kretzschmar Soc Networks 17 299

(1995)[11] M Kretzschmar and M Morris Math Biosci 133 165

(1996)[12] J C Miller and A C Slim arXiv161104800[13] C H Watts and R M May Math Biosci 108 89 (1992)[14] I A Doherty S Shiboski J M Ellen A A Adimora and

N S Padian Sex Transm Dis 33 368 (2006)[15] K Gurski and K Hoffman Math Biosci 282 91 (2016)[16] K T D Eames and M J Keeling Math Biosci 189 115

(2004)[17] C Bauch and D A Rand Proc R Soc B 267 2019 (2000)[18] K Y Leung and M Kretzschmar AIDS 29 1097 (2015)[19] N Perra B Gonccedilalves R Pastor-Satorras and A

Vespignani Sci Rep 2 469 (2012)[20] M Starnini and R Pastor-Satorras Phys Rev E 87 062807

(2013)[21] B Ribeiro N Perra and A Baronchelli Sci Rep 3 3006

(2013)[22] S Liu N Perra M Karsai and A Vespignani Phys Rev

Lett 112 118702 (2014)

[23] L Zino A Rizzo and M Porfiri Phys Rev Lett 117228302 (2016)

[24] A Vazquez B Raacutecz A Lukaacutecs and A L Barabaacutesi PhysRev Lett 98 158702 (2007)

[25] M Karsai M Kivelauml R K Pan K Kaski J Kerteacutesz A LBarabaacutesi and J Saramaumlki Phys Rev E 83 025102(R)(2011)

[26] G Miritello E Moro and R Lara Phys Rev E 83 045102(R) (2011)

[27] J Stehleacute N Voirin A Barrat C Cattuto V Colizza LIsella C Reacutegis J F Pinton N Khanafer W Van denBroeck and P Vanhems BMC Med 9 87 (2011)

[28] D Liberzon Switching in Systems and Control Systemsand Control Foundations and Applications (BirkhaumluserBoston 2003)

[29] N Masuda K Klemm and V M Eguiacuteluz Phys Rev Lett111 188701 (2013)

[30] M Hasler V Belykh and I Belykh SIAM J Appl DynSyst 12 1007 (2013)

[31] L Speidel K Klemm V M Eguiacuteluz and N MasudaNew J Phys 18 073013 (2016)

[32] See Supplemental Material at httplinkapsorgsupplemental101103PhysRevLett119108301 which in-cludes Refs [3334] for details of mathematical derivationsand extensions of the model

[33] M Geacutenois C L Vestergaard J Fournet A Panisson IBonmarin and A Barrat Netw Sci 3 326 (2015)

[34] A Paranjape A R Benson and J Leskovec in Proceed-ings of the Tenth ACM International Conference on WebSearch and Data Mining WSDMrsquo17 (ACM New York2017) pp 601ndash610

[35] H Silk G Demirel M Homer and T Gross New J Phys16 093051 (2014)

[36] MM de Oliveira and R Dickman Phys Rev E 71 016129(2005)

[37] M Karsai N Perra and A Vespignani Sci Rep 4 4001(2014)

[38] M J Keeling and J V Ross J R Soc Interface 5 171(2008)

[39] P L Simon M Taylor and I Z Kiss J Math Biol 62 479(2011)

[40] J Hindes and I B Schwartz Phys Rev Lett 117 028302(2016)

[41] C Tantipathananandh T Berger-Wolf and D Kempe inProceedings of the Thirteenth ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining(ACM New York 2007) pp 717ndash726

[42] J Stehleacute A Barrat and G Bianconi Phys Rev E 81035101(R) (2010)

[43] K Zhao M Karsai and G Bianconi PLoS One 6 e28116(2011)

[44] J W Eaton T B Hallett and G P Garnett AIDS Behav15 687 (2011)

[45] E Valdano L Ferreri C Poletto and V Colizza Phys RevX 5 021005 (2015)

[46] L E C Rocha and N Masuda Sci Rep 6 31456(2016)

PRL 119 108301 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

8 SEPTEMBER 2017

108301-5

Supplemental Material for ldquoConcurrency-induced transitions in epidemic dynamics ontemporal networksrdquo

PREVALENCE ON THE AGGREGATENETWORK

0

β

0

02

01

04

03

05 delta functionpower-law

5 10 15 20

lt gtρβ c β c

FIG S1 Prevalence on the aggregate (hence static) networkwhose adjacency matrix is given (in the limit N rarr infin) byAlowast

ij = m(ai + aj)N [1 2] The lines represent the numericalresults for the delta function (ie all nodes have same activitypotential) and power-law activity distributions The arrows

indicate βc =[m(〈a〉+

radic〈a2〉

)]minus1

We set m = 5 and

〈a〉 = 001

WHEN THE LOW-ACTIVITY ASSUMPTION ISVIOLATED

Here we consider the situation in which the low-activity assumption m2〈a〉 1 is violated Whenm N the expected number of star graphs that a stargraph overlaps with is given by

p = N〈a〉[1minus

(1minus m+ 1

N minus 1

)m]asymp m(m+ 1)〈a〉 (S1)

If p 1 is violated a star graph would overlap withothers such that the actual concurrency is larger thanm In the extreme case of p ge 1 almost all star graphsoverlap with each other such that the concurrency is notsensitive to m In this situation our results overestimatethe epidemic threshold because our analysis does not takeinto account infections across different star graphs Ifp ge 1 the individual-based approximation describes thenumerical results more accurately than our method does(Figs S2(c) and S2(d)) However even at a moderatelylarge value of p (= 05) our method is more accuratethan the individual-based approximation (Figs S2 (a)and S2(b))

τ

0

1

2

3

4

025

0

05

075

0 08060402 1

β

(a)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(b)

τ

0

05

1

025

0

05

075

0 15105 252 3

β

(c)

τ

0

4

3

2

1

5

01

0

02

03

04

0 105 25215 3

β

(d)

lt gtρ

lt gtρ

lt gtρ

lt gtρ

FIG S2 Epidemic threshold and numerically calculatedprevalence when the low-activity assumption is violated Weset m = 1 in (a) and (c) m = 10 in (b) and (d) p = 05 in(a) and (b) and p = 15 in (c) and (d) The solid and dashedlines represent the epidemic threshold obtained from Eq (8)and that obtained from the individual-based approximationrespectively All nodes are assumed to have the same activitypotential a = 025 in (a) a = 00045 in (b) a = 075 in (c)and a = 00136 in (d) We calculated the prevalence averagedover 100 simulations after discarding the first 15000 time stepsof each simulation We set N = 1000 and ∆t = 0002

DERIVATION OF c1 c2 c3 c4 AND c5

We consider SIS dynamics on a star graph with mleaves and derive c1 c2 c3 c4 and c5 Let us denotethe state of the star graph by x y z (x y isin S I 0 lez le mminus 1) where x and y are the states of the hub anda specific leaf node respectively and z is the number ofinfected nodes in the other m minus 1 leaf nodes Althougha general network with m+ 1 nodes allows 2m+1 statesusing this notation we can describe SIS dynamics on astar graph by a continuous-time Markov process with 4mstates [3]

We denote the transition rate matrix of the Markovprocess by M Its element Mxprimeyprimezprimexyz is equal tothe rate of transition from x y z to xprime yprime zprime Thediagonal elements are given by

Mxyzxyz = minussum

xprimeyprimezprime6=xyz

Mxprimeyprimezprimexyz

(S2)The rates of the recovery events are given by

MSyzIyz =1 (S3)

MxSzxIz =1 (S4)

Mxyzminus1xyz =z (z ge 1) (S5)

2

The rates of the infection events are given by

MISzSSz = zβ (S6)

MIIzSIz = (z + 1)β (S7)

MIIzISz = β (S8)

MIyz+1Iyz =(mminus 1minus z)β (z le mminus 2) (S9)

The other elements of M are equal to 0 Let pxyz(t)be the probability for a star graph to be in state x y zat time t Because

p(t) = Mp(t) (S10)

where p(t) is the 4m-dimensional column vector whoseelements are pxyz(t) we obtain

p(t) = exp(M t)p(0) (S11)

Note that c1 and c2 are the probabilities with which x = Iat time τ when the initial state is I S 0 and S I 0respectively and that c3 c4 and c5 are the probabilitiesthat y = I at time τ when the initial state is S I 0I S 0 and S S 1 respectively Therefore we ob-tain

c1c2c3c4c5

=

sumyz [exp(Mτ)]IyzIS0sumyz [exp(Mτ)]IyzSI0sumxz [exp(Mτ)]xIzSI0sumxz [exp(Mτ)]xIzIS0sumxz [exp(Mτ)]xIzSS1

(S12)

When m = 1 Eq (S12) yields

c1 = c3 =

eminusτ

2

[eminusβτ + eminus

1+β2 τ

(cosh

κτ

2+

1 + 3β

κsinh

κτ

2

)]

(S13)

c2 = c4 =

eminusτ

2

[minuseminusβτ + eminus

1+β2 τ

(cosh

κτ

2+

1 + 3β

κsinh

κτ

2

)]

(S14)

where κ =radicβ2 + 6β + 1 and c5 is not defined

When m 1 we can apply an individual-based ap-proximation [1 4 5] We assume that the state of eachnode is statistically independent of each other ie

pxyz asymp P (x)P (y)P (z) (S15)

where P (x) for example is the probability that the hubtakes state x We have suppressed t in Eq (S15) Un-der the individual-based approximation x and y obeyBernoulli distributions with parameters pMF

1 and pMF2

respectively and z obeys a binomial distribution with pa-rameters mminus1 and pMF

3 where pMF equiv (pMF1 pMF

2 pMF3 )gt

is given by

pMF =

P (x = I)P (y = I)〈z〉mminus1

=

sumyz pIyzsumxz pxIz

1mminus1

sumxyz zpxyz

(S16)

By substituting Eq (S10) in the time derivative ofEq (S16) we obtain

pMF =

minuspMF1 + βpMF

2 + (mminus 1)βpMF3

βpMF1 minus pMF

2

βpMF1 (1minus pMF

3 )minus pMF3

(S17)

If pMF3 1 pMF obeys linear dynamics given by

pMF asympMMFpMF (S18)

where

MMF =

minus1 β (mminus 1)ββ minus1 0β 0 minus1

(S19)

In a similar fashion to the derivation of Eq (S12) weobtain

c1c2c3c4c5

asymp

[exp(MMFτ)]11[exp(MMFτ)]12[exp(MMFτ)]22[exp(MMFτ)]211

mminus1 [exp(MMFτ)]23

= eminusτ

cosh(β

radicmτ)

1radicm

sinh(βradicmτ)

1 + cosh(βradicmτ)minus1

m1radicm

sinh(βradicmτ)

1m (cosh(β

radicmτ)minus 1)

(S20)

We estimate the extent to which Eq (S20) is validas follows First we need m 1 because the initialcondition pMF

3 = 1(m minus 1) should satisfy pMF3 1

Second pMF3 must satisfy

pMF3 (τ) le β(1minus eminusτ ) + pMF

3 (0)eminusτ (S21)

because pMF1 le 1 in Eq (S17) To satisfy pMF

3 1 weneed τ lt 1β This condition remains unchanged by re-scaling (τ β) to (cτ βc) These two conditions are suf-ficient for this approximation to be valid If m 1 is vi-olated the individual-based approximation significantlyunderestimates the epidemic threshold for any finite τbecause it ignores the effect of stochastic dying-out Ifτ lt 1β is violated the approximation (dashed lines inFig 2 (b) and (d)) underestimates the epidemic thresh-old because dynamics on the star graph deviate from thelinear regime In particular the epidemic threshold ob-tained from the approximation (Eq (S48)) remains finiteeven in the limit τ rarr infin whereas analytical (Eq (8))and numerical (Fig 2) results diverge at a finite τ

3

DERIVATION OF EQ (8)

At the epidemic threshold the largest eigenvalue of Tis equal to unity Let v = (v1 v2 )

gt be the corre-sponding eigenvector of T We normalize v such thatsuminfinj=1 vj = 1 By substituting Eq (7) in v = Tv we

obtain the system of equations

v1 = cprime3v1 + cprime4

infinsumn=1

〈an〉vn + cprime5

infinsumn=1

〈anminus1〉vn (S22)

v2 = cprime1v1 + cprime3v2 + c2

infinsumn=1

〈anminus1〉vn (S23)

vj = cprime1vjminus1 + cprime3vj (j ge 3) (S24)

