TNETS edition

Susceptible

Infectious

Infectious

Comp!rtment!l models

Infectious

RecoveredSusceptible

Cont!ct p!tterns

ID1 ID2 Time1 2 343 4 551 5 564 2 705 4 776 1 1025 6 1105 7 1226 7 1302 5 1983 4 2054 2 2102 7 230

Sociopatterns gallery

P H Y S I C A L P R O X I M I T Y

Prostitution

Sociopatterns conference

Hospital system

N = 16,730, L = 50,632, T = 6.0y

N = 113, L = 20,818, T = 59h

N = 159(8), L = 6,027(350), T = 7.3(1)h

N = 293,878, L = 64,625,283, T = 3,570dReality miningN = 63, L = 26,260, T = 8.6h

ELECTRONIC COMMUNICATION

N = 57,189, L = 444,162, T = 112.0d Bornholdt’s e-mail

Eckmann’s e-mail

N = 3,188, L = 115,684, T = 81.6d

Filmtipset forum

N = 7,084, L = 1,412,401, T = 8.61y

Filmtipset messages

Pussokram dating

N = 28,972, L = 529,890, T = 512.0d

QX datingN = 80,683, L = 4,337,203, T = 63.7d

N = 35,624, L = 472,496, T = 8.27y

Facebook wall posts

N = 293,878, L = 876,993, T = 1591d

TEMPORAL TOSTATIC

t st!

rt

t sto

p0 5 10 15 20

1

2

3

4

5

6

t

1

2

34

5

6

time-slice networks

t st!

rt

t sto

p0 5 10 15 20

1

2

3

4

5

6

t

1

2

34

5

6

ongoing networks

0 5 10 15 20

1

2

3

4

5

6

t

1

2

34

5

6

1

2

34

5

6

! = 2.5

! = 10

exponential-threshold networks

GOOD REPRESENTATION:RANKING OF IMPORTANTVERTICES CONSERVED

FOR ALL PARAMETER VALUES:MEASURE AVG OUTBREAK SIZE WHEN SPREADING STARTS AT i

FOR ALL PARAMETER VALUES:MEASURE DEGREE OF iFOR ALL PARAMETER VALUES:MEASURE CORENESS OF i

degree 4

coreness 0

coreness 2

coreness 3

coreness 4

static importanceoptimal params.

dyna

mic

impo

rtan

ce

Spearmanrank correlationcoefficent =Quality ofrepresentation

E-mail 1

E-mail 2

Dating

Gallery

Conference

Prostitution

Results, Degree

Time-slice Ongoing Exponential-threshold Accumulated

E-mail 1

E-mail 2

Dating

Gallery

Conference

Prostitution

Time-slice Ongoing Exponential-threshold Accumulated

Results, Coreness

Time sliceTime sliceTime slice OngoingOngoingOngoing Expo. thresholdExpo. thresholdExpo. threshold Acc.

ρmax tstart tstop ρmax tstart tstop ρmax τ ΩΩ ρE-m!il 1 0.73 0 0.42 0.50 0.25 0.25 0.77 0.40 0.30 0.46E-m!il 2 0.91 0 0.25 0.91 0.20 0.20 0.93 1.0 0.26 0.88D!tin" 0.82 0 0.65 0.42 0.25 0.25 0.86 0.10 0.16 0.71G!ller# 0.77 0 0.72 0.53 0.39 0.39 0.87 0.70 0.71 0.76Conference 0.79 0 0.10 0.74 0.10 0.11 0.77 0.04 0.02 0.53Prostitution 0.71 0 0.77 0.30 0.60 0.60 0.72 0.04 0.20 0.49

Perform!nce & p!r!meter v!lues De"ree

P!r!meter dependence of perform!nce

0 0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

00

0.2

0.4

0.6

0.8

1

!

tstart/ T

t sto

p/

T

Time slice

P!r!meter dependence of perform!nce

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

!

tstart/ T

t sto

p/

T

Concurrency

P!r!meter dependence of perform!nce

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0

0.5

1

1.5

2

!

!

" /

T

Exponential threshold

STEP 1 Assign stubs to vertices from a random number distribution.

1

2

3

4

5

6

STEP 2 Connect random pairs of stubs to form a simple graph.

1

2

34

5

6

STEP 3 Create active intervals for each edge.

(1,2)(2,3)(2,4)(2,5)(3,4)(3,5)(4,5)(5,6)

time

STEP 4 Create a time series of contacts from some interevent-time distribution.

time

STEP 5 Split the time series into segments proportional to the intervals and impose the contacts of the segments to the intervals.

