TECHNICAL REPORT 1

Dynamics of Information Spreading in OnlineSocial Networks

Sai Zhang, Ke Xu, Member, IEEE, Xi Chen, Student Member, IEEE, and Xue Liu, Member, IEEE

Abstract—Online social networks (OSNs) are changing theway information spreads throughout the Internet. A deep under-standing of information spreading in OSNs leads to both socialand commercial benefits. In this paper, dynamics of informationspreading (e.g., how fast and widely the information spreadsagainst time) in OSNs are characterized, and a general andaccurate model based on Interactive Markov Chains (IMCs)and mean-field theory is established. This model shows tightrelations between network topology and information spreading inOSNs, e.g., the information spreading ability is positively relatedto the heterogeneity of degree distributions whereas negativelyrelated to the degree-degree correlations in general. Further, themodel is extended to feature the time-varying user behaviorand the ever-changing information popularity. By leveragingthe mean-field theory, the model is able to characterize thecomplicated information spreading process (e.g., the dynamicpatterns of information spreading) with six parameters. Extensiveevaluations based on Renren’s data set illustrate the accuracyof the model, e.g., it can characterize dynamic patterns ofvideo sharing in Renren precisely and predict future spreadingdynamics successfully.

Index Terms—online social networks, complex networks, in-formation spreading, network topology, dynamic patterns.

I. INTRODUCTION

W ITH the fast development of the Internet and Web2.0,online social networks (OSNs) generated by the Web

applications are playing an important role in informationspreading throughout the Internet. Getting, publishing andsharing information with forums, video-sharing sites (VSSes)and social networking sites (SNSes) are becoming increasinglypopular. A recent report1 shows that as of May 2013, almost72% online U.S. adults use SNSes, up from 67% in late 2012and only 8% in February 2005. In August 20132, there wereabout 500 million tweets per day on Twitter and a peak of143,199 tweets per second was observed during the airing

Sai Zhang is with the Institute for Interdisciplinary Information Sciences,Tsinghua University, Beijing, 100084, China (e-mail: zs11235@gmail.com).Ke Xu is with the Department of Computer Science & Technology, TsinghuaUniversity, Beijing, 100084, China (e-mail: xuke@mail.tsinghua.edu.cn). XiChen and Xue Liu are with the School of Computer Science and the Depart-ment of Electrical and Computer Engineering, McGill University, Montreal,QC H3A 0E9, Canada (e-mail: {xchen100,xueliu}@cs.mcgill.ca).

This work was supported in part by the 973 Programs (2011CBA00300,2011CBA00301, 2012CB315803), the 863 Program (2013AA013302), theNSFC Projects (61033001, 61170292, 61361136003), the New GenerationBroadband Wireless Mobile Communication Network of the National Scienceand Technology Major Projects (2012ZX03005001), and the EU MARIECURIE ACTIONS EVANS (PIRSES-GA-2010-269323).

1Pew Research Center’s Internet & American Life Project. http://www.pewinternet.org/2013/08/05/methods-15/.

2New Tweets per second record, and how!. https://blog.twitter.com/2013/new-tweets-per-second-record-and-how.

of Castle in the Sky3. SNSes can influence user’s purchasedecisions. Statistics4 show that 74% consumers make theirpurchase decisions based on SNSes. In emergency situationssuch as fires and earthquakes, information spreading in SNSescan also provide valuable knowledge that is crucial in life-saving. Approximately 96% earthquakes of Japan Meteorolog-ical Agency (JMA) seismic intensity scale 3 or more are de-tected merely by monitoring tweets [1]. Moreover, SNSes evenfacilitate the mobilization of mass movements5. Therefore, itis important and necessary to study the nature of informationspreading in OSNs to promote viral marketing, improve socialbenefits, and maintain social security in practical aspects, aswell as understand the characteristics of complex networks andindividual behaviors in scientific aspects.

Numerous works [2–5] have studied various spreadingprocesses in networks. Due to the complexity of underlyingnetwork topology and the diversity of collective behaviors,these models focus on simplified scenarios where several im-portant factors of information spreading are missing. Previousworks on epidemic spreading [6–13] and rumor diffusion [14–17] have constructed rigorous theories to describe spreadingprocesses in different circumstances. These works use tools ofmean-field and dynamical systems to derive precise resultsin characterizing spreading dynamics. Since the processesand mechanisms of epidemic and rumor spreading are quitedifferent from those of information spreading in OSNs, weshould establish new theoretical models to characterize infor-mation spreading in OSNs. In this paper, we first elaboratea general mechanism of information spreading in realisticOSNs and establish a probabilistic model in the frameworkof Interacting Markov Chains (IMCs) [18]. Then based onthe mean-field principle, we transform the probabilistic modelinto a deterministic version characterized by a non-lineardynamical system. By analyzing this system, we show thatinformation spreading in OSNs is mainly characterized bythe following three arguments: spreading threshold, prevalenceand efficiency. Our work reveals how the defined modelparameters and network topology influence these arguments.

The process of information spreading relies on both thespreading mechanism and the underlying network topology.On one hand, several existing models [4, 5] are limited bythe fact that they ignore the network topology and onlycharacterize the spreading process in global perspectives. On

3Castle in the Sky. http://en.wikipedia.org/wiki/Castle in the Sky.4Social Networks Influence 74% Consumers Buying Decisions. http://

sproutsocial.com/insights/social-networks-influence-buying-decisions/.5Technology and Globalization. http://www.globalization101.org/uploads/

File/Technology/tech2012.pdf.

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2 TECHNICAL REPORT

the other hand, other approaches [2, 3] modeling informationspreading based on the network structure cannot work whenthe network is implicit or even unknown. In this paper, a moreaccurate model is established by considering two key quanti-ties of network topology, i.e., degree distributions (the first-order correlation) and degree-degree correlations (the second-order correlation), which balances the global-local paradox.We derive analytic relations between network heterogeneityand information spreading, and demonstrate that the positivedegree-degree correlations inhibit information spreading ingeneral. This theoretical result has been confirmed by previousexperiments.