Equation (S24) gives

vj =q

〈a〉vjminus1 (j ge 3) (S25)

where

q equiv 〈a〉cprime1

1minus cprime3 (S26)

By combining Eqs (S23) and (S25) we obtain

(q + r)v1 = 〈a〉 [1minus (1 + qS)r] v2 (S27)

where

r equiv 〈a〉cprime2

1minus cprime3 (S28)

S(q) equivinfinsumn=0

〈an+2〉〈a〉n+2

qn =1

〈a〉2

langa2

1minus a〈a〉q

rang (S29)

Because v is normalized we obtain

v =

[〈a〉minusq][1minus(1+qS)r]r+〈a〉+(1+qS)[qminus〈a〉]r

[1minus q〈a〉 ](q+r)

r+〈a〉+(1+qS)[qminus〈a〉]rq

〈a〉 [1minus q〈a〉 ](q+r)

r+〈a〉+(1+qS)[qminus〈a〉]r

( q〈a〉 )

2[1minus q

〈a〉 ](q+r)r+〈a〉+(1+qS)[qminus〈a〉]r

(S30)

Equation (S22) leads to

[1minus sminus u]v1 = 〈a〉 [sS + (1 + qS)u] v2 (S31)

where

s equiv 〈a〉cprime4

1minus cprime3 (S32)

u equiv cprime51minus cprime3

(S33)

By substituting Eq (S30) in Eq (S31) we obtain

f(τ βc) equiv(1minus r)(1minus s)minus (1 + q)u

S(q)

minus qr minus qs+ qrsminus q2uminus rs = 0 (S34)

which is Eq (8) in the main text If all nodes have thesame activity potential a Eq (S34) is reduced to

f(τ βc) = 1minus q minus r minus sminus u = 0 (S35)

CONVERGENCE OF THE MACLAURIN SERIES

We derive the condition under which the Maclaurin se-ries in Eq (5) converges for any t when β le βc First att = 0 the series converges because w(0) = (p0 0 0 )

gtSecond consider a finite t It should be noted that the

series is only defined at t that is a multiple of τ BecauseTij = 0 (i ge j + 2) in Eq (7) we obtain

wn(t) = 0 for n ge 1 +t

τ (S36)

Therefore the series convergesThird we consider the limit trarrinfin If β lt βc because

limtrarrinfin〈ρ〉 = 0 (S37)

we obtain

limtrarrinfin

wn(t) = 0 for n ge 1 (S38)

Therefore the series converges For β = βc we considerthe convergence of the series when

limtrarrinfin

w(t) = bv (S39)

where v is the eigenvector of T given by Eq (S30) andb is a constant Because Eq (S30) yields

limjrarrinfin

vj+1

vj=

q

〈a〉 (S40)

the radius of convergence is equal to 〈a〉q To ensureconvergence we require that

maxi

(ai) lt〈a〉q (S41)

Because ci (1 le i le 5) are probabilities we obtain

c1 le 1 (S42)

c3 le 1 (S43)

By substituting Eqs (S42) and (S43) in the definitionsof cprime1 and cprime2 we obtain

cprime1 le 1minus eminusτ (S44)

cprime3 le eminusτ +m〈a〉(1minus eminusτ ) (S45)

4

By substituting Eqs (S44) and (S45) in Eq (S26) weobtain

q le 〈a〉1minusm〈a〉

(S46)

Inequalities (S42)ndash(S46) hold with equality in the limitβ rarr infin Hence a sufficient condition for convergence isgiven by

maxi

(ai) lt 1minusm〈a〉 (S47)

Equation (S47) holds true in practical situations becausethe assumption m2〈a〉 1 guarantees that m〈a〉 1and ai is a probability

EPIDEMIC THRESHOLD UNDER THEINDIVIDUAL-BASED APPROXIMATION

When m 1 the epidemic threshold can be obtainedby the individual-based approximation [1 4 5] We as-sume that all nodes have the same activity potential aBy substituting Eq (S20) in Eq (S35) we obtain

βc asymp1radicmτ

ln

(1 +

eτ minus 1

2radicma

) (S48)

Equation (S48) agrees with the value derived in [1] Notethat this approximation is valid only for small τ (τ lt1βc)

DERIVATION OF τlowast FOR GENERAL ACTIVITYDISTRIBUTIONS

In the limit β rarrinfin we obtain ci rarr 1 (1 le i le 5) Forgeneral activity distributions f(τlowast βc rarr infin) = 0 leadsto

τlowast = minus ln

(1minus b+

radicb2 + 4d

2

) (S49)

where

b = m〈a〉2 [1minusm〈a〉]minus3 [2minus (m+ 1)〈a〉]S(

〈a〉1minusm〈a〉

)+ m〈a〉 [1minusm〈a〉]minus2

[m+ 1minus (m2 + 1)〈a〉

] (S50)

d = m2〈a〉2 [1minusm〈a〉]minus3 [1minus (m+ 1)〈a〉]S(

〈a〉1minusm〈a〉

)minus m2〈a〉2 [1minusm〈a〉]minus2 (S51)

DERIVATION OF mc FOR GENERAL ACTIVITYDISTRIBUTIONS

At m = mc an infinitesimal increase in τ from 0 to ∆τdoes not change the βc value For general activity distri-butions by setting partfpartτ = 0 for f given by Eq (S34)

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(c)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(d)

lt gtρ lt gtρ

lt gtρ lt gtρ

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(a)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(b)

FIG S3 Epidemic threshold and numerically calculatedprevalence for the activity-driven model with link dynamicsdriven by a reinforcement process [6] We set m = 1 in (a) and(c) and m = 10 in (b) and (d) We used the original activity-driven model in (a) and (b) and the extended model withc = 1 in (c) and (d) The solid lines represent the epidemicthreshold obtained from Eq (8) All nodes have ai = 005(1 le i le N) in (a) and (c) and ai = 0005 (1 le i le N) in(b) and (d) We calculated the prevalence averaged over 100simulations after discarding the first 15000 time steps in eachsimulation We set N = 2000 and ∆t = 0002

we obtain

mc =1 + 2

radic〈a2〉〈a〉

1minus 2radic〈a2〉 minus 2 〈a

2〉〈a〉

(S52)

ACTIVITY-DRIVEN MODEL WITH AREINFORCEMENT PROCESS

We carried out numerical simulations for an extendedactivity-driven model in which link dynamics are drivenby a reinforcement process [6] The original activity-driven model is memoryless [7] In the extended modelan activated node i connects to a node j that i has al-ready contacted with probability 1(ni+c) and to a nodej that i has not contacted with probability c(ni + c)where ni denotes the number of nodes that node i hasalready contacted

The numerically calculated prevalence is compared be-tween the original model (Figs S3(a) and S3(b)) and theextended model with c = 1 (Figs S3(c) and S3(d)) Wereplicate Figs 2(a) and 2(b) in the main text as FigsS3(a) and S3(b) as reference All nodes are assumed tohave the same activity potential ai = 005 (1 le i le N)in (a) and (c) and ai = 0005 (1 le i le N) in (b) and(d) Figure S3 indicates that the extended model onlyslightly changes the epidemic threshold

5

STOCHASTIC m

We consider the case in which the strength of concur-rency m is not constant To analyze this case we changethe definitions of cprime1 cprime2 cprime3 cprime4 and cprime5 to

cprimeprime1 = E[c1 minus eminusτ ] (S53)

cprimeprime2 = E[mc2] (S54)

cprimeprime3 = E[eminusτ +m〈a〉(c3 minus eminusτ )] (S55)

cprimeprime4 = E[mc4] (S56)

cprimeprime5 = E[m(mminus 1)〈a〉c5] (S57)

where E[middot] is the expectation with respect to the distri-bution of m The mean degree is given by 〈k〉 = 2aE[m]Using Eqs (S53)ndash(S57) instead of cprimei (1 le i le 5) we de-rived the epidemic threshold in the same manner as thederivation of Eq (8) The phase diagrams of the epi-demic threshold when m obeys a truncated Poisson dis-tribution and a power-law distribution are shown in FigsS4(a) and S4(b) respectively We obtain βc = 1〈k〉at τ = 0 We set the activity potential of all nodesa = 〈k〉(2E[m]) such that the epidemic threshold is thesame for all E[m] at τ = 0 We numerically calculatedmc at which τc = 0 For the power-law distribution ofm we cannot make E[m] smaller than mc because thedistribution does not have a probability mass at m = 0by definition However the phase diagrams in the case ofboth the truncated Poisson and power-law distributionsof m are qualitatively similar to the case of constant m

To gain analytical insights we calculated the phasediagrams when m is equal to m1 and m2 with probabil-ities p and 1 minus p respectively We varied p between 0and 1 Here again we set the activity potential of allnodes a = 〈k〉(2E[m]) such that the epidemic thresholdis the same for all E[m] at τ = 0 The phase diagram(Fig S4(c)) is again qualitatively similar to that foundin the case of constant m

HETEROGENEOUS ACTIVITY DISTRIBUTIONS

We analyzed the phase diagram for different distribu-tions of activity potentials to confirm the robustness ofthe results shown in the main text We consider an ex-ponential distribution and a power-law distribution withexponent 25 We numerically calculate the epidemicthreshold by solving Eq (8) and derive τlowast and mc fromEqs (S49) and (S52) respectively The phase dia-grams for the exponential and power-law distributionsare shown in Figs S5(a) and S5(b) respectively Theseresults are qualitatively similar to those found when allnodes have the same activity potential value

TEMPORAL NETWORKS COMPOSED OFCLIQUES

We consider the case in which an activated node cre-ates a clique (a fully-connected subgraph) with m ran-domly chosen nodes instead of a star graph This situa-tion models a group conversation among m + 1 peopleWe only consider the case in which all nodes have thesame activity potential a The mean degree for a net-work in a single time window is given by 〈k〉 = m(m+1)aThe aggregate network is the complete graph We imposem2a 1 so that cliques in the same time window do notoverlap

As in the case of the activity-driven model we denotethe state of a clique by x y z (x y isin S I 0 le z lem minus 1) where x and y are the states of the activatednode and another specific node respectively and z isthe number of infected nodes in the other m minus 1 nodesThe transition rate matrix of the SIS dynamics on thistemporal network model is given as follows The ratesof the recovery events are given by Eqs (S3) (S4) and(S5) The rates of the infection events are given by

MISzSSz = zβ (S58)

MSIzSSz = zβ (S59)

MIIzSIz = (z + 1)β (S60)

MIIzISz = (z + 1)β (S61)

MSSz+1SSz = z(mminus 1minus z)β (z le mminus 2)

(S62)

MISz+1ISz =(z + 1)(mminus 1minus z)β (z le mminus 2)

(S63)

MSIz+1SIz =(z + 1)(mminus 1minus z)β (z le mminus 2)

(S64)

MIIz+1IIz =(z + 2)(mminus 1minus z)β (z le mminus 2)

(S65)

We obtain ci (1 le i le 5) from M in the same fashionas in the case of the activity-driven model Because ofthe symmetry inherent in a clique we obtain c1 = c3 andc2 = c4 = c5 Therefore Eq (S35) is reduced to

f(τ βc) = 1minus q minus (m+ 1)r = 0 (S66)

Calculations similar to the case of the activity-drivenmodel lead to

τlowast = ln1minus (1 +m)a

1minus (1 +m)2aasymp 〈k〉 (S67)

mc = 2 (S68)

The phase diagram shown in Fig S6 is qualitativelythe same as those for the activity-driven model (Fig 3)Note that in Fig S6 we selected the activity potentialvalue a to force βc to be independent of m at τ = 0 ie

a =〈k〉

m(m+ 1) (S69)

6

τ

1

10

5

100

0 02 080604 1

β c

10

50

die out

enhanced

suppre

ssed

(c)

5

τ

1

10

0 02 080604 1

die out

enhanced

supp

ress

ed

(b)

5

τ

0

6

4

2

8

0 02 080604 1

die out

supp

ress

ed

(a)enhanced

E[m] E[m] E[m]

mc

mc

mc

FIG S4 Phase diagram of the epidemic threshold when m is stochastic m obeys (a) a truncated Poisson distribution(0 le m le mmax) and (b) a power-law distribution with exponent 3 (mmin le m le mmax) and (c) a bimodal distribution inwhich m takes m1 and m2 with probabilities p and 1minus p respectively In (a) and (b) we set mmax = 11 In (a) we truncateda Poisson distribution with varying the mean between 001 and 8 to modulate E[m] In (b) We vary mmin between 1 to 9 In(c) we set (m1m2) = (10 1) and varied p to modulate E[m] We set 〈k〉 = 01 The dashed line represents τc In the grayregions βc gt 100

