Rumor Dynamics with Inoculations for CorrelatedScale Free Networks

Anurag Singh, Yatindra Nath SinghDepartment of Electrical Engineering

Indian Institute od TechnologyKanpur, India-208016

Email: {anuragsg, ynsingh}@iitk.ac.in

Abstract—We study the rumor spreading dynamics in complexnetworks, where degree correlations between the pair of nodesexist. An analytical description is provided that includes twovertices, degree-degree correlations in the dynamical evolution ofthe rumor spreading. In any social network, rumors can spreadand may have undesirable effect. To control rumor spread, acertain fraction of nodes are inoculated against rumors. In thispaper, the inoculation of nodes are done by random and targetedfashion with the variation of assortative coefficient in scale freenetworks. The new degree distributions of scale free networkshave been calculated for both kind of inoculations. It has beenobserved that rumor threshold in targeted inoculation schemeis greater than the rumor threshold in the random inoculationscheme. In targeted inoculation scheme, rumor threshold forcorrelated networks is found to be smaller than in uncorrelatednetworks. Random inoculation is not that much effective for scalefree networks. It is found that degree-degree correlation betweennodes doesn’t play any role in random inoculations. However, itis not true for targeted inoculation. The proposed hypothesis hasalso been verified by the simulation results.

I. INTRODUCTION

In today’s world, Internet has become the most importantmedium to circulate information. We use online social networksites daily to express our altitude, emotions and communicatewith friends. For the most of the events, information firstspreads over Internet than any other medium. Twitter andFacebook have emerged as most important mechanisms forinformation spread. Currently, Twitter has more than 500 mil-lion registered users and Facebook has more than 955 millionregistered users. Huge number of users share informationon Twitter and Facebook. Lot of research has been carriedout to get valuable insight in the information diffusion oversocial networks. If any information circulates without officiallypublicized confirmation, it is called rumor [1]. In other words,rumors are unreliable information.

The rumor spread phenomenon is similar to epidemicspread, in which all the informed nodes spread rumor byinforming their neighboring nodes [2], [3]. Recent researchin complex network theory has given new direction to theepidemic spreading model [4], [5]. With the developmentof computer technology now we can study the topology ofmany real systems which may contains millions of nodes andit is found that most of these have three main properties(small world, scale free, high clustering). The susceptible-

infected-refractory (SIR) model for epidemic dynamic process,a susceptible node can be infected by an infected neighborwith some spreading rate and introduces a new refractorystate in which nodes cannot be infected. The SIR model forrumor spreading, was introduced many years ago by Daleyand Kendal [6] and its variants by Maki-Thomsan [7]. In DK(Daley-Kendal) model, homogeneous population is subdividedinto three groups: ignorants, spreaders and stifler. The rumoris propagated throughout the population by pairwise contactsbetween spreaders and other individuals in the population.Any spreader involved in a pairwise meeting attempts toinfect other individual with the rumor. In Maki Thomsan(MK) model when spreader contacts another spreader, onlyinitiating spreader becomes a stifler. DK and MK modelshave an important shortcoming, these models do not take intoaccount the topology of the underlying social interconnectionnetworks along which the rumors spread. To consider thetopology of network, the rumor spreading model on smallworld network and scale free (SF) networks [8], [9], [10] havebeen defined. There is a threshold value on rumor spreadingrate below which the rumor or disease cannot propagate inthe network. Using mean field theory, Nekovee et al. [8]discovered that rumor threshold was small in homogeneousnetworks e.g. small world networks, random networks. Onthe other hand, the heterogeneous networks e.g. SF networks,Liu et. al. [10] have found, are more robust against spreadingof rumors as compared to homogeneous model. In previousstudies on rumor spreading in SF networks, it has been foundthat larger the nodal degree, the greater the rumor spreadfrom the informed node. Therefore, in SF networks withsufficiently large size, the rumor threshold, 𝜆𝑐 can be zero [11].Few studies have been done to study how to stop the rumorspreading [12], [13], [14] in small world and SF networks.These studies are more important since false and fatal rumorshave negative impacts on the society during disasters.Rumor spreads by pairwise contacts between nodes in SFnetworks with some spreading rate. Previous research [15],[16] have shown that complex networks display degree-degreecorrelations, implying that highly connected vertices prefer-ably connect to vertices which are also highly connectedand less connected vertices with less connected ones. Recentstudies have shown that social networks display assortativedegree correlations e.g. Facebook, implying that highly con-978-1-4673-5952-8/13/$31.00 c⃝ 2013 IEEE