Previous studies [19–22] have shown that the dynamics ofonline media exhibit great burstiness, diurnal patterns andperiodicity. However, existing “static” models of epidemicand rumor spreading cannot fully characterizes these features.To fill this gap, we propose another spreading model withtime-varying parameters, which describes user behavior andinformation popularity in time domain. We further studyhow the temporal dynamics of model parameters affect thespreading process. Through analytic deductions, we find thatthe temporal dynamics of model parameters largely determinehow information spreading evolves in OSNs. This facilitatesus to learn realistic patterns with six reasonable parameters.To our knowledge, this is the first spreading model whichcharacterizes the dynamic patterns of information spreadingin realistic complex networks in closed forms. By testing on adata set of video sharing in Renren6, one of the largest SNSesin China, we show that our time-varying model can depictthe spreading dynamics precisely. Furthermore, we verify thatthe time-varying model can successfully predict long-rangeddynamic patterns of information spreading given history data.

The rest of the paper is structured as follows. Section IIdescribes our probabilistic model and gives the deterministicdynamical system derived by the model. Section III sums upthe terminologies and notations used in the following sectionsand defines the spreading threshold, prevalence and efficiency.In Section IV, based on defined arguments we comprehen-sively study the impact of underlying network topology oninformation spreading in both uncorrelated and correlated het-erogeneous networks. The spreading model with time-varyingparameters and its analysis are given in Section V. SectionVI carries out several numerical studies to verify previousanalytic conclusions, and uses the Levenberg-Marquardt (LM)algorithm [23] to learn the dynamic patterns of video sharingin Renren based on the time-varying model. We also use theimproved model to predict future dynamics of video sharing.Finally, we summarize related work in Section VII, concludethis paper and discuss our future work in Section VIII.

II. MODEL DESCRIPTION

Consider an undirected network N = (V,E), where Vdenotes the set of vertices and |V | = n, and E denotes theset of edges. Note that we sometimes refer to vertices as usersin OSNs. The information I (e.g., a message, photo or video)spreads along its edges. We first divide the states of vertices

6Renren. http://www.renren.com/

i r

a q

•

1− λ

1− β

(1− α)λ

αλ

β

1

1

Fig. 1. State transition diagram. i, a, r, q are short for ignorant, active,indifferent and quiet, respectively.

into three kinds: ignorant, active and indifferent. Ignorantvertices are those who are unaware of I, active vertices arethose who are aware of I and actually spread (e.g., forwardingor sharing in SNSes) it, and indifferent vertices are those whoare aware of it but do not spread it. Since active vertices havespread I, their ignorant neighbors can be activated with acertain probability. Based on the fact that active vertices canonly affect their neighbors in a limited time duration due to thepopularity decay, we assume that active vertices become quiet(a new state in which they get no influence on their neighborsany more) with a certain probability spontaneously.

Figure 1 illustrates our model. In network N , there are fourkinds of vertices, ignorant, active, indifferent and quiet. Infor-mation I spreads in N based on the following mechanisms:

(1) While keeping their states unchanged with probability 1−λ, ignorant vertices with active neighbors can be aware ofI to become active and indifferent with probability λ.

(2) Vertices who are aware of I and are as yet inactive willbecome active with probability α or be indifferent withprobability 1− α.

(3) Active vertices become quiet spontaneously with proba-bility β.

From the above rules, we find that indifference and quiet-ness are two final states of vertices. In other words, the processof information spreading is also the process of vertex statedecay with indifference and quietness as its final states. Thuswe can add another state “•” to uniformly describe all verticesknowing I. Since indifferent and quiet vertices both belongto this kind of vertices and cannot affect their neighbors, theybecome “•” absolutely (see Fig. 1). Note that the process ofinformation spreading or vertex state decay terminates whenthere is no active vertices in the network.

We have shown that each vertex in N changes its stateaccording to Fig. 1 and its neighbor’s states. This processis what IMCs [18] deal with. Therefore, keeping the abovespreading mechanisms in mind, we get7 the following set ofcoupled differential equations (non-linear dynamical system)that describe the dynamics of information spreading on a

7Derivation details are given in Appendix A.

SAI ZHANG et al.: DYNAMICS OF INFORMATION SPREADING IN ONLINE SOCIAL NETWORKS 3

mean-field level [16].

dik(t)

dt= −λkik(t)

∑k′

P (k′ | k)ak′(t), (1a)

dak(t)

dt= αλkik(t)

∑k′

P (k′ | k)ak′(t)− βak(t), (1b)

drk(t)

dt= (1− α)λkik(t)

∑k′

P (k′ | k)ak′(t), (1c)

dqk(t)

dt= βak(t), (1d)

where ik(t), ak(t), rk(t), qk(t) denote the fractions of verticeswith degree k in the states of ignorant, active, indifferent andquiet at time t, respectively, and integer k′ ranges from 0 tokc, which is the largest degree8 in network N . Obviously,we have ik(t) + ak(t) + rk(t) + qk(t) = 1. Note that we willclarify more definitions, e.g., vertex degree, degree distributionand P (k′ | k), in the next section. If we denote i(t), a(t),r(t), q(t) as the vertex fractions of four states, we also havei(t)+a(t)+r(t)+q(t) = 1. In addition, we assume that at t =0, the initial conditions of the system are i(0) = (n−1)/n ≈ 1,a(0) = 1/n ≈ 0, r(0) = 0 and q(0) = 0, where “≈” makessense if n is large enough.

Note that the defined network N is a kind of undirectedMarkovian random network [6], whose topology is completelydetermined by the degree distribution P (k) and the condi-tional probability P (k′ | k). In other words, we assume inapproximation that all higher-order correlation functions canbe obtained as random combinations of P (k) and P (k′ | k).

III. TERMINOLOGIES, NOTATIONS AND NEW DEFINITIONS

In this section, we list some common terminologies andnotations in network science, which are frequently used inthis paper. We also give the exact definitions of three impor-tant arguments that characterize the dynamics of informationspreading in networks.

A. Relevant Terminologies and Notations

The degree [24] of a vertex in an undirected network isdefined as the the number of edges attached to it. We denotethe largest degree in a network as kc and always use lower-case k to denote degrees. The degree distribution P (k) [24]describes the fraction of vertices with degree k. We denote theaverage degree as 〈k〉, which equals to

∑k kP (k). Similarly,

the average square degree 〈k2〉 equals to∑k k

2P (k). Notethat we always approximate the summation of sequence by itsintegral to simplify computations.