τ

1

5

10

0 02 080604 15

100

β c

10

50

die out

enhanced

supp

ress

ed

(a)

m

τ

1

5

10

0 02 080604 1

die out

enhanced

suppre

ssed

(b)

m

mc

mc

FIG S5 Phase diagram of the epidemic threshold when theactivity potential obeys (a) an exponential distribution witha rate parameter λ (0 le ai le 09) and (b) a power-law dis-tribution with exponent 25 (ε le ai le 09) We set 〈k〉 = 01at m = 1 and adjust the value of λ and ε such that βc takesthe same value for all m at τ = 0 The solid and dashedlines represent τlowast and τc respectively In the gray regionsβc gt 100

τ

1

5

mc

10

5

100

0 005 01 015

m

10

50

die out

enhanced

suppressed

β c

FIG S6 Phase diagram of the epidemic threshold for tempo-ral networks composed of cliques The solid and dashed linesrepresent τlowast (Eq (S67)) and τc respectively All nodes are as-sumed to have the same activity potential given by Eq (S69)We set 〈k〉 = 01

Although Eq (S67) coincides with the expression of τlowastfor the activity-driven model (Eq (10)) τlowast as a functionof m is different between the activity-driven model (solidline in Fig 3(a)) and the present clique network model(solid line in Fig S6) This is because the values of a aredifferent between the two cases when m ge 2

EMPIRICAL ACTIVITY DISTRIBUTIONS

The epidemic threshold and prevalence when F (a) isconstructed from empirical contact data at a workplaceobtained from the SocioPatterns project [8] are shownin Figs S7(a) and S7(b) for m = 1 and m = 10 re-spectively The results for F (a) constructed from emailcommunication data at a research institution obtainedfrom the Stanford Network Analysis Platform [9] areshown in Figs S7(c) and S7(d) for m = 1 and m = 10respectively These results are qualitatively similar tothose shown in Fig 2

[1] L Speidel K Klemm V M Eguıluz and N MasudaNew J Phys 18 073013 (2016)

[2] N Masuda and R Lambiotte A Guide to Temporal Net-works (World Scientific Singapore 2016)

[3] P L Simon M Taylor and I Z Kiss J Math Biol 62479 (2011)

[4] E Valdano L Ferreri C Poletto and V Colizza PhysRev X 5 021005 (2015)

[5] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani Rev Mod Phys 87 925 (2015)

[6] M Karsai N Perra and A Vespignani Sci Rep 4 4001(2014)

[7] N Perra B Goncalves R Pastor-Satorras andA Vespignani Sci Rep 2 469 (2012)

[8] M Genois C L Vestergaard J Fournet A PanissonI Bonmarin and A Barrat Netw Sci 3 326 (2015)

[9] A Paranjape A R Benson and J Leskovec in Proc ofthe Tenth ACM International Conference on Web Search

7

τ

0

05

1

15

025

0

05

075

0 321 4 times10-3

times103

β

(c)

τ

0

05

025

075

1

01

0

02

03

04

0 105 times10-2

times103

15

β

(d)

τ

0

200

400

600

025

0

05

075

0 05 1

β

(a)

τ

0

50

100

150

200

250

01

0

02

03

0 21 3

β

(b)

lt gtρ

lt gtρ

lt gtρ

lt gtρ

times10-2

times10-2

FIG S7 Results for activity potentials derived from empiri-cal data The epidemic threshold and numerically simulatedprevalence are shown for m = 1 ((a) and (c)) and m = 10((b) and (d)) In (a) and (b) the activity potential is con-structed from contact data obtained from the SocioPatternsproject [8] This data set contains contacts between pairs ofN = 92 individuals measured every 20 seconds In (c) and(d) the activity potential is constructed from email communi-cation data at a research institution obtained from the Stan-ford Network Analysis Platform [9] Although the originaledges are directed we treat them as undirected We assumethat each email exchange event corresponds to a one-minutecontact We calculate the degree of each node per minuteaveraged over time denoted by 〈ki〉 and define the activitypotential as ai = [〈ki〉 minus 〈k〉2] m In (c) and (d) we usedN = 439 individuals satisfying ai gt 0 (some individuals ex-changed few emails such that ai lt 0) We set ∆t = 0001

and Data Mining WSDMrsquo17 (ACM New York 2017) pp601ndash610

2

The rates of the infection events are given by

MISzSSz = zβ (S6)

MIIzSIz = (z + 1)β (S7)

MIIzISz = β (S8)

MIyz+1Iyz =(mminus 1minus z)β (z le mminus 2) (S9)

The other elements of M are equal to 0 Let pxyz(t)be the probability for a star graph to be in state x y zat time t Because

p(t) = Mp(t) (S10)

where p(t) is the 4m-dimensional column vector whoseelements are pxyz(t) we obtain

p(t) = exp(M t)p(0) (S11)

Note that c1 and c2 are the probabilities with which x = Iat time τ when the initial state is I S 0 and S I 0respectively and that c3 c4 and c5 are the probabilitiesthat y = I at time τ when the initial state is S I 0I S 0 and S S 1 respectively Therefore we ob-tain

c1c2c3c4c5

=

sumyz [exp(Mτ)]IyzIS0sumyz [exp(Mτ)]IyzSI0sumxz [exp(Mτ)]xIzSI0sumxz [exp(Mτ)]xIzIS0sumxz [exp(Mτ)]xIzSS1

(S12)

When m = 1 Eq (S12) yields

c1 = c3 =

eminusτ

2

[eminusβτ + eminus

1+β2 τ

(cosh

κτ

2+

1 + 3β

κsinh

κτ

2

)]

(S13)

c2 = c4 =

eminusτ

2

[minuseminusβτ + eminus

1+β2 τ

(cosh

κτ

2+

1 + 3β

κsinh

κτ

2

)]

(S14)

where κ =radicβ2 + 6β + 1 and c5 is not defined

When m 1 we can apply an individual-based ap-proximation [1 4 5] We assume that the state of eachnode is statistically independent of each other ie

pxyz asymp P (x)P (y)P (z) (S15)

where P (x) for example is the probability that the hubtakes state x We have suppressed t in Eq (S15) Un-der the individual-based approximation x and y obeyBernoulli distributions with parameters pMF

1 and pMF2

respectively and z obeys a binomial distribution with pa-rameters mminus1 and pMF

3 where pMF equiv (pMF1 pMF

2 pMF3 )gt

is given by

pMF =

P (x = I)P (y = I)〈z〉mminus1

=

sumyz pIyzsumxz pxIz

1mminus1

sumxyz zpxyz

(S16)

By substituting Eq (S10) in the time derivative ofEq (S16) we obtain

pMF =

minuspMF1 + βpMF

2 + (mminus 1)βpMF3

βpMF1 minus pMF

2

βpMF1 (1minus pMF

3 )minus pMF3

(S17)

If pMF3 1 pMF obeys linear dynamics given by

pMF asympMMFpMF (S18)

where

MMF =

minus1 β (mminus 1)ββ minus1 0β 0 minus1

(S19)

In a similar fashion to the derivation of Eq (S12) weobtain

c1c2c3c4c5

asymp

[exp(MMFτ)]11[exp(MMFτ)]12[exp(MMFτ)]22[exp(MMFτ)]211

mminus1 [exp(MMFτ)]23

= eminusτ

cosh(β

radicmτ)

1radicm

sinh(βradicmτ)

1 + cosh(βradicmτ)minus1

m1radicm

sinh(βradicmτ)

1m (cosh(β

radicmτ)minus 1)

(S20)

We estimate the extent to which Eq (S20) is validas follows First we need m 1 because the initialcondition pMF

3 = 1(m minus 1) should satisfy pMF3 1

Second pMF3 must satisfy

pMF3 (τ) le β(1minus eminusτ ) + pMF

3 (0)eminusτ (S21)

because pMF1 le 1 in Eq (S17) To satisfy pMF

3 1 weneed τ lt 1β This condition remains unchanged by re-scaling (τ β) to (cτ βc) These two conditions are suf-ficient for this approximation to be valid If m 1 is vi-olated the individual-based approximation significantlyunderestimates the epidemic threshold for any finite τbecause it ignores the effect of stochastic dying-out Ifτ lt 1β is violated the approximation (dashed lines inFig 2 (b) and (d)) underestimates the epidemic thresh-old because dynamics on the star graph deviate from thelinear regime In particular the epidemic threshold ob-tained from the approximation (Eq (S48)) remains finiteeven in the limit τ rarr infin whereas analytical (Eq (8))and numerical (Fig 2) results diverge at a finite τ

3

DERIVATION OF EQ (8)

At the epidemic threshold the largest eigenvalue of Tis equal to unity Let v = (v1 v2 )

gt be the corre-sponding eigenvector of T We normalize v such thatsuminfinj=1 vj = 1 By substituting Eq (7) in v = Tv we

obtain the system of equations

v1 = cprime3v1 + cprime4

infinsumn=1

〈an〉vn + cprime5

infinsumn=1

〈anminus1〉vn (S22)

v2 = cprime1v1 + cprime3v2 + c2

infinsumn=1

〈anminus1〉vn (S23)

vj = cprime1vjminus1 + cprime3vj (j ge 3) (S24)

Equation (S24) gives

vj =q

〈a〉vjminus1 (j ge 3) (S25)

where

q equiv 〈a〉cprime1

1minus cprime3 (S26)

By combining Eqs (S23) and (S25) we obtain

(q + r)v1 = 〈a〉 [1minus (1 + qS)r] v2 (S27)

where

r equiv 〈a〉cprime2

1minus cprime3 (S28)

S(q) equivinfinsumn=0

〈an+2〉〈a〉n+2

qn =1

〈a〉2

langa2

1minus a〈a〉q

rang (S29)

Because v is normalized we obtain

v =

[〈a〉minusq][1minus(1+qS)r]r+〈a〉+(1+qS)[qminus〈a〉]r

[1minus q〈a〉 ](q+r)

r+〈a〉+(1+qS)[qminus〈a〉]rq

〈a〉 [1minus q〈a〉 ](q+r)

r+〈a〉+(1+qS)[qminus〈a〉]r

( q〈a〉 )

2[1minus q

〈a〉 ](q+r)r+〈a〉+(1+qS)[qminus〈a〉]r

(S30)

Equation (S22) leads to

[1minus sminus u]v1 = 〈a〉 [sS + (1 + qS)u] v2 (S31)

where

s equiv 〈a〉cprime4

1minus cprime3 (S32)

u equiv cprime51minus cprime3

(S33)

By substituting Eq (S30) in Eq (S31) we obtain

f(τ βc) equiv(1minus r)(1minus s)minus (1 + q)u

S(q)

minus qr minus qs+ qrsminus q2uminus rs = 0 (S34)

which is Eq (8) in the main text If all nodes have thesame activity potential a Eq (S34) is reduced to

f(τ βc) = 1minus q minus r minus sminus u = 0 (S35)

CONVERGENCE OF THE MACLAURIN SERIES

We derive the condition under which the Maclaurin se-ries in Eq (5) converges for any t when β le βc First att = 0 the series converges because w(0) = (p0 0 0 )

gtSecond consider a finite t It should be noted that the

series is only defined at t that is a multiple of τ BecauseTij = 0 (i ge j + 2) in Eq (7) we obtain

wn(t) = 0 for n ge 1 +t

τ (S36)

Therefore the series convergesThird we consider the limit trarrinfin If β lt βc because

limtrarrinfin〈ρ〉 = 0 (S37)

we obtain

limtrarrinfin

wn(t) = 0 for n ge 1 (S38)

Therefore the series converges For β = βc we considerthe convergence of the series when

limtrarrinfin

w(t) = bv (S39)

where v is the eigenvector of T given by Eq (S30) andb is a constant Because Eq (S30) yields

limjrarrinfin

vj+1

vj=

q

〈a〉 (S40)

the radius of convergence is equal to 〈a〉q To ensureconvergence we require that

maxi

(ai) lt〈a〉q (S41)

Because ci (1 le i le 5) are probabilities we obtain

c1 le 1 (S42)

c3 le 1 (S43)