nected vertices preferably connect to vertices which are alsohighly connected [15]. In order to study the impact of suchcorrelations on the dynamics of the rumor spreading model,the local assumption has been used for the degree-degreecorrelations function in order to numerically investigate theimpact of assortative degree correlations on the speed andsize of the rumor spreading in SF networks. It is interestingto find their impact, that the final size of rumor dependsvery much on the rumor spreading rate. In ths paper, thedependence of the rumor threshold in the SF networks on theirassortativity properties have been studied. Considering the SIRmodel proposed by Nekovee et.al. [8] for rumor spreading. Inthe SF networks, degree distribution [17] is given by,

𝑃 (𝑘) ∝ 𝑘−𝛾

where, 2 < 𝛾 ≤ 3. It alone does not define the topology of thenetwork completely. It does not say anything about the degreecorrelation of vertices.

II. MODIFIED RUMOR SPREADING MODEL

Nekovee et al. [8] have given a general stochastic modelfor the rumor spreading. In their model, the population isdivided into three compartments based on SIR model: ig-norant individuals, spreaders and stifler. Ignorant populationis susceptible to being informed, spreader spreads the rumorand stifler know the rumor but they are not interested inspreading it. When a spreader meets with an ignorant node,ignorant nodes becomes a spreader, with rate 𝜆. When aspreader meets with a spreader or stifler, the initiating spreaderbecomes a stifler with rate 𝜎. The stifler no longer spread arumor when they know that the rumor is out dated or wrong.However, when a rumor starts to propagate on networks fromspreaders, stifling is not the only way to stop it, thus we shouldconsider the factor of forgetting mechanism with rate 𝛿. Inthis paper 𝐼(𝑘, 𝑡), 𝑆(𝑘, 𝑡), 𝑅(𝑘, 𝑡) are defined as the fractionof ignorants, spreaders and stifler with degree k at time trespectively. These fraction of nodes satisfy the normalizationcondition 𝐼(𝑘, 𝑡)+𝑆(𝑘, 𝑡)+𝑅(𝑘, 𝑡) = 1. The rumor spreadingprocess can be summarized by the following set of pairwiseinteractions,

𝑆1 + 𝐼2𝜆−→ 𝑆1 + 𝑆2,

(when spreader meets with the ignorant, ignorants

becomes spreader at rate 𝜆)

𝑆1 +𝑅2𝜎−→ 𝑅1 +𝑅2,

(when a spreader contacts a stifler, the spreader

becomes a stifler at the rate 𝜎)

𝑆1 + 𝑆2𝜎−→ 𝑅1 + 𝑆2,

(when a spreader contacts with another spreader,

initiating spreader becomes a stifler at the rate 𝜎)

𝑆𝛿−→ 𝑅.

(𝛿 is the rate at which spreader stops spreading of a

rumor spontaneously)

An ignorant node with degree k is influenced by∑𝑙 𝑃 (𝑙∣𝑘)𝑆(𝑙, 𝑡) neighboring spreaders over different con-

nectivity classes [8]. 𝑃 (𝑙∣𝑘) is the degree-degree correlationfunction that a randomly chosen edge emanating from a nodeof degree k leads to a node of degree l. For uncorrelatednetworks conditional probability satisfies 𝑃 (𝑙∣𝑘) = 𝑙𝑃 (𝑙)/⟨𝑘⟩.The rate equations for the rumor diffusion model are,

𝑑𝐼(𝑘, 𝑡)

𝑑𝑡= −𝑘𝜆𝐼(𝑘, 𝑡)

∑𝑙

𝑆(𝑙, 𝑡)𝑃 (𝑙∣𝑘)