Degree distribution characterizes the network topology froma local perspective, and different networks with the samedegree distribution share similar topologies. In this paper,we consider the scale-free (SF) networks, the most commonnetworks in real world [24]. The degree distribution of an

8For brevity, we make kc approach infinity in certain cases. We will seethis approximation is reasonable as well as computationally helpful in thefollowing sections.

SF network follows a power law. Formally speaking, in SFnetworks, the degree distribution P (k) is defined as [24]

P (k) =

{Zk−γ if k ≥ m,0 otherwise,

(2)

where m is the minimum degree of the network and is set to 1in this paper, and the normalization constant Z approximatelyequals to (γ−1)mγ−1. In order to ensure finite average degreesin SF networks, we shall keep γ > 2. Particularly, the case ofγ = 3 is the degree distribution of networks generated by theBarabasi-Albert (BA) model [25].

A network is heterogeneous [13] if there are vertices whosedegrees are significantly larger than the average degree. Other-wise, the network is homogeneous [13]. We can use 〈k2〉/〈k〉to quantify the network heterogeneity approximately, i.e., anetwork usually becomes more heterogeneous if its 〈k2〉/〈k〉gets larger. Thus, heterogeneity characterizes the irregularityof network topology from the degree distribution perspective.It is not hard to find that in an SF network, as the exponent γgets smaller, the network is more heterogeneous (large degreeappears with higher probability).

Most realistic networks [24] exhibit non-trivial connectivitypatterns. For instance, most OSNs [26–28] present assortativ-ity, while the Web networks and the Internet [29] are alwaysdisassortative. These connectivity trends among vertices withdifferent degrees cannot be characterized by degree distri-butions. Actually, various connectivity patterns influence thenetwork topology drastically and result in different degreedistributions.

Degree distribution is often called the first-order correlation[10] in complex networks. Generally speaking, we define theconditional probabilities P (k′, k′′, ..., k(n) | k), the probabil-ities of vertices with degree k attaching simultaneously toother n vertices with corresponding degrees k′, k′′, ..., k(n),to characterize higher-order (the nth-order) correlations. Thenetwork is referred as uncorrelated [10] if there are no con-ditional probabilities in the network connectivity. Otherwise,the network is correlated [10]. It is known that in Markoviannetworks [6], which is the main concern in the followingsections, only the degree distribution P (k) and the degree-degree correlations P (k′ | k) are taken into account. If wedefine the joint probability distribution P (k, k′) [24] as theprobability of randomly selecting an edge that connects avertex with degree k and another vertex with degree k′, anddefine the excess degree distribution q(k) ,

∑k′ P (k, k′) [24]

that equals to kP (k)/〈k〉, we have

P (k, k′) = q(k)P (k′ | k). (3)

Therefore, the joint distribution P (k, k′) totally determinesthe topology of the Markovian network. In addition, we haveP (k, k′) = q(k)q(k′) in uncorrelated networks, and this leadsto P (k′ | k) = q(k′). These results are frequently used in thefollowing sections.

Another measure that characterizes the second-order cor-relations is the average nearest neighbors degree (ANND)〈knn〉 ,

∑k′ k′P (k′ | k) [30], i.e., the average neighbors

degree of vertices with degree k. In uncorrelated networks,we have 〈knn〉 = 〈k2〉/〈k〉 independent on k.

4 TECHNICAL REPORT

B. Three Arguments Characterizing Information Spreading

As t → ∞, the process of information spreading reachesits equilibrium when there is no active vertices in the net-work. What matters most is the condition under which theinformation I spreads out, and the range and the velocity ofthe spreading process. We now define three arguments thatdescribe these aspects.

Definition 1: The spreading prevalence P is defined asr∞ + q∞, where r∞ , limt→∞ r(t) and q∞ , limt→∞ q(t).

The spreading prevalence P characterizes the final spread-ing range, i.e., the fraction of vertices aware of the informationwhen the spreading process ends. It is obvious that 0 < P ≤ 1.

Consider the model parameter function ρ(α, β, γ) = αγ/β,which is denoted as ρ for short. In fact, ρ reflects theability and potential of information spreading. Concretely, ifvertices in network N get and spread the information I easily(correspond to large λ and large α, respectively), and at thesame time, I loses its popularity slowly (corresponds to smallβ), then I will spread in a wide range. Otherwise, I will betrapped in a narrow area.

Definition 2: If there is a real number ρc ≥ 0 satisfyingP > 0 when ρ > ρc, we refer ρc as the spreading threshold.

If a function f(t) evolves with a dominant term et/τ , it isclear that the growth time scale τ determines the evolutionvelocity of f , i.e., the smaller τ is, the faster f evolves. Inorder to describe the velocity of the spreading process, wedefine the spreading efficiency as follows.

Definition 3: The spreading efficiency E is defined as thereciprocal of the growth time scale τ in information spreading,i.e., E , 1/τ .

Note that in Definition 3 we treat the information spreadingprocess as the evolution of a(t). We will clarify this definitionwith the help of some analytic results derived in Section IV.

IV. INFORMATION SPREADING CHARACTERISTICS

In this section, we study the characteristics of informationspreading in uncorrelated and correlated heterogeneous net-works, based on the arguments defined in Section III. We focuson how the model parameters and the network topology, i.e.,the degree distributions and the degree-degree correlations,influence the information spreading process. The philosophyhere is to study naive networks first, add higher-order correla-tions gradually, and investigate how the correlations influencethe spreading process.

A. Uncorrelated Heterogeneous Networks

We begin our study with the uncorrelated heterogeneousnetworks, where we concern how the degree distributionsaffect the spreading dynamics. To clarify this, we focus onSF networks and analyze exponent γ’s function on spreading.We have the following theorems to characterize the dynamicsof information spreading.