By substituting Eqs (S42) and (S43) in the definitionsof cprime1 and cprime2 we obtain

cprime1 le 1minus eminusτ (S44)

cprime3 le eminusτ +m〈a〉(1minus eminusτ ) (S45)

4

By substituting Eqs (S44) and (S45) in Eq (S26) weobtain

q le 〈a〉1minusm〈a〉

(S46)

Inequalities (S42)ndash(S46) hold with equality in the limitβ rarr infin Hence a sufficient condition for convergence isgiven by

maxi

(ai) lt 1minusm〈a〉 (S47)

Equation (S47) holds true in practical situations becausethe assumption m2〈a〉 1 guarantees that m〈a〉 1and ai is a probability

EPIDEMIC THRESHOLD UNDER THEINDIVIDUAL-BASED APPROXIMATION

When m 1 the epidemic threshold can be obtainedby the individual-based approximation [1 4 5] We as-sume that all nodes have the same activity potential aBy substituting Eq (S20) in Eq (S35) we obtain

βc asymp1radicmτ

ln

(1 +

eτ minus 1

2radicma

) (S48)

Equation (S48) agrees with the value derived in [1] Notethat this approximation is valid only for small τ (τ lt1βc)

DERIVATION OF τlowast FOR GENERAL ACTIVITYDISTRIBUTIONS

In the limit β rarrinfin we obtain ci rarr 1 (1 le i le 5) Forgeneral activity distributions f(τlowast βc rarr infin) = 0 leadsto

τlowast = minus ln

(1minus b+

radicb2 + 4d

2

) (S49)

where

b = m〈a〉2 [1minusm〈a〉]minus3 [2minus (m+ 1)〈a〉]S(

〈a〉1minusm〈a〉

)+ m〈a〉 [1minusm〈a〉]minus2

[m+ 1minus (m2 + 1)〈a〉

] (S50)

d = m2〈a〉2 [1minusm〈a〉]minus3 [1minus (m+ 1)〈a〉]S(

〈a〉1minusm〈a〉

)minus m2〈a〉2 [1minusm〈a〉]minus2 (S51)

DERIVATION OF mc FOR GENERAL ACTIVITYDISTRIBUTIONS

At m = mc an infinitesimal increase in τ from 0 to ∆τdoes not change the βc value For general activity distri-butions by setting partfpartτ = 0 for f given by Eq (S34)

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(c)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(d)

lt gtρ lt gtρ

lt gtρ lt gtρ

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(a)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(b)

FIG S3 Epidemic threshold and numerically calculatedprevalence for the activity-driven model with link dynamicsdriven by a reinforcement process [6] We set m = 1 in (a) and(c) and m = 10 in (b) and (d) We used the original activity-driven model in (a) and (b) and the extended model withc = 1 in (c) and (d) The solid lines represent the epidemicthreshold obtained from Eq (8) All nodes have ai = 005(1 le i le N) in (a) and (c) and ai = 0005 (1 le i le N) in(b) and (d) We calculated the prevalence averaged over 100simulations after discarding the first 15000 time steps in eachsimulation We set N = 2000 and ∆t = 0002

we obtain

mc =1 + 2

radic〈a2〉〈a〉

1minus 2radic〈a2〉 minus 2 〈a

2〉〈a〉

(S52)

ACTIVITY-DRIVEN MODEL WITH AREINFORCEMENT PROCESS

We carried out numerical simulations for an extendedactivity-driven model in which link dynamics are drivenby a reinforcement process [6] The original activity-driven model is memoryless [7] In the extended modelan activated node i connects to a node j that i has al-ready contacted with probability 1(ni+c) and to a nodej that i has not contacted with probability c(ni + c)where ni denotes the number of nodes that node i hasalready contacted

The numerically calculated prevalence is compared be-tween the original model (Figs S3(a) and S3(b)) and theextended model with c = 1 (Figs S3(c) and S3(d)) Wereplicate Figs 2(a) and 2(b) in the main text as FigsS3(a) and S3(b) as reference All nodes are assumed tohave the same activity potential ai = 005 (1 le i le N)in (a) and (c) and ai = 0005 (1 le i le N) in (b) and(d) Figure S3 indicates that the extended model onlyslightly changes the epidemic threshold

5

STOCHASTIC m

We consider the case in which the strength of concur-rency m is not constant To analyze this case we changethe definitions of cprime1 cprime2 cprime3 cprime4 and cprime5 to

cprimeprime1 = E[c1 minus eminusτ ] (S53)

cprimeprime2 = E[mc2] (S54)

cprimeprime3 = E[eminusτ +m〈a〉(c3 minus eminusτ )] (S55)

cprimeprime4 = E[mc4] (S56)

cprimeprime5 = E[m(mminus 1)〈a〉c5] (S57)

where E[middot] is the expectation with respect to the distri-bution of m The mean degree is given by 〈k〉 = 2aE[m]Using Eqs (S53)ndash(S57) instead of cprimei (1 le i le 5) we de-rived the epidemic threshold in the same manner as thederivation of Eq (8) The phase diagrams of the epi-demic threshold when m obeys a truncated Poisson dis-tribution and a power-law distribution are shown in FigsS4(a) and S4(b) respectively We obtain βc = 1〈k〉at τ = 0 We set the activity potential of all nodesa = 〈k〉(2E[m]) such that the epidemic threshold is thesame for all E[m] at τ = 0 We numerically calculatedmc at which τc = 0 For the power-law distribution ofm we cannot make E[m] smaller than mc because thedistribution does not have a probability mass at m = 0by definition However the phase diagrams in the case ofboth the truncated Poisson and power-law distributionsof m are qualitatively similar to the case of constant m

To gain analytical insights we calculated the phasediagrams when m is equal to m1 and m2 with probabil-ities p and 1 minus p respectively We varied p between 0and 1 Here again we set the activity potential of allnodes a = 〈k〉(2E[m]) such that the epidemic thresholdis the same for all E[m] at τ = 0 The phase diagram(Fig S4(c)) is again qualitatively similar to that foundin the case of constant m

HETEROGENEOUS ACTIVITY DISTRIBUTIONS

We analyzed the phase diagram for different distribu-tions of activity potentials to confirm the robustness ofthe results shown in the main text We consider an ex-ponential distribution and a power-law distribution withexponent 25 We numerically calculate the epidemicthreshold by solving Eq (8) and derive τlowast and mc fromEqs (S49) and (S52) respectively The phase dia-grams for the exponential and power-law distributionsare shown in Figs S5(a) and S5(b) respectively Theseresults are qualitatively similar to those found when allnodes have the same activity potential value

TEMPORAL NETWORKS COMPOSED OFCLIQUES

We consider the case in which an activated node cre-ates a clique (a fully-connected subgraph) with m ran-domly chosen nodes instead of a star graph This situa-tion models a group conversation among m + 1 peopleWe only consider the case in which all nodes have thesame activity potential a The mean degree for a net-work in a single time window is given by 〈k〉 = m(m+1)aThe aggregate network is the complete graph We imposem2a 1 so that cliques in the same time window do notoverlap

As in the case of the activity-driven model we denotethe state of a clique by x y z (x y isin S I 0 le z lem minus 1) where x and y are the states of the activatednode and another specific node respectively and z isthe number of infected nodes in the other m minus 1 nodesThe transition rate matrix of the SIS dynamics on thistemporal network model is given as follows The ratesof the recovery events are given by Eqs (S3) (S4) and(S5) The rates of the infection events are given by

MISzSSz = zβ (S58)

MSIzSSz = zβ (S59)

MIIzSIz = (z + 1)β (S60)

MIIzISz = (z + 1)β (S61)

MSSz+1SSz = z(mminus 1minus z)β (z le mminus 2)

(S62)

MISz+1ISz =(z + 1)(mminus 1minus z)β (z le mminus 2)

(S63)

MSIz+1SIz =(z + 1)(mminus 1minus z)β (z le mminus 2)

(S64)

MIIz+1IIz =(z + 2)(mminus 1minus z)β (z le mminus 2)

(S65)

We obtain ci (1 le i le 5) from M in the same fashionas in the case of the activity-driven model Because ofthe symmetry inherent in a clique we obtain c1 = c3 andc2 = c4 = c5 Therefore Eq (S35) is reduced to

f(τ βc) = 1minus q minus (m+ 1)r = 0 (S66)

Calculations similar to the case of the activity-drivenmodel lead to

τlowast = ln1minus (1 +m)a

1minus (1 +m)2aasymp 〈k〉 (S67)

mc = 2 (S68)

The phase diagram shown in Fig S6 is qualitativelythe same as those for the activity-driven model (Fig 3)Note that in Fig S6 we selected the activity potentialvalue a to force βc to be independent of m at τ = 0 ie

a =〈k〉

m(m+ 1) (S69)

6

τ

1

10

5

100

0 02 080604 1

β c

10

50

die out

enhanced

suppre

ssed

(c)

5

τ

1

10

0 02 080604 1

die out

enhanced

supp

ress

ed

(b)

5

τ

0

6

4

2

8

0 02 080604 1

die out

supp

ress

ed

(a)enhanced

E[m] E[m] E[m]

mc

mc

mc

FIG S4 Phase diagram of the epidemic threshold when m is stochastic m obeys (a) a truncated Poisson distribution(0 le m le mmax) and (b) a power-law distribution with exponent 3 (mmin le m le mmax) and (c) a bimodal distribution inwhich m takes m1 and m2 with probabilities p and 1minus p respectively In (a) and (b) we set mmax = 11 In (a) we truncateda Poisson distribution with varying the mean between 001 and 8 to modulate E[m] In (b) We vary mmin between 1 to 9 In(c) we set (m1m2) = (10 1) and varied p to modulate E[m] We set 〈k〉 = 01 The dashed line represents τc In the grayregions βc gt 100

τ

1

5

10

0 02 080604 15

100

β c

10

50

die out

enhanced

supp

ress

ed

(a)

m

τ

1

5

10

0 02 080604 1

die out

enhanced

suppre

ssed

(b)

m

mc

mc

FIG S5 Phase diagram of the epidemic threshold when theactivity potential obeys (a) an exponential distribution witha rate parameter λ (0 le ai le 09) and (b) a power-law dis-tribution with exponent 25 (ε le ai le 09) We set 〈k〉 = 01at m = 1 and adjust the value of λ and ε such that βc takesthe same value for all m at τ = 0 The solid and dashedlines represent τlowast and τc respectively In the gray regionsβc gt 100

τ

1

5

mc

10

5

100

0 005 01 015

m

10

50

die out

enhanced

suppressed

β c

FIG S6 Phase diagram of the epidemic threshold for tempo-ral networks composed of cliques The solid and dashed linesrepresent τlowast (Eq (S67)) and τc respectively All nodes are as-sumed to have the same activity potential given by Eq (S69)We set 〈k〉 = 01

Although Eq (S67) coincides with the expression of τlowastfor the activity-driven model (Eq (10)) τlowast as a functionof m is different between the activity-driven model (solidline in Fig 3(a)) and the present clique network model(solid line in Fig S6) This is because the values of a aredifferent between the two cases when m ge 2

EMPIRICAL ACTIVITY DISTRIBUTIONS

The epidemic threshold and prevalence when F (a) isconstructed from empirical contact data at a workplaceobtained from the SocioPatterns project [8] are shownin Figs S7(a) and S7(b) for m = 1 and m = 10 re-spectively The results for F (a) constructed from emailcommunication data at a research institution obtainedfrom the Stanford Network Analysis Platform [9] areshown in Figs S7(c) and S7(d) for m = 1 and m = 10respectively These results are qualitatively similar tothose shown in Fig 2

[1] L Speidel K Klemm V M Eguıluz and N MasudaNew J Phys 18 073013 (2016)

[2] N Masuda and R Lambiotte A Guide to Temporal Net-works (World Scientific Singapore 2016)

[3] P L Simon M Taylor and I Z Kiss J Math Biol 62479 (2011)

[4] E Valdano L Ferreri C Poletto and V Colizza PhysRev X 5 021005 (2015)

[5] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani Rev Mod Phys 87 925 (2015)

[6] M Karsai N Perra and A Vespignani Sci Rep 4 4001(2014)

[7] N Perra B Goncalves R Pastor-Satorras andA Vespignani Sci Rep 2 469 (2012)

[8] M Genois C L Vestergaard J Fournet A PanissonI Bonmarin and A Barrat Netw Sci 3 326 (2015)