𝑑𝑆(𝑘, 𝑡)

𝑑𝑡= 𝑘𝜆𝐼(𝑘, 𝑡)

∑𝑙

𝑆(𝑙, 𝑡)𝑃 (𝑙∣𝑘)− 𝑘𝜎𝑆(𝑘, 𝑡)×∑𝑙

[𝑆(𝑙, 𝑡) +𝑅(𝑙, 𝑡)]𝑃 (𝑙∣𝑘)− 𝛿𝑆(𝑘, 𝑡)

𝑑𝑅(𝑘, 𝑡)

𝑑𝑡= 𝑘𝜎𝑆(𝑘, 𝑡)

∑𝑙

[𝑆(𝑙, 𝑡) +𝑅(𝑙, 𝑡)]𝑃 (𝑙∣𝑘) + 𝛿𝑆(𝑘, 𝑡)

(1)

For SF networks, degree distribution, 𝑃 (𝑘) ∝ 𝑘−𝛾 alonedoes not define the topology of the network completely. It doesnot say anything about the vertices that are connected to eachother. A correlated network is completely defined by its degreedistribution 𝑃 (𝑘) and its degree-degree correlation matrix𝑃 (𝑘, 𝑙) defines the probability of finding an edge emergingfrom a k degree node to a l degree node,

⎛⎜⎜⎜⎜⎝

𝑃 (1, 1) 𝑃 (1, 2) ... 𝑃 (1, 𝑘𝑚𝑎𝑥)𝑃 (2, 1) 𝑃 (2, 2) ... 𝑃 (2, 𝑘𝑚𝑎𝑥). . . .. . . .

𝑃 (𝑘𝑚𝑎𝑥, 1) 𝑃 (𝑘𝑚𝑎𝑥, 2) ... 𝑃 (𝑘𝑚𝑎𝑥, 𝑘𝑚𝑎𝑥)

⎞⎟⎟⎟⎟⎠

(2 − 𝛿𝑘𝑙)𝑃 (𝑘, 𝑙) is the probability that a randomly chosenedge connects two vertices of degree k and l respectively [18]and,

𝑃 (𝑙∣𝑘) = 𝑃 (𝑘, 𝑙)𝑞𝑘

(2)

where, 𝑞𝑘 = 𝑘𝑃 (𝑘)⟨𝑘⟩ and,

𝑃 (𝑘, 𝑙) = 𝑞𝑘[𝑟𝛿𝑘𝑙 + (1− 𝑟)𝑞𝑙] (3)

𝑃 (𝑙∣𝑘) = 𝑃 (𝑘, 𝑙)𝑞𝑘

= 𝑟𝛿𝑘𝑙 + (1− 𝑟)𝑞𝑙 (4)

Where, 𝛿𝑘𝑙 is the Kronecker delta function and r is theassortativity coefficient. It is 0 ≤ 𝑟 ≤ 1 for assortativenetworks, 𝑟 = 0 for uncorrelated network and 𝑟 = 1 forfull assorted network. Therefore for social network 𝑟 will bepositive [15].

Now, rumor equations from Eq. (1) can be modified as,

𝑑𝐼(𝑘, 𝑡)

𝑑𝑡= − 𝜆⟨𝑘⟩

𝑃 (𝑘)𝐼(𝑘, 𝑡)

∑𝑙

𝑆(𝑙, 𝑡)𝑃 (𝑘, 𝑙)

𝑑𝑆(𝑘, 𝑡)

𝑑𝑡=𝜆⟨𝑘⟩𝑃 (𝑘)

𝐼(𝑘, 𝑡)∑𝑙

𝑆(𝑙, 𝑡)𝑃 (𝑘, 𝑙)− 𝜎⟨𝑘⟩𝑃 (𝑘)

×

𝑆(𝑘, 𝑡)∑𝑙

[𝑆(𝑙, 𝑡) +𝑅(𝑙, 𝑡)]𝑃 (𝑘, 𝑙)− 𝛿𝑆(𝑘, 𝑡)

𝑑𝑅(𝑘, 𝑡)