Theorem 1: In uncorrelated heterogeneous networks, thespreading threshold ρc equals to 〈k〉/〈k2〉. In other words, theprevalence P > 0 if and only if ρ > 〈k〉/〈k2〉. In particular,

in SF networks, we have

ρc =

γ − 3

γ − 2if γ > 3,

0 if 2 < γ ≤ 3.(4)

Proof: See Appendix B.Remark 1: We have shown that 〈k2〉/〈k〉 can be regarded

as the measure of network heterogeneity. Hence the thresholdin uncorrelated heterogeneous networks is negatively relatedto the network heterogeneity. In other words, the more het-erogeneous the uncorrelated network is, the more easily theinformation spreads out. According to Theorem 1, when2 < γ ≤ 3, SF networks are heterogeneous enough to makethe spreading threshold disappear.

Given the explicit expression of degree distribution of SFnetworks, we can get the following theorem that characterizesthe spreading range.

Theorem 2: In uncorrelated SF networks, the spreadingprevalence P has the following expressions in different cases.

(1) 2 < γ < 3:P ∼ ρ1/(3−γ), (5)

(2) 3 < γ < 4:

P ∼(

1− ρcρ

)1/(γ−3)

, (6)

(3) γ > 4 and γ /∈ N:

P ∼ 1− ρcρ, (7)

where “∼” means “be approximately proportional to”. Whenγ ∈ N, the expression of P should be analyzed case by case.Particularly, in the case of γ = 3, we have

P ∼ e−1/ρ. (8)

Proof: See Appendix C.Remark 2: We find from Theorem 2 that the spreading

prevalence in SF networks is determined by both the model pa-rameters (i.e., ρ) and the network topology (i.e., the exponentγ). For example, in heterogeneous networks where 2 < γ < 3,the heterogeneity increases the spreading prevalence if ρ < 1,while decreases it otherwise. Therefore, the network topology,e.g., heterogeneity, influences the spreading process in compli-cated manners. We will discuss more details in Section IV-B.

Theorem 1 and Theorem 2 are about the stable state of thespreading process, whereas the following theorem describesthe temporal behavior of information spreading.

Theorem 3: In uncorrelated heterogeneous networks, thegrowth time scale τ of information spreading is negativelyrelated to the network heterogeneity, i.e., 〈k2〉/〈k〉, which isthe reciprocal of threshold ρc. More precisely, we have

τ =1/β

ρ/ρc − 1. (9)

Particularly, in SF networks, the efficiency E has the expres-sion

E =

αλ(γ − 2)− β(γ − 3)

γ − 3if γ > 3,

∞ if 2 < γ ≤ 3.(10)

SAI ZHANG et al.: DYNAMICS OF INFORMATION SPREADING IN ONLINE SOCIAL NETWORKS 5

Proof: See Appendix D.Remark 3: It is shown in Theorem 3 that an increasing

heterogeneity improves the efficiency of information spread-ing, i.e., it improves the spreading speed. For instance, theinfinite efficiency contributes to almost instantaneous rise ofspreading incidence in more heterogeneous SF networks where2 < γ ≤ 3.

B. Correlated Heterogeneous Networks

Previous empirical studies [26–28] have found that OSNsexhibit assortativity, i.e., vertices with large degrees tend toconnect to vertices with large degrees, while vertices withsmall degrees prefer to connect to vertices with small degrees.This phenomenon is not well explained by researchers, and itis conjectured for several reasons, e.g., preferential attachment[25] and proximity bias [31]. No matter what makes theassortativity in OSNs, it has a significant impact on theedge creation, network evolution and network topology. Thusit is necessary to study how these (positive) second-ordercorrelations affect the dynamics of information spreading. Weconduct our analysis of Markovian networks equipped withfirst-order and second-order correlations in this subsection.

We know that ik(t) = 1−ak(t)−rk(t)−qk(t). By omitting9

terms of O(a2), Eq. 1b can be written as

dak(t)

dt≈∑k′

Lkk′ak′(t), (11)

orda

dt= La, (12)

where the Jacobian matrix L = {Lkk′} is defined as

Lkk′ = −βδkk′ + αλkP (k′ | k), (13)

and δkk′ is the Kronecker delta symbol. The solutions ofEq. 12 imply that the behavior of ak(t) is given by thelinear combination of exponential functions with the form eλit,where λi is the eigenvalue of L.

By linear stability analysis [10] of the system (12), weconclude the following theorem that determines the spreadingthreshold of Markovian networks.

Theorem 4: The connectivity matrix C = {Ckk′} is de-fined as Ckk′ = kP (k′ | k). The spreading threshold of theMarkovian networks is

ρc =1

Λm, (14)

where Λm is the largest eigenvalue of C.Proof: See Appendix E.

Remark 4: Comparing to Theorem 1, Theorem 4 statesa more general result on spreading threshold in Markoviannetworks where the degree-degree correlations are considered.At the same time, Eq. 14 agrees with the result of Theorem 1in unstructured networks with no second-order correlations,where C has the unique eigenvalue Λm = 〈k2〉/〈k〉.

9We cannot perform this approximation all the time as ak(t)2, rk(t)ak(t)and qk(t)ak(t) are not always small enough to be omitted during thespreading process. But this approximation works well in this part of analysis.

In order to study the influence of positive degree-degreecorrelations on the spreading process precisely, we here definethe conditional probability P (k′ | k) satisfying [32]

P (k′ | k) = (1− θ)q(k′) + θδkk′ , (15)

where 0 ≤ θ < 1. Note that if θ = 0, there is no degree-degreecorrelations in the network. As θ increases, the network ownshigher positive correlations, i.e., stronger assortativity.

Before stating main results on the influence of degree-degreecorrelations on the spreading dynamics, we have the followingcorollary.

Corollary 1: The spreading threshold of correlated het-erogeneous networks, where degree-degree correlations aredefined as Eq. 15, is negatively related to degree-degreecorrelations, i.e.,

ρc ∼ 1

/(〈k2〉〈k〉

(1− θ)). (16)

Proof: The connection matrix C of SF networks can bewritten as C = {(1− θ)kq(k′) + θkδkk′}. Thus we have

det(C − ΛI) = f(Λ) ·kc∏k=2

(θk − Λ),

where

f(Λ) = (θ − Λ)

(1 + (1− θ)

kc∑k=1

kq(k)

θk − Λ

).