[9] A Paranjape A R Benson and J Leskovec in Proc ofthe Tenth ACM International Conference on Web Search

7

τ

0

05

1

15

025

0

05

075

0 321 4 times10-3

times103

β

(c)

τ

0

05

025

075

1

01

0

02

03

04

0 105 times10-2

times103

15

β

(d)

τ

0

200

400

600

025

0

05

075

0 05 1

β

(a)

τ

0

50

100

150

200

250

01

0

02

03

0 21 3

β

(b)

lt gtρ

lt gtρ

lt gtρ

lt gtρ

times10-2

times10-2

FIG S7 Results for activity potentials derived from empiri-cal data The epidemic threshold and numerically simulatedprevalence are shown for m = 1 ((a) and (c)) and m = 10((b) and (d)) In (a) and (b) the activity potential is con-structed from contact data obtained from the SocioPatternsproject [8] This data set contains contacts between pairs ofN = 92 individuals measured every 20 seconds In (c) and(d) the activity potential is constructed from email communi-cation data at a research institution obtained from the Stan-ford Network Analysis Platform [9] Although the originaledges are directed we treat them as undirected We assumethat each email exchange event corresponds to a one-minutecontact We calculate the degree of each node per minuteaveraged over time denoted by 〈ki〉 and define the activitypotential as ai = [〈ki〉 minus 〈k〉2] m In (c) and (d) we usedN = 439 individuals satisfying ai gt 0 (some individuals ex-changed few emails such that ai lt 0) We set ∆t = 0001

and Data Mining WSDMrsquo17 (ACM New York 2017) pp601ndash610

3

DERIVATION OF EQ (8)

At the epidemic threshold the largest eigenvalue of Tis equal to unity Let v = (v1 v2 )

gt be the corre-sponding eigenvector of T We normalize v such thatsuminfinj=1 vj = 1 By substituting Eq (7) in v = Tv we

obtain the system of equations

v1 = cprime3v1 + cprime4

infinsumn=1

〈an〉vn + cprime5

infinsumn=1

〈anminus1〉vn (S22)

v2 = cprime1v1 + cprime3v2 + c2

infinsumn=1

〈anminus1〉vn (S23)

vj = cprime1vjminus1 + cprime3vj (j ge 3) (S24)

Equation (S24) gives

vj =q

〈a〉vjminus1 (j ge 3) (S25)

where

q equiv 〈a〉cprime1

1minus cprime3 (S26)

By combining Eqs (S23) and (S25) we obtain

(q + r)v1 = 〈a〉 [1minus (1 + qS)r] v2 (S27)

where

r equiv 〈a〉cprime2

1minus cprime3 (S28)

S(q) equivinfinsumn=0

〈an+2〉〈a〉n+2

qn =1

〈a〉2

langa2

1minus a〈a〉q

rang (S29)

Because v is normalized we obtain

v =

[〈a〉minusq][1minus(1+qS)r]r+〈a〉+(1+qS)[qminus〈a〉]r

[1minus q〈a〉 ](q+r)

r+〈a〉+(1+qS)[qminus〈a〉]rq

〈a〉 [1minus q〈a〉 ](q+r)

r+〈a〉+(1+qS)[qminus〈a〉]r

( q〈a〉 )

2[1minus q

〈a〉 ](q+r)r+〈a〉+(1+qS)[qminus〈a〉]r

(S30)

Equation (S22) leads to

[1minus sminus u]v1 = 〈a〉 [sS + (1 + qS)u] v2 (S31)

where

s equiv 〈a〉cprime4

1minus cprime3 (S32)

u equiv cprime51minus cprime3

(S33)

By substituting Eq (S30) in Eq (S31) we obtain

f(τ βc) equiv(1minus r)(1minus s)minus (1 + q)u

S(q)

minus qr minus qs+ qrsminus q2uminus rs = 0 (S34)

which is Eq (8) in the main text If all nodes have thesame activity potential a Eq (S34) is reduced to

f(τ βc) = 1minus q minus r minus sminus u = 0 (S35)

CONVERGENCE OF THE MACLAURIN SERIES

We derive the condition under which the Maclaurin se-ries in Eq (5) converges for any t when β le βc First att = 0 the series converges because w(0) = (p0 0 0 )

gtSecond consider a finite t It should be noted that the

series is only defined at t that is a multiple of τ BecauseTij = 0 (i ge j + 2) in Eq (7) we obtain

wn(t) = 0 for n ge 1 +t

τ (S36)

Therefore the series convergesThird we consider the limit trarrinfin If β lt βc because

limtrarrinfin〈ρ〉 = 0 (S37)

we obtain

limtrarrinfin

wn(t) = 0 for n ge 1 (S38)

Therefore the series converges For β = βc we considerthe convergence of the series when

limtrarrinfin

w(t) = bv (S39)

where v is the eigenvector of T given by Eq (S30) andb is a constant Because Eq (S30) yields

limjrarrinfin

vj+1

vj=

q

〈a〉 (S40)

the radius of convergence is equal to 〈a〉q To ensureconvergence we require that

maxi

(ai) lt〈a〉q (S41)

Because ci (1 le i le 5) are probabilities we obtain

c1 le 1 (S42)

c3 le 1 (S43)

By substituting Eqs (S42) and (S43) in the definitionsof cprime1 and cprime2 we obtain

cprime1 le 1minus eminusτ (S44)

cprime3 le eminusτ +m〈a〉(1minus eminusτ ) (S45)

4

By substituting Eqs (S44) and (S45) in Eq (S26) weobtain

q le 〈a〉1minusm〈a〉

(S46)

Inequalities (S42)ndash(S46) hold with equality in the limitβ rarr infin Hence a sufficient condition for convergence isgiven by

maxi

(ai) lt 1minusm〈a〉 (S47)

Equation (S47) holds true in practical situations becausethe assumption m2〈a〉 1 guarantees that m〈a〉 1and ai is a probability

EPIDEMIC THRESHOLD UNDER THEINDIVIDUAL-BASED APPROXIMATION

When m 1 the epidemic threshold can be obtainedby the individual-based approximation [1 4 5] We as-sume that all nodes have the same activity potential aBy substituting Eq (S20) in Eq (S35) we obtain

βc asymp1radicmτ

ln

(1 +

eτ minus 1

2radicma

) (S48)

Equation (S48) agrees with the value derived in [1] Notethat this approximation is valid only for small τ (τ lt1βc)

DERIVATION OF τlowast FOR GENERAL ACTIVITYDISTRIBUTIONS

In the limit β rarrinfin we obtain ci rarr 1 (1 le i le 5) Forgeneral activity distributions f(τlowast βc rarr infin) = 0 leadsto

τlowast = minus ln

(1minus b+

radicb2 + 4d

2

) (S49)

where

b = m〈a〉2 [1minusm〈a〉]minus3 [2minus (m+ 1)〈a〉]S(

〈a〉1minusm〈a〉

)+ m〈a〉 [1minusm〈a〉]minus2

[m+ 1minus (m2 + 1)〈a〉

] (S50)

d = m2〈a〉2 [1minusm〈a〉]minus3 [1minus (m+ 1)〈a〉]S(

〈a〉1minusm〈a〉

)minus m2〈a〉2 [1minusm〈a〉]minus2 (S51)

DERIVATION OF mc FOR GENERAL ACTIVITYDISTRIBUTIONS

At m = mc an infinitesimal increase in τ from 0 to ∆τdoes not change the βc value For general activity distri-butions by setting partfpartτ = 0 for f given by Eq (S34)

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(c)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(d)

lt gtρ lt gtρ

lt gtρ lt gtρ

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(a)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(b)

FIG S3 Epidemic threshold and numerically calculatedprevalence for the activity-driven model with link dynamicsdriven by a reinforcement process [6] We set m = 1 in (a) and(c) and m = 10 in (b) and (d) We used the original activity-driven model in (a) and (b) and the extended model withc = 1 in (c) and (d) The solid lines represent the epidemicthreshold obtained from Eq (8) All nodes have ai = 005(1 le i le N) in (a) and (c) and ai = 0005 (1 le i le N) in(b) and (d) We calculated the prevalence averaged over 100simulations after discarding the first 15000 time steps in eachsimulation We set N = 2000 and ∆t = 0002

we obtain

mc =1 + 2

radic〈a2〉〈a〉

1minus 2radic〈a2〉 minus 2 〈a

2〉〈a〉

(S52)

ACTIVITY-DRIVEN MODEL WITH AREINFORCEMENT PROCESS

We carried out numerical simulations for an extendedactivity-driven model in which link dynamics are drivenby a reinforcement process [6] The original activity-driven model is memoryless [7] In the extended modelan activated node i connects to a node j that i has al-ready contacted with probability 1(ni+c) and to a nodej that i has not contacted with probability c(ni + c)where ni denotes the number of nodes that node i hasalready contacted

The numerically calculated prevalence is compared be-tween the original model (Figs S3(a) and S3(b)) and theextended model with c = 1 (Figs S3(c) and S3(d)) Wereplicate Figs 2(a) and 2(b) in the main text as FigsS3(a) and S3(b) as reference All nodes are assumed tohave the same activity potential ai = 005 (1 le i le N)in (a) and (c) and ai = 0005 (1 le i le N) in (b) and(d) Figure S3 indicates that the extended model onlyslightly changes the epidemic threshold

5

STOCHASTIC m

We consider the case in which the strength of concur-rency m is not constant To analyze this case we changethe definitions of cprime1 cprime2 cprime3 cprime4 and cprime5 to

cprimeprime1 = E[c1 minus eminusτ ] (S53)

cprimeprime2 = E[mc2] (S54)

cprimeprime3 = E[eminusτ +m〈a〉(c3 minus eminusτ )] (S55)

cprimeprime4 = E[mc4] (S56)

cprimeprime5 = E[m(mminus 1)〈a〉c5] (S57)

where E[middot] is the expectation with respect to the distri-bution of m The mean degree is given by 〈k〉 = 2aE[m]Using Eqs (S53)ndash(S57) instead of cprimei (1 le i le 5) we de-rived the epidemic threshold in the same manner as thederivation of Eq (8) The phase diagrams of the epi-demic threshold when m obeys a truncated Poisson dis-tribution and a power-law distribution are shown in FigsS4(a) and S4(b) respectively We obtain βc = 1〈k〉at τ = 0 We set the activity potential of all nodesa = 〈k〉(2E[m]) such that the epidemic threshold is thesame for all E[m] at τ = 0 We numerically calculatedmc at which τc = 0 For the power-law distribution ofm we cannot make E[m] smaller than mc because thedistribution does not have a probability mass at m = 0by definition However the phase diagrams in the case ofboth the truncated Poisson and power-law distributionsof m are qualitatively similar to the case of constant m

To gain analytical insights we calculated the phasediagrams when m is equal to m1 and m2 with probabil-ities p and 1 minus p respectively We varied p between 0and 1 Here again we set the activity potential of allnodes a = 〈k〉(2E[m]) such that the epidemic thresholdis the same for all E[m] at τ = 0 The phase diagram(Fig S4(c)) is again qualitatively similar to that foundin the case of constant m

HETEROGENEOUS ACTIVITY DISTRIBUTIONS

We analyzed the phase diagram for different distribu-tions of activity potentials to confirm the robustness ofthe results shown in the main text We consider an ex-ponential distribution and a power-law distribution withexponent 25 We numerically calculate the epidemicthreshold by solving Eq (8) and derive τlowast and mc fromEqs (S49) and (S52) respectively The phase dia-grams for the exponential and power-law distributionsare shown in Figs S5(a) and S5(b) respectively Theseresults are qualitatively similar to those found when allnodes have the same activity potential value

TEMPORAL NETWORKS COMPOSED OFCLIQUES

We consider the case in which an activated node cre-ates a clique (a fully-connected subgraph) with m ran-domly chosen nodes instead of a star graph This situa-tion models a group conversation among m + 1 peopleWe only consider the case in which all nodes have thesame activity potential a The mean degree for a net-work in a single time window is given by 〈k〉 = m(m+1)aThe aggregate network is the complete graph We imposem2a 1 so that cliques in the same time window do notoverlap

As in the case of the activity-driven model we denotethe state of a clique by x y z (x y isin S I 0 le z lem minus 1) where x and y are the states of the activatednode and another specific node respectively and z isthe number of infected nodes in the other m minus 1 nodesThe transition rate matrix of the SIS dynamics on thistemporal network model is given as follows The ratesof the recovery events are given by Eqs (S3) (S4) and(S5) The rates of the infection events are given by