𝑑𝑡=𝜎⟨𝑘⟩𝑃 (𝑘)

𝑆(𝑘, 𝑡)∑𝑙

[𝑆(𝑙, 𝑡) +𝑅(𝑙, 𝑡)]𝑃 (𝑘, 𝑙)

+𝛿𝑆(𝑘, 𝑡) (5)

III. TARGETED INOCULATION

It is assumed that fraction 𝑔𝑘 of nodes with degree k aresuccessfully inoculated. All nodes with degree 𝑘 > 𝑘𝑡 getsinoculated i.e. 𝑔𝑘 = 1. The fraction of inoculated nodes givenby,

𝑔𝑘 =

⎧⎨⎩

1, 𝑘 > 𝑘𝑡,𝑓, 𝑘 = 𝑘𝑡,0, 𝑘 < 𝑘𝑡.

where, 0 < 𝑓 ≤ 1 and 𝑘𝑡 is the cut-off degree.Initial degree distribution is 𝑃 (𝑘) with network size 𝑁 .The new degree distribution after the application of targetedinoculation is [19],

𝑃 ′(𝑘) =∞∑𝑞=𝑘

(𝑞

𝑘

)Φ𝑞−𝑘

𝑞 (1− Φ𝑞)𝑘𝑝𝑞 (6)

Where, 𝑝𝑘 is the probability to find a node of degree k in thenon removed nodes (during inoculations) before disconnectingthe network. Φ𝑙 is the probability of finding edge from the 𝑙degree node,

Φ𝑙 =𝑁⟨𝑘⟩(1− 𝑔𝑙)

∑𝑘 𝑃 (𝑙, 𝑘)𝑔𝑘

𝑙𝑁𝑃 (𝑙)(1− 𝑔𝑙) (7)

𝑝𝑘 =(1− 𝑔𝑘)𝑃 (𝑘)1−∑

𝑖 𝑃 (𝑖)𝑔𝑖(8)

IV. RANDOM INOCULATION

In random inoculation strategy, randomly selected nodewill be inoculated. This approach inoculates a fraction of thenodes randomly, without any information of the network. Herevariable 𝑔(0 ≤ 𝑔 ≤ 1) defines the fraction of inoculativenodes. For random inoculation put 𝑔𝑘 = 𝑔 in Eq (8), the finaldegree distribution in correlated network is found to be sameas in the uncorrelated network.

V. SIMULATIONS AND RESULTS

The numerical simulations have been done to observe thecomplete dynamical process with and without inoculationstrategies for different spreading rates (𝜆) and with the vari-ation of assortativity coefficient (𝑟) for correlated networks.Stifling rate (𝜎) has been fixed to be 0.25 and spontaneouslyrumor forgetting rate (𝛿) is fixed to be 1. After spread of rumor

Algorithm 1 New degree distribution after Random inocula-tion

Input: A 2-d edge array after inoculationInput: Total number of nodes in the network(N) beforeinoculationInput: Number of nodes after deletion (𝑇 𝑛𝑜𝑑𝑒𝑠)Input: Fraction of nodes deleted in the network (𝑑𝑒𝑙 𝑓𝑟)Input: Number of edges in the network (𝑇 𝑒𝑑𝑔𝑒𝑠) afterinoculationOutput: Degree distribution after applying random inocu-lationdeg, 𝑑𝑒𝑔 𝑑𝑖𝑠𝑡𝑟𝑖𝑏, 𝑛𝑜𝑟𝑚 𝑑𝑒𝑔 𝑑𝑖𝑠𝑡𝑏, count, 𝑘 = 0, 𝑖 = 0;𝑐𝑜𝑢𝑛𝑡 = 0;while 𝑖 < 𝑇 𝑒𝑑𝑔𝑒𝑠 do