With regard to f(Λ), we find that for any θ and k =1, . . . , kc, there is null point Λk locating in (θk, θk + εk),where θk is the pole of f(Λ) and εk > 0. Denote Λm ∈(θkc, θkc + εkc) as the maximum null point of f(Λ), hence itis the largest eigenvalue of C. To approximate Λm, we have

1 + (1− θ)kc∑k=1

kq(k)

θk − Λm= 0,

which further gives

Λm − (1− θ)kc∑k=1

kq(k)

1− θkΛm

≈ Λm −〈k2〉〈k〉

(1− θ) = 0.

This implies Eq. 16.Remark 4a: Corollary 1 implies that higher positive degree-

degree correlations increase spreading threshold. In otherwords, correlations inhibit information spreading to some ex-tent. Note that when the network is uncorrelated (i.e., θ = 0),we also get Λm = 〈k2〉/〈k〉 in the proof of Corollary 1. Hencewe can treat Eq. 16 as the generalized result of Theorem 1 incorrelated networks.

Based on Eq. 15, the system of Eqs. 1a to 1d becomes

dik(t)

dt= −(1− θ)λkik(t)

∑k′

q(k′)ak′(t)

− θλkik(t)ak(t), (17a)dak(t)

dt= (1− θ)αλkik(t)

∑k′

q(k′)ak′(t)

+ (θαλkik(t)− β) ak(t), (17b)

6 TECHNICAL REPORT

drk(t)

dt= (1− θ)(1− α)λkik(t)

∑k′

q(k′)ak′(t)

+ θ(1− α)λkik(t)ak(t), (17c)dqk(t)

dt= βak(t), (17d)

In this case, the relations between the network topology andspreading prevalence are more complicated than those inuncorrelated networks. We have the following theorem.

Theorem 5: In correlated SF networks, the function of thedegree-degree correlations (i.e., θ) on the spreading prevalenceP is affected by both the model parameters (i.e., ρ) and theexponent γ of degree distributions.

Proof: See Appendix F.Remark 5: Several previous studies [11, 16] concluded

incomplete results about the impact of degree-degree correla-tions on the epidemic and rumor spreading. In [16], Nekoveeet al. experimentally found that these correlations’ influenceon the final fractions of vertices hearing a rumor dependsmuch on the rate of rumor diffusion. Theorem 5 gives thetheoretical explanation behind this phenomenon in the frame-work of our model. In fact, the influence of degree-degreecorrelations on information spreading varies a lot along withdifferent information spreading potentials and network degreedistributions. Note that we do not concern about the impactof the largest degree kc or the network size here. However, kcand the finite network size indeed affect the spreading processas well [8]. In Section VI, we will find numerically that largekc and network heterogeneity can balance out the torsion ofthe degree-degree correlations’ influence on spreading causedby model parameters.

We have shown that if L has n distinct eigenvalues, thesystem da/dt has eλ1tr1, . . . , eλntrn as its fundamentalsolutions, where ri is the eigenvector of L with respect to theeigenvalue λi where i = 1, . . . , n. Thus, a(t) is determinedmainly by the maximum eigenvalue λm, i.e.,

ak(t) ∼ eλmt.

We have proven part of the following theorem.Theorem 6: In correlated networks, the spreading effi-

ciency E is determined by the maximum eigenvalue of Ja-cobian matrix L. In SF networks, we have

E ∼ 〈k2〉〈k〉

(1− θ). (18)

Proof: We have proved the first part of the theorem.For the remaining part, by the deduction similarly to that inCorollary 1, the maximum eigenvalue λm of L equals approx-imately to αλ(1− θ)〈k2〉/〈k〉. According to Definition 3, weget Eq. 18.

Remark 6: First, in uncorrelated networks, L has the onlyeigenvalue αλ〈k2〉/〈k〉−β, thus the conclusion of Theorem 6agrees with Theorem 3. Second, the degree-degree correlationsalso inhibit the spreading process in the aspect of spreadingefficiency. At the same time, the largest degree kc that con-tributes to network heterogeneity also has great influence onthe growth time scale of the spreading process. When kc →∞,which usually results in the infinity of 〈k2〉/〈k〉, the growth

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

Time (number of iterations)

Num

ber o

f act

ive

verti

ces

per i

tera

tion

(a) Simulated information spreading based on static model

0 50 100 150 200 250 300 3500

20

40

60

Time (unit interval is 0.5h)

Shar

ing

times

pe

r uni

t int

erva

l

(b) A video sharing in Renren

Fig. 2. Comparison of simulated and realistic spreading dynamics. (a) Timeseries (blue curve) of Monte Carlo simulated information spreading based onthe model in Section II. To reduce rapid fluctuations in series, we apply theGaussian smoothing (red curve). We generate the underlying SF network of1,000 vertices according to the BA model. We set the model parameters asα = 0.5, λ = 0.5 and β = 0.5. (b) Time series of a video sharing in Renrenafter the Gaussian smoothing.

time scale disappears. This implies an irresistible spreadingin correlated heterogeneous networks. Actually, by boundingλm more precisely, we can get this conclusion without therestriction of specific degree-degree correlations. Given thePerron-Frobenius theorem, we have

λ2m ≥ min

k

1

k

∑k′

k′∑l

LklLlk′

= mink

∑l

∑k′

(α2λ2lP (l | k)k′P (k′ | l) + β2 k

′

kδklδlk′

− αλβ lk′

kδklP (k′ | l)− αλβk′δlk′P (l | k)

)= min

k

(∑l

(α2λ2lP (l | k)〈lnn〉(kc)

)− 2αλβ〈knn〉(kc)

+ β2

)≥ min

kβ2((ρ2〈knn〉min − 2ρ)〈knn〉(kc) + 1

),

where 〈knn〉min denotes the minimum ANND in the network.Boguna et al. [9, 10] have shown that the function 〈knn〉(kc)diverges as kc → ∞ in SF networks with 2 < γ ≤ 3. Thisleads to the divergence of λm and further the infinite E .