MISzSSz = zβ (S58)

MSIzSSz = zβ (S59)

MIIzSIz = (z + 1)β (S60)

MIIzISz = (z + 1)β (S61)

MSSz+1SSz = z(mminus 1minus z)β (z le mminus 2)

(S62)

MISz+1ISz =(z + 1)(mminus 1minus z)β (z le mminus 2)

(S63)

MSIz+1SIz =(z + 1)(mminus 1minus z)β (z le mminus 2)

(S64)

MIIz+1IIz =(z + 2)(mminus 1minus z)β (z le mminus 2)

(S65)

We obtain ci (1 le i le 5) from M in the same fashionas in the case of the activity-driven model Because ofthe symmetry inherent in a clique we obtain c1 = c3 andc2 = c4 = c5 Therefore Eq (S35) is reduced to

f(τ βc) = 1minus q minus (m+ 1)r = 0 (S66)

Calculations similar to the case of the activity-drivenmodel lead to

τlowast = ln1minus (1 +m)a

1minus (1 +m)2aasymp 〈k〉 (S67)

mc = 2 (S68)

The phase diagram shown in Fig S6 is qualitativelythe same as those for the activity-driven model (Fig 3)Note that in Fig S6 we selected the activity potentialvalue a to force βc to be independent of m at τ = 0 ie

a =〈k〉

m(m+ 1) (S69)

6

τ

1

10

5

100

0 02 080604 1

β c

10

50

die out

enhanced

suppre

ssed

(c)

5

τ

1

10

0 02 080604 1

die out

enhanced

supp

ress

ed

(b)

5

τ

0

6

4

2

8

0 02 080604 1

die out

supp

ress

ed

(a)enhanced

E[m] E[m] E[m]

mc

mc

mc

FIG S4 Phase diagram of the epidemic threshold when m is stochastic m obeys (a) a truncated Poisson distribution(0 le m le mmax) and (b) a power-law distribution with exponent 3 (mmin le m le mmax) and (c) a bimodal distribution inwhich m takes m1 and m2 with probabilities p and 1minus p respectively In (a) and (b) we set mmax = 11 In (a) we truncateda Poisson distribution with varying the mean between 001 and 8 to modulate E[m] In (b) We vary mmin between 1 to 9 In(c) we set (m1m2) = (10 1) and varied p to modulate E[m] We set 〈k〉 = 01 The dashed line represents τc In the grayregions βc gt 100

τ

1

5

10

0 02 080604 15

100

β c

10

50

die out

enhanced

supp

ress

ed

(a)

m

τ

1

5

10

0 02 080604 1

die out

enhanced

suppre

ssed

(b)

m

mc

mc

FIG S5 Phase diagram of the epidemic threshold when theactivity potential obeys (a) an exponential distribution witha rate parameter λ (0 le ai le 09) and (b) a power-law dis-tribution with exponent 25 (ε le ai le 09) We set 〈k〉 = 01at m = 1 and adjust the value of λ and ε such that βc takesthe same value for all m at τ = 0 The solid and dashedlines represent τlowast and τc respectively In the gray regionsβc gt 100

τ

1

5

mc

10

5

100

0 005 01 015

m

10

50

die out

enhanced

suppressed

β c

FIG S6 Phase diagram of the epidemic threshold for tempo-ral networks composed of cliques The solid and dashed linesrepresent τlowast (Eq (S67)) and τc respectively All nodes are as-sumed to have the same activity potential given by Eq (S69)We set 〈k〉 = 01

Although Eq (S67) coincides with the expression of τlowastfor the activity-driven model (Eq (10)) τlowast as a functionof m is different between the activity-driven model (solidline in Fig 3(a)) and the present clique network model(solid line in Fig S6) This is because the values of a aredifferent between the two cases when m ge 2

EMPIRICAL ACTIVITY DISTRIBUTIONS

The epidemic threshold and prevalence when F (a) isconstructed from empirical contact data at a workplaceobtained from the SocioPatterns project [8] are shownin Figs S7(a) and S7(b) for m = 1 and m = 10 re-spectively The results for F (a) constructed from emailcommunication data at a research institution obtainedfrom the Stanford Network Analysis Platform [9] areshown in Figs S7(c) and S7(d) for m = 1 and m = 10respectively These results are qualitatively similar tothose shown in Fig 2

[1] L Speidel K Klemm V M Eguıluz and N MasudaNew J Phys 18 073013 (2016)

[2] N Masuda and R Lambiotte A Guide to Temporal Net-works (World Scientific Singapore 2016)

[3] P L Simon M Taylor and I Z Kiss J Math Biol 62479 (2011)

[4] E Valdano L Ferreri C Poletto and V Colizza PhysRev X 5 021005 (2015)

[5] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani Rev Mod Phys 87 925 (2015)

[6] M Karsai N Perra and A Vespignani Sci Rep 4 4001(2014)

[7] N Perra B Goncalves R Pastor-Satorras andA Vespignani Sci Rep 2 469 (2012)

[8] M Genois C L Vestergaard J Fournet A PanissonI Bonmarin and A Barrat Netw Sci 3 326 (2015)

[9] A Paranjape A R Benson and J Leskovec in Proc ofthe Tenth ACM International Conference on Web Search

7

τ

0

05

1

15

025

0

05

075

0 321 4 times10-3

times103

β

(c)

τ

0

05

025

075

1

01

0

02

03

04

0 105 times10-2

times103

15

β

(d)

τ

0

200

400

600

025

0

05

075

0 05 1

β

(a)

τ

0

50

100

150

200

250

01

0

02

03

0 21 3

β

(b)

lt gtρ

lt gtρ

lt gtρ

lt gtρ

times10-2

times10-2

FIG S7 Results for activity potentials derived from empiri-cal data The epidemic threshold and numerically simulatedprevalence are shown for m = 1 ((a) and (c)) and m = 10((b) and (d)) In (a) and (b) the activity potential is con-structed from contact data obtained from the SocioPatternsproject [8] This data set contains contacts between pairs ofN = 92 individuals measured every 20 seconds In (c) and(d) the activity potential is constructed from email communi-cation data at a research institution obtained from the Stan-ford Network Analysis Platform [9] Although the originaledges are directed we treat them as undirected We assumethat each email exchange event corresponds to a one-minutecontact We calculate the degree of each node per minuteaveraged over time denoted by 〈ki〉 and define the activitypotential as ai = [〈ki〉 minus 〈k〉2] m In (c) and (d) we usedN = 439 individuals satisfying ai gt 0 (some individuals ex-changed few emails such that ai lt 0) We set ∆t = 0001

and Data Mining WSDMrsquo17 (ACM New York 2017) pp601ndash610

4

By substituting Eqs (S44) and (S45) in Eq (S26) weobtain

q le 〈a〉1minusm〈a〉

(S46)

Inequalities (S42)ndash(S46) hold with equality in the limitβ rarr infin Hence a sufficient condition for convergence isgiven by

maxi

(ai) lt 1minusm〈a〉 (S47)

Equation (S47) holds true in practical situations becausethe assumption m2〈a〉 1 guarantees that m〈a〉 1and ai is a probability

EPIDEMIC THRESHOLD UNDER THEINDIVIDUAL-BASED APPROXIMATION

When m 1 the epidemic threshold can be obtainedby the individual-based approximation [1 4 5] We as-sume that all nodes have the same activity potential aBy substituting Eq (S20) in Eq (S35) we obtain

βc asymp1radicmτ

ln

(1 +

eτ minus 1

2radicma

) (S48)

Equation (S48) agrees with the value derived in [1] Notethat this approximation is valid only for small τ (τ lt1βc)

DERIVATION OF τlowast FOR GENERAL ACTIVITYDISTRIBUTIONS

In the limit β rarrinfin we obtain ci rarr 1 (1 le i le 5) Forgeneral activity distributions f(τlowast βc rarr infin) = 0 leadsto

τlowast = minus ln

(1minus b+

radicb2 + 4d

2

) (S49)

where

b = m〈a〉2 [1minusm〈a〉]minus3 [2minus (m+ 1)〈a〉]S(

〈a〉1minusm〈a〉

)+ m〈a〉 [1minusm〈a〉]minus2

[m+ 1minus (m2 + 1)〈a〉

] (S50)

d = m2〈a〉2 [1minusm〈a〉]minus3 [1minus (m+ 1)〈a〉]S(

〈a〉1minusm〈a〉

)minus m2〈a〉2 [1minusm〈a〉]minus2 (S51)

DERIVATION OF mc FOR GENERAL ACTIVITYDISTRIBUTIONS

At m = mc an infinitesimal increase in τ from 0 to ∆τdoes not change the βc value For general activity distri-butions by setting partfpartτ = 0 for f given by Eq (S34)

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(c)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(d)

lt gtρ lt gtρ

lt gtρ lt gtρ

τ

0

10

20

30

40

025

0

05

075

0 01501005 02

β

(a)

τ

0

5

10

15

01

0

02

03

04

0 02 080604 1

β

(b)

FIG S3 Epidemic threshold and numerically calculatedprevalence for the activity-driven model with link dynamicsdriven by a reinforcement process [6] We set m = 1 in (a) and(c) and m = 10 in (b) and (d) We used the original activity-driven model in (a) and (b) and the extended model withc = 1 in (c) and (d) The solid lines represent the epidemicthreshold obtained from Eq (8) All nodes have ai = 005(1 le i le N) in (a) and (c) and ai = 0005 (1 le i le N) in(b) and (d) We calculated the prevalence averaged over 100simulations after discarding the first 15000 time steps in eachsimulation We set N = 2000 and ∆t = 0002

we obtain

mc =1 + 2

radic〈a2〉〈a〉

1minus 2radic〈a2〉 minus 2 〈a

2〉〈a〉

(S52)

ACTIVITY-DRIVEN MODEL WITH AREINFORCEMENT PROCESS

We carried out numerical simulations for an extendedactivity-driven model in which link dynamics are drivenby a reinforcement process [6] The original activity-driven model is memoryless [7] In the extended modelan activated node i connects to a node j that i has al-ready contacted with probability 1(ni+c) and to a nodej that i has not contacted with probability c(ni + c)where ni denotes the number of nodes that node i hasalready contacted

The numerically calculated prevalence is compared be-tween the original model (Figs S3(a) and S3(b)) and theextended model with c = 1 (Figs S3(c) and S3(d)) Wereplicate Figs 2(a) and 2(b) in the main text as FigsS3(a) and S3(b) as reference All nodes are assumed tohave the same activity potential ai = 005 (1 le i le N)in (a) and (c) and ai = 0005 (1 le i le N) in (b) and(d) Figure S3 indicates that the extended model onlyslightly changes the epidemic threshold

5

STOCHASTIC m

We consider the case in which the strength of concur-rency m is not constant To analyze this case we changethe definitions of cprime1 cprime2 cprime3 cprime4 and cprime5 to

cprimeprime1 = E[c1 minus eminusτ ] (S53)

cprimeprime2 = E[mc2] (S54)

cprimeprime3 = E[eminusτ +m〈a〉(c3 minus eminusτ )] (S55)

cprimeprime4 = E[mc4] (S56)

cprimeprime5 = E[m(mminus 1)〈a〉c5] (S57)

where E[middot] is the expectation with respect to the distri-bution of m The mean degree is given by 〈k〉 = 2aE[m]Using Eqs (S53)ndash(S57) instead of cprimei (1 le i le 5) we de-rived the epidemic threshold in the same manner as thederivation of Eq (8) The phase diagrams of the epi-demic threshold when m obeys a truncated Poisson dis-tribution and a power-law distribution are shown in FigsS4(a) and S4(b) respectively We obtain βc = 1〈k〉at τ = 0 We set the activity potential of all nodesa = 〈k〉(2E[m]) such that the epidemic threshold is thesame for all E[m] at τ = 0 We numerically calculatedmc at which τc = 0 For the power-law distribution ofm we cannot make E[m] smaller than mc because thedistribution does not have a probability mass at m = 0by definition However the phase diagrams in the case ofboth the truncated Poisson and power-law distributionsof m are qualitatively similar to the case of constant m

To gain analytical insights we calculated the phasediagrams when m is equal to m1 and m2 with probabil-ities p and 1 minus p respectively We varied p between 0and 1 Here again we set the activity potential of allnodes a = 〈k〉(2E[m]) such that the epidemic thresholdis the same for all E[m] at τ = 0 The phase diagram(Fig S4(c)) is again qualitatively similar to that foundin the case of constant m