(1,2)(2,3)(2,4)(2,5)(3,4)(3,5)(4,5)(5,6)

time

STEP 6 Forget the active intervals.

(1,2)(2,3)(2,4)(2,5)(3,4)(3,5)(4,5)(5,6)

time

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.1 1µ

! max

0.50.05

Exponential threshold

Time-slice

Accumulated

Ongoing

bre!k

STATIC TOTEMPORAL

(1,2)(2,3)(2,4)(2,5)(3,4)(3,5)(4,5)(5,6)

(1,2)

(2,3)(2,4)(2,5)(3,4)(3,5)(4,5)(5,6)

time

time

(1,2)(2,3)(2,4)(2,5)(3,4)(3,5)(4,5)(5,6)

ON

GO

ING

LIN

K P

ICT

UR

E

time

(2,3)(2,4)(2,5)(3,4)(3,5)(4,5)(5,6)

(1,2)

(1,2)(2,3)(2,4)(2,5)(3,4)(3,5)(4,5)(5,6)

time

LIN

K T

UR

NO

VER

PIC

TU

RE

timeBeginning time Interevent times End time

0 Tt1 t2 t3 t4t5 t6t7 t8tB tE

DEFI NITI ONS

Compensate for the size bias on intervals because of finite

T0

t’

t

sampling time (t’ would only be recorded if it starts within [0,T–t’])

Compensate for the size bias on intervals because of finite

T0

t’

t

sampling time (t’ would only be recorded if it starts within [0,T–t’])

Compensate for the chance an interevent time t is active

0

tat the start of the sampling is proportional to t

Compensate for the size bias on intervals because of finite

T0

t’

t

sampling time (t’ would only be recorded if it starts within [0,T–t’])

Compensate for the chance an interevent time t is active

0

tat the start of the sampling is proportional to t

tiT–tii: ti!t

! / tiT–tii

!Sum up and normalize

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000time t (days)

predicted frominterevent timesend timesbeginning times

PROSTITUTION

P (t

)B

Dating 2

1

0

0.5

0.75

0.25

ࢥ

Dating 1

Forum

1

0

0.5

0.75

0.25

ࢥ

1

0

0.5

0.75

0.25

ࢥ

Prostitution

Hospital

1

0

0.5

0.75

0.25

ࢥ

1

0

0.5

0.75

0.25

ࢥ

E-mail 2

Facebook

1

0

0.5

0.75

0.25

ࢥ

1

0

0.5

0.75

0.25

ࢥE-mail 1

Film

1

0

0.5

0.75

0.25

ࢥ

1

0

0.5

0.75

0.25

ࢥ

Conference

Gallery

1

0

0.5

0.75

0.25

ࢥ

1

0

0.5

0.75

0.25

ࢥ

End Times

Beginning Times

Predictableedges w.r.t.beginning /end times

(1,2)

(1,3)

(1,4)

(2,3)

reference networks

(1,2)

(1,3)

(1,4)

(2,3)

(1,2)

(1,3)

(1,4)

(2,3)

reference network:identic!l interevent times

(1,2)

(1,3)

(1,4)

(2,3)

(1,2)

(1,3)

(1,4)

(2,3)

reference network:identic!l be"innin" times

(1,2)

(1,3)

(1,4)

(2,3)

(1,2)

(1,3)

(1,4)

(2,3)

reference network:identic!l end times

0

0.1

0.2

0.3

0.1 0.2 0.90.8 10.70.60.50.40.3

0.1

1

0.01

0.001

per-cont!ct tr!nsmission prob!bilit"

dur!

tio

n of

infe

ctiv

e st!#

e

fr!

ctio

n of

infe

ctiv

es

O r i ! i n " l d " t " S I R

0

0.1

0.2

0.3

0.1 0.2 0.90.8 10.70.60.50.40.3

0.1

1

0.01

0.001

per-cont!ct tr!nsmission prob!bilit"

dur!

tio

n of

infe

ctiv

e st!#

e

fr!

ctio

n of

infe

ctiv

es

I n t e r e v e n t t i m e s S I R

0

0.1

0.2

0.3

0.1 0.2 0.90.8 10.70.60.50.40.3

0.1

1

0.01

0.001

per-cont!ct tr!nsmission prob!bilit"

dur!

tio

n of

infe

ctiv

e st!#

e

fr!