V. IMPROVED MODEL WITH TIME-VARYING PARAMETERS

In previous sections, we assume that model parameters α,λ and β are invariant during the spreading process. Thisassumption is imprecise considering realistic circumstancesand we can indeed take one step forward. In fact, there areonly three kinds of vertex states in the information spreadingprocess in OSNs, i.e., ignorant, active and indifferent. We

SAI ZHANG et al.: DYNAMICS OF INFORMATION SPREADING IN ONLINE SOCIAL NETWORKS 7

i

ra

1− λ(t)

(1− α(t))λ(t)α(t)λ(t)

Fig. 3. State transition diagram of improved model. i, a, r are short forignorant, active and indifferent, respectively.

add the quiet state derived from the active state to emphasizethe nonpersistence of the impact of active vertices on theirneighbors. It is unreasonable to assume that the probabilities ofcontacting and spreading the information are invariable fromthe beginning to the end. In realistic OSNs [21], probabilityλ that describes the likelihood of user being aware of theinformation is relevant to temporal patterns of user’s onlinebehaviors, e.g., λ varies over time in a day, or even overdays of a week. Moreover, as probability α represents user’sinterest in the information, it is also changing depending onthe fluctuation of information popularity.

Constant parameters fail to characterize temporal featuresin information spreading such as the dynamic patterns, i.e.,the number change of new coming active vertices over time.Figure 2 illustrates the dynamic patterns of information spread-ing in an SF network based on our former static model andof a video sharing in Renren. From Fig. 2b, we find that inRenren the dynamics of information spreading own significantpatterns, e.g., diurnal patterns, periodicity, sudden spikes andgradual decay, while the time series in Fig. 2a generated byour previous model seems to be random and has less notabletendencies. This distinct gap between the realistic exampleand the simulated result leads us to improve our model withtime-varying parameters.

In this section, we first describe our new model with time-varying parameters, and then study the impact of varyingparameters on information spreading.

A. Description of Improved Model

In our improved spreading model, where α , α(t) andλ , λ(t), we delete the state of quiet and the correspondingtransition probability β. In this case, the information spread-ing process comes to an end along with the vanish of theinformation popularity, i.e., limt→∞ α(t) = 0.

Figure 3 describes the process of vertex state decay. Notethat as α(t) → 0, no vertex would spread the informationand the process terminates. The system Eqs. 1a to 1d can beimproved as

dik(t)

dt= −λ(t)kik(t)

∑k′

P (k′ | k)ak′(t), (19a)

dak(t)

dt= α(t)λ(t)kik(t)

∑k′

P (k′ | k)ak′(t), (19b)

drk(t)

dt= (1− α(t))λ(t)kik(t)

∑k′

P (k′ | k)ak′(t),(19c)

where 0 ≤ α(t) ≤ 1, 0 ≤ λ(t) ≤ 1 and α∞ = 0.

B. Impacts of Time-Varying Parameters on InformationSpreading

The active vertex is the key to the spreading process.Hence we focus on how time-varying parameters influencethe change of the number of active vertices. The followingtheorem characterizes the spreading dynamics.

Theorem 7: In correlated heterogeneous networks, thechanging rate of the number of active vertices is

da(t)

dt= α(t)λ(t) exp

(∫ t

0

α(x)λ(x)dx

)Ca(0), (20)

where a(0) is the initial fraction of active vertices in thenetwork.

Proof: Similarly to Eq. 12, we can write Eq. 19b as

da(t)

dt= L(t)a(t), (21)

where the Jacobian matrix L(t) = {Lkk′(t)} is defined by

Lkk′(t) = α(t)λ(t)kP (k′ | k).

Note that L(t) = α(t)λ(t)C, where C is the connectivitymatrix defined in Theorem 4. Thus, L(t) is commutative, i.e.,L(t1)L(t2) = L(t2)L(t1) for all t1, t2 ∈ (0,∞). Define

L(t) ,∫ t

0

L(x)dx,

and we have L(t1)L(t2) = L(t2)L(t1). Based on these facts,we can easily get

dLn(t)

dt= nL

′(t)L

n−1(t).

We claim that a(t) = exp L(t) · a(0) is the solution ofEq. 21. To prove this, take the derivative of a(t), we have

da(t)

dt=

d

dt

( ∞∑n=0

Ln(t)

n!· a(0)

)=

∞∑n=0

L′(t)

Ln(t)

n!· a(0)

= L′(t) exp L(t) · a(0) = L(t)a(t),

which is exactly Eq. 21.Since C is a real symmetric matrix, it can be diagonalized

by orthogonal matrix. Furthermore, C is nonsingular accord-ing to the proof of Corollary 1. Thus it is similar to theidentity matrix, i.e., there exists orthogonal matrix P suchthat P−1CP = I . Therefore, we can rewrite the solution ofEq. 21 as

a(t) = exp L(t) · a(0) = exp

(∫ t

0

L(x)dx

)a(0)

= exp

(∫ t

0

α(x)λ(x)dx ·C)a(0)

= exp

(∫ t

0

α(x)λ(x)dx · PIP−1

)a(0)

8 TECHNICAL REPORT

=

∞∑n=0

1

n!

(P

(∫ t

0

α(x)λ(x)dx · I)P−1

)na(0)

= P exp

(∫ t

0

α(x)λ(x)dx · I)P−1a(0)

= exp

(∫ t

0

α(x)λ(x)dx

)PIP−1a(0)

= exp

(∫ t

0

α(x)λ(x)dx

)a(0). (22)

Therefore, together with Eq. 21, we finally get Eq. 20.Remark 7: We here concern about the changing rate of

the number of active vertices, i.e., da(t)/dt, where a(t) =∑k P (k)ak(t). According to Theorem 7, we have

da(t)

dt=

d

dt

∑k

P (k)ak(t) =∑k

P (k)dak(t)

dt

=∑k

(P (k)α(t)λ(t) exp

(∫ t

0

α(x)λ(x)dx

)∑k′

kP (k′ | k)ak′(0)

)= α(t)λ(t) exp

(∫ t

0

α(x)λ(x)dx

)C, (23)

where C = 〈k〉∑k

∑k′ P (k, k′)ak′(0), which is given by

the initial state and the underlying network topology of in-formation spreading. Eq. 23 implies that the changing rate ofthe number of active vertices, or the spreading velocity, isa function of time-varying parameters α(t) and λ(t) in theimproved model.