HETEROGENEOUS ACTIVITY DISTRIBUTIONS

We analyzed the phase diagram for different distribu-tions of activity potentials to confirm the robustness ofthe results shown in the main text We consider an ex-ponential distribution and a power-law distribution withexponent 25 We numerically calculate the epidemicthreshold by solving Eq (8) and derive τlowast and mc fromEqs (S49) and (S52) respectively The phase dia-grams for the exponential and power-law distributionsare shown in Figs S5(a) and S5(b) respectively Theseresults are qualitatively similar to those found when allnodes have the same activity potential value

TEMPORAL NETWORKS COMPOSED OFCLIQUES

We consider the case in which an activated node cre-ates a clique (a fully-connected subgraph) with m ran-domly chosen nodes instead of a star graph This situa-tion models a group conversation among m + 1 peopleWe only consider the case in which all nodes have thesame activity potential a The mean degree for a net-work in a single time window is given by 〈k〉 = m(m+1)aThe aggregate network is the complete graph We imposem2a 1 so that cliques in the same time window do notoverlap

As in the case of the activity-driven model we denotethe state of a clique by x y z (x y isin S I 0 le z lem minus 1) where x and y are the states of the activatednode and another specific node respectively and z isthe number of infected nodes in the other m minus 1 nodesThe transition rate matrix of the SIS dynamics on thistemporal network model is given as follows The ratesof the recovery events are given by Eqs (S3) (S4) and(S5) The rates of the infection events are given by

MISzSSz = zβ (S58)

MSIzSSz = zβ (S59)

MIIzSIz = (z + 1)β (S60)

MIIzISz = (z + 1)β (S61)

MSSz+1SSz = z(mminus 1minus z)β (z le mminus 2)

(S62)

MISz+1ISz =(z + 1)(mminus 1minus z)β (z le mminus 2)

(S63)

MSIz+1SIz =(z + 1)(mminus 1minus z)β (z le mminus 2)

(S64)

MIIz+1IIz =(z + 2)(mminus 1minus z)β (z le mminus 2)

(S65)

We obtain ci (1 le i le 5) from M in the same fashionas in the case of the activity-driven model Because ofthe symmetry inherent in a clique we obtain c1 = c3 andc2 = c4 = c5 Therefore Eq (S35) is reduced to

f(τ βc) = 1minus q minus (m+ 1)r = 0 (S66)

Calculations similar to the case of the activity-drivenmodel lead to

τlowast = ln1minus (1 +m)a

1minus (1 +m)2aasymp 〈k〉 (S67)

mc = 2 (S68)

The phase diagram shown in Fig S6 is qualitativelythe same as those for the activity-driven model (Fig 3)Note that in Fig S6 we selected the activity potentialvalue a to force βc to be independent of m at τ = 0 ie

a =〈k〉

m(m+ 1) (S69)

6

τ

1

10

5

100

0 02 080604 1

β c

10

50

die out

enhanced

suppre

ssed

(c)

5

τ

1

10

0 02 080604 1

die out

enhanced

supp

ress

ed

(b)

5

τ

0

6

4

2

8

0 02 080604 1

die out

supp

ress

ed

(a)enhanced

E[m] E[m] E[m]

mc

mc

mc

FIG S4 Phase diagram of the epidemic threshold when m is stochastic m obeys (a) a truncated Poisson distribution(0 le m le mmax) and (b) a power-law distribution with exponent 3 (mmin le m le mmax) and (c) a bimodal distribution inwhich m takes m1 and m2 with probabilities p and 1minus p respectively In (a) and (b) we set mmax = 11 In (a) we truncateda Poisson distribution with varying the mean between 001 and 8 to modulate E[m] In (b) We vary mmin between 1 to 9 In(c) we set (m1m2) = (10 1) and varied p to modulate E[m] We set 〈k〉 = 01 The dashed line represents τc In the grayregions βc gt 100

τ

1

5

10

0 02 080604 15

100

β c

10

50

die out

enhanced

supp

ress

ed

(a)

m

τ

1

5

10

0 02 080604 1

die out

enhanced

suppre

ssed

(b)

m

mc

mc

FIG S5 Phase diagram of the epidemic threshold when theactivity potential obeys (a) an exponential distribution witha rate parameter λ (0 le ai le 09) and (b) a power-law dis-tribution with exponent 25 (ε le ai le 09) We set 〈k〉 = 01at m = 1 and adjust the value of λ and ε such that βc takesthe same value for all m at τ = 0 The solid and dashedlines represent τlowast and τc respectively In the gray regionsβc gt 100

τ

1

5

mc

10

5

100

0 005 01 015

m

10

50

die out

enhanced

suppressed

β c

FIG S6 Phase diagram of the epidemic threshold for tempo-ral networks composed of cliques The solid and dashed linesrepresent τlowast (Eq (S67)) and τc respectively All nodes are as-sumed to have the same activity potential given by Eq (S69)We set 〈k〉 = 01

Although Eq (S67) coincides with the expression of τlowastfor the activity-driven model (Eq (10)) τlowast as a functionof m is different between the activity-driven model (solidline in Fig 3(a)) and the present clique network model(solid line in Fig S6) This is because the values of a aredifferent between the two cases when m ge 2

EMPIRICAL ACTIVITY DISTRIBUTIONS

The epidemic threshold and prevalence when F (a) isconstructed from empirical contact data at a workplaceobtained from the SocioPatterns project [8] are shownin Figs S7(a) and S7(b) for m = 1 and m = 10 re-spectively The results for F (a) constructed from emailcommunication data at a research institution obtainedfrom the Stanford Network Analysis Platform [9] areshown in Figs S7(c) and S7(d) for m = 1 and m = 10respectively These results are qualitatively similar tothose shown in Fig 2

[1] L Speidel K Klemm V M Eguıluz and N MasudaNew J Phys 18 073013 (2016)

[2] N Masuda and R Lambiotte A Guide to Temporal Net-works (World Scientific Singapore 2016)

[3] P L Simon M Taylor and I Z Kiss J Math Biol 62479 (2011)

[4] E Valdano L Ferreri C Poletto and V Colizza PhysRev X 5 021005 (2015)

[5] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani Rev Mod Phys 87 925 (2015)

[6] M Karsai N Perra and A Vespignani Sci Rep 4 4001(2014)

[7] N Perra B Goncalves R Pastor-Satorras andA Vespignani Sci Rep 2 469 (2012)

[8] M Genois C L Vestergaard J Fournet A PanissonI Bonmarin and A Barrat Netw Sci 3 326 (2015)

[9] A Paranjape A R Benson and J Leskovec in Proc ofthe Tenth ACM International Conference on Web Search

7

τ

0

05

1

15

025

0

05

075

0 321 4 times10-3

times103

β

(c)

τ

0

05

025

075

1

01

0

02

03

04

0 105 times10-2

times103

15

β

(d)

τ

0

200

400

600

025

0

05

075

0 05 1

β

(a)

τ

0

50

100

150

200

250

01

0

02

03

0 21 3

β

(b)

lt gtρ

lt gtρ

lt gtρ

lt gtρ

times10-2

times10-2

FIG S7 Results for activity potentials derived from empiri-cal data The epidemic threshold and numerically simulatedprevalence are shown for m = 1 ((a) and (c)) and m = 10((b) and (d)) In (a) and (b) the activity potential is con-structed from contact data obtained from the SocioPatternsproject [8] This data set contains contacts between pairs ofN = 92 individuals measured every 20 seconds In (c) and(d) the activity potential is constructed from email communi-cation data at a research institution obtained from the Stan-ford Network Analysis Platform [9] Although the originaledges are directed we treat them as undirected We assumethat each email exchange event corresponds to a one-minutecontact We calculate the degree of each node per minuteaveraged over time denoted by 〈ki〉 and define the activitypotential as ai = [〈ki〉 minus 〈k〉2] m In (c) and (d) we usedN = 439 individuals satisfying ai gt 0 (some individuals ex-changed few emails such that ai lt 0) We set ∆t = 0001

and Data Mining WSDMrsquo17 (ACM New York 2017) pp601ndash610

5

STOCHASTIC m

We consider the case in which the strength of concur-rency m is not constant To analyze this case we changethe definitions of cprime1 cprime2 cprime3 cprime4 and cprime5 to

cprimeprime1 = E[c1 minus eminusτ ] (S53)

cprimeprime2 = E[mc2] (S54)

cprimeprime3 = E[eminusτ +m〈a〉(c3 minus eminusτ )] (S55)

cprimeprime4 = E[mc4] (S56)

cprimeprime5 = E[m(mminus 1)〈a〉c5] (S57)

where E[middot] is the expectation with respect to the distri-bution of m The mean degree is given by 〈k〉 = 2aE[m]Using Eqs (S53)ndash(S57) instead of cprimei (1 le i le 5) we de-rived the epidemic threshold in the same manner as thederivation of Eq (8) The phase diagrams of the epi-demic threshold when m obeys a truncated Poisson dis-tribution and a power-law distribution are shown in FigsS4(a) and S4(b) respectively We obtain βc = 1〈k〉at τ = 0 We set the activity potential of all nodesa = 〈k〉(2E[m]) such that the epidemic threshold is thesame for all E[m] at τ = 0 We numerically calculatedmc at which τc = 0 For the power-law distribution ofm we cannot make E[m] smaller than mc because thedistribution does not have a probability mass at m = 0by definition However the phase diagrams in the case ofboth the truncated Poisson and power-law distributionsof m are qualitatively similar to the case of constant m

To gain analytical insights we calculated the phasediagrams when m is equal to m1 and m2 with probabil-ities p and 1 minus p respectively We varied p between 0and 1 Here again we set the activity potential of allnodes a = 〈k〉(2E[m]) such that the epidemic thresholdis the same for all E[m] at τ = 0 The phase diagram(Fig S4(c)) is again qualitatively similar to that foundin the case of constant m

HETEROGENEOUS ACTIVITY DISTRIBUTIONS

We analyzed the phase diagram for different distribu-tions of activity potentials to confirm the robustness ofthe results shown in the main text We consider an ex-ponential distribution and a power-law distribution withexponent 25 We numerically calculate the epidemicthreshold by solving Eq (8) and derive τlowast and mc fromEqs (S49) and (S52) respectively The phase dia-grams for the exponential and power-law distributionsare shown in Figs S5(a) and S5(b) respectively Theseresults are qualitatively similar to those found when allnodes have the same activity potential value

TEMPORAL NETWORKS COMPOSED OFCLIQUES

We consider the case in which an activated node cre-ates a clique (a fully-connected subgraph) with m ran-domly chosen nodes instead of a star graph This situa-tion models a group conversation among m + 1 peopleWe only consider the case in which all nodes have thesame activity potential a The mean degree for a net-work in a single time window is given by 〈k〉 = m(m+1)aThe aggregate network is the complete graph We imposem2a 1 so that cliques in the same time window do notoverlap

As in the case of the activity-driven model we denotethe state of a clique by x y z (x y isin S I 0 le z lem minus 1) where x and y are the states of the activatednode and another specific node respectively and z isthe number of infected nodes in the other m minus 1 nodesThe transition rate matrix of the SIS dynamics on thistemporal network model is given as follows The ratesof the recovery events are given by Eqs (S3) (S4) and(S5) The rates of the infection events are given by

MISzSSz = zβ (S58)

MSIzSSz = zβ (S59)

MIIzSIz = (z + 1)β (S60)

MIIzISz = (z + 1)β (S61)

MSSz+1SSz = z(mminus 1minus z)β (z le mminus 2)

(S62)

MISz+1ISz =(z + 1)(mminus 1minus z)β (z le mminus 2)

(S63)

MSIz+1SIz =(z + 1)(mminus 1minus z)β (z le mminus 2)

(S64)

MIIz+1IIz =(z + 2)(mminus 1minus z)β (z le mminus 2)

(S65)

We obtain ci (1 le i le 5) from M in the same fashionas in the case of the activity-driven model Because ofthe symmetry inherent in a clique we obtain c1 = c3 andc2 = c4 = c5 Therefore Eq (S35) is reduced to

f(τ βc) = 1minus q minus (m+ 1)r = 0 (S66)