Put 𝑖𝑡ℎ node in the match=Edges[𝑖][0]while 𝑖 < 𝑇 𝑒𝑑𝑔𝑒𝑠 do

if match(𝑖)==Edges[𝑖][0] thencount++ ; 𝑖++ ;

elsebreak

end ifend whiledegree of the match(𝑖) in degree array, deg(match)=count

end whilefor 𝑖 < 𝑇 𝑛𝑜𝑑𝑒 do

for 𝑗 < 𝑁 doif deg[𝑗]==𝑖 then𝑘++

end ifend for𝑑𝑒𝑔 𝑑𝑖𝑠𝑡𝑏[ i ]=𝑘𝑑𝑒𝑔 𝑑𝑖𝑠𝑡𝑏 will contain the node of same degree𝑛𝑜𝑟𝑚 𝑑𝑒𝑔 𝑑𝑖𝑠𝑡𝑏 [𝑖]=𝑘/𝑁

end forTABLE I: No. of deleted edges after inoculation

Removed Fraction(g) r 𝑒𝑑𝑒𝑙(𝑅𝐼) 𝑒𝑑𝑒𝑙(𝑇𝐼)0.03 0 1461 111940.03 0.3 1160 106280.03 0.5 1163 108660.03 0.7 928 105750.06 0 1915 119280.06 0.3 1623 116540.06 0.5 1611 114910.06 0.7 1576 113290.2 0 7527 143420.2 0.3 6746 140920.2 0.5 6428 140450.2 0.7 6271 14077

in a time step, the spreader will become stifler in next timestep. At the starting of each simulation, initially spreader nodesare chosen randomly for rumor spreading model, in Eq. (5),while all the other nodes are ignorants. In each time step,all the N nodes interact with each other for rumor passing.After N nodes update their states according to the proposedrumor model, time step is incremented. Scale free networksare used for the contact process. The SF networks have beengenerated according to the power law, 𝑃 (𝑘) = 𝑘−𝛾 , where

100

101

102

103

10−5

10−4

10−3

10−2

10−1

100

k

P(k

)

Initial degree distribution

Final degree distribution (Theory)

Final degree distribution (Simulation)

100

101

102

103

10−4

10−3

10−2

10−1

100

k

P(k

)

Initial degree distributionFinal degree distribution (Simulation)Final degree distribution (Theory)

(a) (b)Fig. 1: Degree distribution of correlated SF network, beforeand after applying (a) targeted inoculation with cutoff de-gree=60 and (b) random inoculation 𝑔 = 40 %

2 < 𝛾 ≤ 3. We have taken 𝑁 = 10000 and 𝛾 = 2.1. Therandom inoculation is implemented by randomly selecting 𝑔𝑁nodes in the network.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

λ

Fina

l siz

e of

rum

or(R

)

r=0

r=0.1

r=0.3

r=0.5

r=0.7

Fig. 2: Final rumor size R vs 𝜆 plot for correlated anduncorrelated SF networks without inoculations

Fig. 1 shows the degree distribution of the correlated SFnetworks before and after the application of random (Fig. 1(a))and targeted inoculation (Fig. 1(b)). The degree distributionsbefore and after the inoculation schemes has been plotted fromtheory and simulation (using Algorithm 1). Theoretical and thesimulation results were found to be similar. Table I shows thenumber of removed edges from the SF network after applyingrandom and targeted inoculations for different fraction ofremoved nodes and assortativity coefficients (𝑟). Initially totalnumber of edges are 16500 without any inoculations. In Figs.2-4, the final size of rumor spread has been plotted againstthe rumor transmission rate, 𝜆 for no inoculations, randomand targeted inoculation schemes for various assortativitycoefficients (𝑟 = 0, 0.1, 0.3, 0.5, 0.7).

The final size of rumor has been observed to be morewith high values of 𝑟 in correlated networks for spreadingrate 𝜆 ≥ 0.5 without inoculations. Before this rate, therumor size is higher in uncorrelated network (𝑟=0). Similarpatterns are observed in the random (Fig. 3) and targetedinoculation (Fig. 4). Interestingly rumor threshold is very lessfor random inoculation scheme in Fig. 3 for both correlatedand uncorrelated networks and it is same for all values of

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

λ

Fin

al s

ize

of

rum

or(

R)

r=0r=0.1r=0.3r=0.5r=0.7

Fig. 3: Final rumor size R vs 𝜆 plot for 25% random in-oculation of nodes in correlated and uncorrelated scale freenetworks