C. Explicit Expressions of α(t) and λ(t)

The previous section studies the impact of time-varyingparameters on information spreading in general. In this section,we give explicit mathematical expressions of α(t) and λ(t)based on empirical studies.

Previous studies [5, 27, 33, 34] have shown that the pop-ularity tendency of different kinds of information in OSNsfollows the long-tailed or power-law pattern. Probability α(t)quantifies the popularity of one particular information in thefollowing way,

α(t) , Pr{E1 happens at time t}

where E1 denotes the event that a user aware of the infor-mation becomes active, i.e., the user spreads the information.Note that E1 happens at time ∞ means that the user becomesindifferent, i.e., the user is aware of the information butdoes not spread it. Thus α(t) satisfies 0 ≤ α(t) ≤ 1 and∫∞

0α(t)dt = 1, and is a probability distribution. To model

realistic popularity tendency, we let α(t) follow the Gammadistribution Γ(p, η) whose density function satisfying

p(x; p, η) =xp−1e−x/η

ηpΓ(p), (24)

where x, p, η > 0 and Γ(p) is the Gamma function. Weapproximate α(t) by the value of its density function at t andhave α(t) = p(t; p, η).

Several studies [21, 34, 35] have found that in the Weband OSNs, user’s online behaviors are of daily cycle and thediurnal pattern has a gentle peak. Assume that immediatelyafter they log in, users in SNSes can get all user generatedcontents (UGCs) that their neighbors have published andshared. Thus λ(t) is determined by the temporal pattern ofuser’s log-in behavior. Similarly, we regard λ(t) as

λ(t) , Pr{E2 happens at time t},where E2 denotes the event that a user logs in, or is awareof the information spread by his/her active neighbors. Wealso assume that user logs in every day and denote Cpas the measure of one day period10. Hence λ(t) satisfies0 ≤ λ(t) ≤ 1,

∫ Cp0

λ(t)dt = 1 and λ(t) = λ(t+Cp), and is aperiodic distribution. We make λ(t) follow a popular circulardistribution, the von Mises distribution, or the circular normal[36], whose density distribution is

p(x; z, ϑ) =1

2πI0(z)exp (z cos(x− ϑ)) , (25)

where ϑ corresponds to the mean of the distribution, z isknown as the concentration parameter and

I0(z) =1

2π

∫ 2π

0

exp(z cosϑ)dϑ.

Note that the von Mises distribution is of period 2π. To modelλ(t) with period Cp, we perform variable transformation toEq. 25 and get

λ(t) =1

CpI0(z)exp

(z cos(

2π

Cpt− ϑ)

). (26)

In general, user’s activity pattern represented by λ(t) isfixed on average, and the popularity tendency quantified byα(t) varies to different information. Given the constant inEq. 23, there are six unknown parameters to be determined,i.e., Θ = {p, η, z, ϑ, Cp, C}. According to Eq. 23, given theinitial conditions, we find that the shape of the time seriescurve da/dt is controlled by α(t)λ(t) exp

(∫ t0α(x)λ(x)dx

).

Since this equation is intractable, we adopt the LM algorithmto learn Θ in Section VI.

VI. EXPERIMENTS

In this section, we first present several numerical solutionson the system of Eqs. 1a to 1d based on the model proposed inSection II. These numerical results verify analytic results givenin Section IV. Furthermore, for the improved time-varyingmodel, we use LM to learn key parameters derived in SectionV, based on a data set of video sharing in Renren. We find thatthe synthetic curves generated by the time-varying model afterlearning match the realistic time series precisely. This verifiesthe accuracy of our model, as well as gives the underlyingexplanation of the dynamic patterns of information spreadingin OSNs. At last, we use Eq. 23 to predict the remaining part ofvideo sharing time series given the initial spreading dynamics.Compared to traditional time series analysis approach, ourmodel works much better and succeeds in predicting futuredynamics in the long range.

10Since the time measure in the model does not correspond to real time,we here set the time period as another parameter.

SAI ZHANG et al.: DYNAMICS OF INFORMATION SPREADING IN ONLINE SOCIAL NETWORKS 9

0.8

1

1.2

1.4

1.6x 10-3

θ (step size is 0.01)

1/tim

e

0 0.25 0.5 0.75 10

0.5

1

α (step size is 0.01)

Pre

vale

nce

(a)

0 0.25 0.5 0.75 10

0.5

1

α (step size is 0.01)

Pre

vale

nce

(b)

0 0.25 0.5 0.75 10

0.5

1

α (step size is 0.01)

Pre

vale

nce

(c)

0 0.25 0.5 0.75 10.5

1

1.5x 10-3

(d)θ (step size is 0.01)

1/tim

e

γ=2.5γ=5

θ=0.01θ=0.5θ=0.99

θ=0.01θ=0.5θ=0.99

θ=0.01θ=0.5θ=0.99

threshold

cross

cross

linear

non-linear

Fig. 4. Spreading dynamics in correlated SF networks. (a) Prevalence Pin the SF network with γ = 2.5 and kc = 473 is plotted as a function ofα. We change α for every 0.01. The results are shown for three values ofcorrelation parameter θ. (b) Prevalence P in the SF network with γ = 5 isplotted as a function of α. We find clear turning points and crosses in curves.(c) Prevalence P in the SF network with γ = 2.5 but kc = 22 is plotted as afunction of α. The crosses in curves come back in the network with small kcbut the same exponent γ as that in (a). (d) Reciprocal of numerical iterationnumber of system Eqs. 1a to 1d, regarded as the spreading efficiency, is shownfor different correlation parameter θ. Two degree distributions of γ = 2.5 andγ = 5 are considered. We find approximate negative linear relation betweenefficiency and degree-degree correlations.