Calculations similar to the case of the activity-drivenmodel lead to

τlowast = ln1minus (1 +m)a

1minus (1 +m)2aasymp 〈k〉 (S67)

mc = 2 (S68)

The phase diagram shown in Fig S6 is qualitativelythe same as those for the activity-driven model (Fig 3)Note that in Fig S6 we selected the activity potentialvalue a to force βc to be independent of m at τ = 0 ie

a =〈k〉

m(m+ 1) (S69)

6

τ

1

10

5

100

0 02 080604 1

β c

10

50

die out

enhanced

suppre

ssed

(c)

5

τ

1

10

0 02 080604 1

die out

enhanced

supp

ress

ed

(b)

5

τ

0

6

4

2

8

0 02 080604 1

die out

supp

ress

ed

(a)enhanced

E[m] E[m] E[m]

mc

mc

mc

FIG S4 Phase diagram of the epidemic threshold when m is stochastic m obeys (a) a truncated Poisson distribution(0 le m le mmax) and (b) a power-law distribution with exponent 3 (mmin le m le mmax) and (c) a bimodal distribution inwhich m takes m1 and m2 with probabilities p and 1minus p respectively In (a) and (b) we set mmax = 11 In (a) we truncateda Poisson distribution with varying the mean between 001 and 8 to modulate E[m] In (b) We vary mmin between 1 to 9 In(c) we set (m1m2) = (10 1) and varied p to modulate E[m] We set 〈k〉 = 01 The dashed line represents τc In the grayregions βc gt 100

τ

1

5

10

0 02 080604 15

100

β c

10

50

die out

enhanced

supp

ress

ed

(a)

m

τ

1

5

10

0 02 080604 1

die out

enhanced

suppre

ssed

(b)

m

mc

mc

FIG S5 Phase diagram of the epidemic threshold when theactivity potential obeys (a) an exponential distribution witha rate parameter λ (0 le ai le 09) and (b) a power-law dis-tribution with exponent 25 (ε le ai le 09) We set 〈k〉 = 01at m = 1 and adjust the value of λ and ε such that βc takesthe same value for all m at τ = 0 The solid and dashedlines represent τlowast and τc respectively In the gray regionsβc gt 100

τ

1

5

mc

10

5

100

0 005 01 015

m

10

50

die out

enhanced

suppressed

β c

FIG S6 Phase diagram of the epidemic threshold for tempo-ral networks composed of cliques The solid and dashed linesrepresent τlowast (Eq (S67)) and τc respectively All nodes are as-sumed to have the same activity potential given by Eq (S69)We set 〈k〉 = 01

Although Eq (S67) coincides with the expression of τlowastfor the activity-driven model (Eq (10)) τlowast as a functionof m is different between the activity-driven model (solidline in Fig 3(a)) and the present clique network model(solid line in Fig S6) This is because the values of a aredifferent between the two cases when m ge 2

EMPIRICAL ACTIVITY DISTRIBUTIONS

The epidemic threshold and prevalence when F (a) isconstructed from empirical contact data at a workplaceobtained from the SocioPatterns project [8] are shownin Figs S7(a) and S7(b) for m = 1 and m = 10 re-spectively The results for F (a) constructed from emailcommunication data at a research institution obtainedfrom the Stanford Network Analysis Platform [9] areshown in Figs S7(c) and S7(d) for m = 1 and m = 10respectively These results are qualitatively similar tothose shown in Fig 2

[1] L Speidel K Klemm V M Eguıluz and N MasudaNew J Phys 18 073013 (2016)

[2] N Masuda and R Lambiotte A Guide to Temporal Net-works (World Scientific Singapore 2016)

[3] P L Simon M Taylor and I Z Kiss J Math Biol 62479 (2011)

[4] E Valdano L Ferreri C Poletto and V Colizza PhysRev X 5 021005 (2015)

[5] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani Rev Mod Phys 87 925 (2015)

[6] M Karsai N Perra and A Vespignani Sci Rep 4 4001(2014)

[7] N Perra B Goncalves R Pastor-Satorras andA Vespignani Sci Rep 2 469 (2012)

[8] M Genois C L Vestergaard J Fournet A PanissonI Bonmarin and A Barrat Netw Sci 3 326 (2015)

[9] A Paranjape A R Benson and J Leskovec in Proc ofthe Tenth ACM International Conference on Web Search

7

τ

0

05

1

15

025

0

05

075

0 321 4 times10-3

times103

β

(c)

τ

0

05

025

075

1

01

0

02

03

04

0 105 times10-2

times103

15

β

(d)

τ

0

200

400

600

025

0

05

075

0 05 1

β

(a)

τ

0

50

100

150

200

250

01

0

02

03

0 21 3

β

(b)

lt gtρ

lt gtρ

lt gtρ

lt gtρ

times10-2

times10-2

FIG S7 Results for activity potentials derived from empiri-cal data The epidemic threshold and numerically simulatedprevalence are shown for m = 1 ((a) and (c)) and m = 10((b) and (d)) In (a) and (b) the activity potential is con-structed from contact data obtained from the SocioPatternsproject [8] This data set contains contacts between pairs ofN = 92 individuals measured every 20 seconds In (c) and(d) the activity potential is constructed from email communi-cation data at a research institution obtained from the Stan-ford Network Analysis Platform [9] Although the originaledges are directed we treat them as undirected We assumethat each email exchange event corresponds to a one-minutecontact We calculate the degree of each node per minuteaveraged over time denoted by 〈ki〉 and define the activitypotential as ai = [〈ki〉 minus 〈k〉2] m In (c) and (d) we usedN = 439 individuals satisfying ai gt 0 (some individuals ex-changed few emails such that ai lt 0) We set ∆t = 0001

and Data Mining WSDMrsquo17 (ACM New York 2017) pp601ndash610

6

τ

1

10

5

100

0 02 080604 1

β c

10

50

die out

enhanced

suppre

ssed

(c)

5

τ

1

10

0 02 080604 1

die out

enhanced

supp

ress

ed

(b)

5

τ

0

6

4

2

8

0 02 080604 1

die out

supp

ress

ed

(a)enhanced

E[m] E[m] E[m]

mc

mc

mc

FIG S4 Phase diagram of the epidemic threshold when m is stochastic m obeys (a) a truncated Poisson distribution(0 le m le mmax) and (b) a power-law distribution with exponent 3 (mmin le m le mmax) and (c) a bimodal distribution inwhich m takes m1 and m2 with probabilities p and 1minus p respectively In (a) and (b) we set mmax = 11 In (a) we truncateda Poisson distribution with varying the mean between 001 and 8 to modulate E[m] In (b) We vary mmin between 1 to 9 In(c) we set (m1m2) = (10 1) and varied p to modulate E[m] We set 〈k〉 = 01 The dashed line represents τc In the grayregions βc gt 100

τ

1

5

10

0 02 080604 15

100

β c

10

50

die out

enhanced

supp

ress

ed

(a)

m

τ

1

5

10

0 02 080604 1

die out

enhanced

suppre

ssed

(b)

m

mc

mc

FIG S5 Phase diagram of the epidemic threshold when theactivity potential obeys (a) an exponential distribution witha rate parameter λ (0 le ai le 09) and (b) a power-law dis-tribution with exponent 25 (ε le ai le 09) We set 〈k〉 = 01at m = 1 and adjust the value of λ and ε such that βc takesthe same value for all m at τ = 0 The solid and dashedlines represent τlowast and τc respectively In the gray regionsβc gt 100

τ

1

5

mc

10

5

100

0 005 01 015

m

10

50

die out

enhanced

suppressed

β c

FIG S6 Phase diagram of the epidemic threshold for tempo-ral networks composed of cliques The solid and dashed linesrepresent τlowast (Eq (S67)) and τc respectively All nodes are as-sumed to have the same activity potential given by Eq (S69)We set 〈k〉 = 01

Although Eq (S67) coincides with the expression of τlowastfor the activity-driven model (Eq (10)) τlowast as a functionof m is different between the activity-driven model (solidline in Fig 3(a)) and the present clique network model(solid line in Fig S6) This is because the values of a aredifferent between the two cases when m ge 2

EMPIRICAL ACTIVITY DISTRIBUTIONS

The epidemic threshold and prevalence when F (a) isconstructed from empirical contact data at a workplaceobtained from the SocioPatterns project [8] are shownin Figs S7(a) and S7(b) for m = 1 and m = 10 re-spectively The results for F (a) constructed from emailcommunication data at a research institution obtainedfrom the Stanford Network Analysis Platform [9] areshown in Figs S7(c) and S7(d) for m = 1 and m = 10respectively These results are qualitatively similar tothose shown in Fig 2

[1] L Speidel K Klemm V M Eguıluz and N MasudaNew J Phys 18 073013 (2016)

[2] N Masuda and R Lambiotte A Guide to Temporal Net-works (World Scientific Singapore 2016)

[3] P L Simon M Taylor and I Z Kiss J Math Biol 62479 (2011)

[4] E Valdano L Ferreri C Poletto and V Colizza PhysRev X 5 021005 (2015)

[5] R Pastor-Satorras C Castellano P Van Mieghem andA Vespignani Rev Mod Phys 87 925 (2015)

[6] M Karsai N Perra and A Vespignani Sci Rep 4 4001(2014)

[7] N Perra B Goncalves R Pastor-Satorras andA Vespignani Sci Rep 2 469 (2012)

[8] M Genois C L Vestergaard J Fournet A PanissonI Bonmarin and A Barrat Netw Sci 3 326 (2015)

[9] A Paranjape A R Benson and J Leskovec in Proc ofthe Tenth ACM International Conference on Web Search

7

τ

0

05

1

15

025

0

05

075

0 321 4 times10-3

times103

β

(c)

τ

0

05

025

075

1

01

0

02

03

04

0 105 times10-2

times103

15

β

(d)

τ

0

200

400

600

025

0

05

075

0 05 1

β

(a)

τ

0

50

100

150

200

250

01

0

02

03

0 21 3

β

(b)

lt gtρ

lt gtρ

lt gtρ

lt gtρ

times10-2

times10-2

FIG S7 Results for activity potentials derived from empiri-cal data The epidemic threshold and numerically simulatedprevalence are shown for m = 1 ((a) and (c)) and m = 10((b) and (d)) In (a) and (b) the activity potential is con-structed from contact data obtained from the SocioPatternsproject [8] This data set contains contacts between pairs ofN = 92 individuals measured every 20 seconds In (c) and(d) the activity potential is constructed from email communi-cation data at a research institution obtained from the Stan-ford Network Analysis Platform [9] Although the originaledges are directed we treat them as undirected We assumethat each email exchange event corresponds to a one-minutecontact We calculate the degree of each node per minuteaveraged over time denoted by 〈ki〉 and define the activitypotential as ai = [〈ki〉 minus 〈k〉2] m In (c) and (d) we usedN = 439 individuals satisfying ai gt 0 (some individuals ex-changed few emails such that ai lt 0) We set ∆t = 0001

and Data Mining WSDMrsquo17 (ACM New York 2017) pp601ndash610

7

τ

0

05

1

15

025

0

05

075

0 321 4 times10-3

times103

β

(c)

τ

0

05

025

075

1

01

0

02

03

04

0 105 times10-2

times103

15

β

(d)

τ

0

200

400

600

025

0

05

075

0 05 1

β

(a)

τ

0

50

100

150

200

250

01

0

02

03

0 21 3

β

(b)

lt gtρ

lt gtρ

lt gtρ

lt gtρ

times10-2

times10-2

FIG S7 Results for activity potentials derived from empiri-cal data The epidemic threshold and numerically simulatedprevalence are shown for m = 1 ((a) and (c)) and m = 10((b) and (d)) In (a) and (b) the activity potential is con-structed from contact data obtained from the SocioPatternsproject [8] This data set contains contacts between pairs ofN = 92 individuals measured every 20 seconds In (c) and(d) the activity potential is constructed from email communi-cation data at a research institution obtained from the Stan-ford Network Analysis Platform [9] Although the originaledges are directed we treat them as undirected We assumethat each email exchange event corresponds to a one-minutecontact We calculate the degree of each node per minuteaveraged over time denoted by 〈ki〉 and define the activitypotential as ai = [〈ki〉 minus 〈k〉2] m In (c) and (d) we usedN = 439 individuals satisfying ai gt 0 (some individuals ex-changed few emails such that ai lt 0) We set ∆t = 0001

and Data Mining WSDMrsquo17 (ACM New York 2017) pp601ndash610