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

λFi

nal s

ize

of r

umor

(R)

r=0

r=0.1

r=0.3

r=0.5

r=0.7

Fig. 4: Final rumor size R vs 𝜆 plot for targeted inoculationwith cut off degree =10 in correlated and uncorrelated scalefree networks

assortative coefficients (𝑟). On the other hand for targetedinoculation in Fig. 4 rumor thresholds are higher than randomand it increases with the decrease of assortative coefficient (𝑟)and found to be maximum in uncorrelated SF network.

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

t

R(t)

, S(t)

r=0

r=0.1

r=0.3

r=0.5

r=0.7

R(t)

S(t)

Fig. 5: Size of rumor and spreaders with time for correlatednetworks with 𝑔 = 25% random inoculations for 𝜆 = 0.5

The sizes of informed nodes and spreader nodes observedwith the time are shown in Figs. 5-7. It is observed that size ofinformed nodes initially increases exponentially with the timeand after some time they achieved steady state and remainforever in this state. The size of spreader initially increaseswith the increase of time but after some time when spreadersturn into stifler then the number of spreaders decrease withthe increase of time. After some time the size of spreaderswill be zero as system is in get the steady state. In Fig.

0 2 4 6 8 10 120

1

2

3

4

5

6

7

8

9x 10

−5

t

R(t)

, S(t)

r=0

r=0.1

r=0.3

r=0.5

r=0.7

R(t)

S(t)

Fig. 6: Size of rumor and spreaders with time for correlatednetworks with 𝑔 = 25% targeted inoculations for 𝜆 = 0.5

5, 𝑅(𝑡) and 𝑆(𝑡) are plotted against time for SF networkwith different assortativity coefficient (𝑟) with 25% randominoculation and in Fig. 6 with 25% targeted inoculation. It canbe easily observed that final size of rumor increases with theincrease of 𝑟. Initially spreading rate of rumor increases withthe increment of value 𝑟 and dies out early when correlationsare stronger and rumor spreading found almost zero in thetargeted inoculations in Fig. 6. Similar patterns found forthe 10 % inoculation of nodes in Fig. 7. Here in size ofrumor and spreaders, some perturbations are observed. When𝑔𝑘 is less in targeted inoculations, the rumor spreads veryfast initially because of hub neighbors. After some time stepsif spreaders will become more then the number of stiflerwill increase. At that time number of spreaders will go less.But again in next time steps hub ignorants neighbors canbe found and same process of increasing and decreasing ofspreaders happens. Therefore, we will get perturbations in thespreaders and stifler sizes with time. For the higher 𝑔𝑘 intargeted inoculation scheme, correlated SF networks will loosethe heterogeneity and this scheme will be similar as randominoculation. Therefore no perturbations found in the size ofspreaders and stifler.

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

0.3

t

R(t

), S

(t)

r=0

r=0.1

r=0.3

r=0.5

r=0.7

R(t)

S(t)

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

t

R(t

), S

(t)

r=0

r=0.1

r=0.3

r=0.5

r=0.7

R(t)

S(t)

(a) Random (b) TargetedFig. 7: Size of rumor and spreaders with time for correlatednetworks with 10% inoculations and 𝜆 = 0.5

VI. CONCLUSIONS

In the proposed rumor spreading model for correlatednetworks, it is observed that rumor can be stopped moreeffectively in correlated networks using targeted inoculation.

It is found that degree-degree correlation doesn’t play anyrole to change degree distribution in random inoculations.However, it doesn’t hold for targeted inoculations. The newdegree distribution is generated after targeted and randominoculation schemes; in random inoculation, deformed degreedistribution for correlated networks is same as in uncorrelatednetworks. It is interesting to observe that for small values ofrumor transmission rate (𝜆) final size of rumor in uncorrelatednetworks is larger than in correlated one. On the other handfor higher values of transmission rate rumor size is lower inuncorrelated SF networks. It can be conclude that removalof connections among hub nodes can decrease the rumorspreading.

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