ctio

n of

infe

ctiv

es

B e ! i n n i n ! t i m e s S I R

0

0.1

0.2

0.3

0.1 0.2 0.90.8 10.70.60.50.40.3

0.1

1

0.01

0.001

per-cont!ct tr!nsmission prob!bilit"

dur!

tio

n of

infe

ctiv

e st!#

e

fr!

ctio

n of

infe

ctiv

es

E n d t i m e s S I R

0

0.02

0.04

0.06

E-mail 1

0.1

0

0.05

Film

0

0.05

0.1

Dating 1

0.05

0.1

0.15

0.2

0Forum

0

0.02

0.04

0.06

E-mail 2

0

0.02

0.06

0.08

0.04

Facebook0

0.01

0.02

0.03

0.04

Prostitution

0

0.1

0.2

0.3

Hospital

0

0.04

0.06

0.08

0.02

Gallery

0

0.02

0.04

0.06

Conference

0.05

0.1

0Dating 2

end

times

begin

nin

g times

intereven

t times

0.1 0.2 0.90.8 10.70.60.50.40.3

0.1

1

0.01

0.001

per-cont!ct tr!nsmission prob!bilit"

dur!

tio

n of

infe

ctiv

e st!#

e

O r i ! i n " l d " t "

0

0.2

0.3

0.4

!ve

r!#

e nu

mbe

r of

infe

ctio

ns

S I S

0.1

0.1 0.2 0.90.8 10.70.60.50.40.3

0.1

1

0.01

0.001

per-cont!ct tr!nsmission prob!bilit"

dur!

tio

n of

infe

ctiv

e st!#

e

I n t e r e v e n t t i m e s

0

0.2

0.3

0.4

!ve

r!#

e nu

mbe

r of

infe

ctio

ns

S I S

0.1

0.1 0.2 0.90.8 10.70.60.50.40.3

0.1

1

0.01

0.001

per-cont!ct tr!nsmission prob!bilit"

dur!

tio

n of

infe

ctiv

e st!#

e

B e ! i n n i n ! t i m e s

0

0.2

0.3

0.4

!ve

r!#

e nu

mbe

r of

infe

ctio

ns

S I S

0.1

0

0.2

0.3

0.4

0.1 0.2 0.90.8 10.70.60.50.40.3

0.1

1

0.01

0.001

per-cont!ct tr!nsmission prob!bilit"

dur!

tio

n of

infe

ctiv

e st!#

e

!ve

r!#

e nu

mbe

r of

infe

ctio

ns

E n d t i m e s S I S

0.1

0

0.01

0.02

0

0.0005

0.001

0.0015

0

0.05

0.1

0

0.01

0.02

0.03

0.04

0

0.0005

0.001

0.0005

0.001

0 0

0.05

0.1

0

0.001

0.0015

0.0005

0.05

0.1

0.15

0.2

0 0

0.02

0.04

0.06

0.08

0

0.01

0.03

0.04

0.02

E-mail 1

Film

Dating 1

Forum

E-mail 2

Facebook Prostitution

Hospital

Gallery

Conference

Dating 2

end

times

begin

nin

g times

intereven

t times

Science by: Illustrations by:

Petter Holme Fredrik Liljeros Mi Jin Lee

P Holme, 2013, PLoS Comp. Biol. 9:e1003142. P Holme, F Liljeros, 2013, arxiv:1307.6436.

0

0.2

0.4

0.6

0.8

E-mail 1

0

0.2

0.6

0.4

Film

0

0.1

0.2

0.3

0.4

E-mail 2

0

0.2

0.4

Facebook

0

0.4

0.6

0.2

Dating 1

0.8

0.2

0.4

0.6

0

ForumDating 2

0.2

0.8

0

0.4

0.6

–0.4

–0.2

0.2

0.8

0

0.4

0.6

Conference

Hospital

0

0.2

0.4

Prostitution

0

0.1

0.2

Gallery0

0.2

0.8

0.4

0.6

end

times

begin

nin

g times

intereven

t times

0

0.1

0.2

0.3

0.4

E-mail 1

0

0.2

0.4

0.6

E-mail 2

0

0.4

0.6

0.2

Dating 1

0

0.005

0.01

Facebook

0.8

0.2

0.4

0ForumDating 2

–0.005

0

0.005

end

times

–0.5

0

0.5

Conference

Hospital

–0.001

0

0.001

Prostitution

0

0.1

0.2

begin

nin

g times

intereven

t times

Gallery0

0.2

0.8

0.4

0.6

0

0.2

0.5

0.4

Film

0.3

0.1