A. Numerical Results of Model with Constant Parameters

In this subsection, we numerically study the dynamics ofinformation spreading in correlated SF networks based onthe model with constant parameters. We present numericalsolutions using standard finite difference scheme on two SFnetworks of γ = 2.5 (heterogeneous) and γ = 5 (homoge-neous) with kc = 473 and kc = 11, respectively. The power-law degree sequences of the exponent-varying networks aregenerated by the algorithm proposed in [37]. The sizes of thesenetworks are both n = 104. We set the model parametersas λ = 1 and β = 0.3, and increase the value of α from0.01 to 1 with the step size of 0.01. We investigate how thespreading prevalence P changes along with the varying α.Note that values of P are averaged over 100 random equivalentsolutions, i.e., we randomly choose one initial active vertexand leave others ignorant each time, and at the same timekeep all the model and network parameters unchanged.

In Fig. 4a, the results for the SF network with γ = 2.5are shown. As λ and β are fixed, the spreading threshold isonly related to α. In this case, we observe the absence ofthe threshold, which agrees with the conclusion in Remark 4.Whereas in the case of less heterogeneous network withγ = 5, as shown in Fig. 4b, we observe explicit thresholds ofinformation spreading. Moreover, the threshold in the networkwith higher positive degree-degree correlations (larger θ) islarger. This verifies the conclusion of Corollary 1 and impliesthat positive correlations inhibit the spreading process to someextent.

As for the spreading prevalence, Theorem 5 shows thatthe impact of correlations on prevalence relates to modelparameters and the degree distribution. We find that the crosses

of curves plotted in Fig. 4b correspond to the conclusion ofTheorem 5, i.e., positive correlations increase the spreadingprevalence P in some values of α while decrease P in othervalues. Theorem 5 also holds in the SF network of γ = 2.5where kc is set to be small enough11, e.g., set kc = 22 asthat in Fig. 4c. At the same time, in more heterogeneousnetworks (e.g., kc = 473 in Fig. 4a), the positive correlationsinhibit prevalence independent with model parameters. Thesecomplicated relations among the spreading prevalence, modelparameters and the network topology (e.g., degree distribution,the largest degree and degree-degree correlations) are partlydescribed by Theorem 5. These results also confirm the essen-tial influence of the largest degree on information spreading.

In Fig. 4d, we set model parameters as α = 0.7, λ = 1and β = 0.3, and increase the value of θ from 0 to 0.99with step size of 0.01. We study how correlations influencethe spreading efficiency. Note that in Fig. 4d, we take theiteration number of solving system Eqs. 1a to 1d before thespreading process terminates (i.e., there are no active verticesin the network) as the spreading time, whose reciprocal isregarded as the spreading efficiency. The results in Fig. 4d arealso averaged over 100 equivalent random solutions. We findapproximately negative linear relations when θ < 0.9 betweencorrelations and the efficiency in SF networks of both γ = 2.5and γ = 5. Moreover, with the same θ, information spreadsfaster in more heterogeneous network (smaller γ and larger〈k2〉/〈k〉). These numerical results conform to the conclusionin Theorem 6. Note that the non-linear tails of two curvesin Fig. 4d result from approximation errors in the proof ofTheorem 6 as θ becomes large.

The relations between the spreading process and the un-derlying network topology (heterogeneity and correlations)are complicated according to both our theoretical analysis inprevious sections and numerical studies in this subsection.However, we find that in general, the network heterogeneityimproves information spreading in OSNs whereas positivedegree-degree correlations inhibit it.

B. Learning Dynamic Patterns of Video Sharing in Renren

From Fig. 2, we have seen that the model with constantparameters cannot characterize dynamic patterns of infor-mation spreading in networks subtly. Thus we propose animproved model with time-varying parameters, which canreflex changes in the information popularity and user behavior.This subsection provides detailed experiments on how well thetime-varying model characterizes the information spreadingprocess in realistic OSNs.

Our video sharing data set is provided by Renren, one of themost popular SNSes in China. When a user shares a uniformresource locator (URL) of a video from an outside video-sharing site (VSS) to Renren, it becomes a seed and wouldspread on the network through being re-shared by neighbors ofthe initial sharing user and neighbors of neighbors of the initialuser and so on. This process terminates once no one re-shares

11Although generally speaking, small exponents of SF networks lead tothe existence of large degree vertices with high probabilities, they are notnegatively related in a deterministic manner.

10 TECHNICAL REPORT

the URL any more. We find that this is exactly the processdescribed in Section II where our model comes from. Notethat all kinds of information (e.g., messages, blogs, photos) inRenren spread in the network in the same manner.

The data set consists of over 7.5 millions logs on videosharing for a week from September 10th to 16th, 2012. Each logrecords sharing time, user identity numbers and video URLs.And the whole logs contain 335,283 video URLs of whichthe largest sharing number is 154,955. Note that the numberof videos shared in this period should be smaller than thenumber of URLs, as the same video may share different URLs.Limited by the data set, we consider each URL as a uniquevideo in this subsection.

To preprocess the data set, we first select 1000 top sharingURLs with at least 861 shares. This process is reasonablesince popular URLs spread in a larger scale than unpopularones, and thus they can tell us more about the features ofinformation spreading in OSNs. We then construct time seriesthat record sharing number every thirty minutes based on theselected logs. These time series finally correspond to curvescharacterized by Eq. 23 and are objects we decide to model.The time-varying model is verified if these two curves fitwell. In any case, we prefer somewhat smooth curves thanfluctuated ones as we concern more on general shapes of timeseries curves, which reflect the characteristics of spreadingdynamics in essence. However, realistic time series oscillateintensely due to countless uncertainties. To reduce fluctuationsand eliminate noises in curve fitting, we apply the Gaussiansmoothing on each time series .

Up to now, the experimental verification becomes the prob-lem of curve fitting between preprocessed time series of videosharing and the curves determined by Eq. 23. To test thedescription ability of Eq. 23, we need to perform curve fittingon different time series. It is a waste of time to carry it outon each time series as many examples share common shapes.This reminds us to first cluster time series and then fit Eq. 23to the centroid of each cluster. We here adopt the K-SpectralCentroid (K-SC) clustering algorithm12 proposed by Yang andLeskovec [22], which focuses on pure curve patterns invariantto scaling and translation. In K-SC clustering, the distanced(x,y) between time series x and y is defined as [38]

d(x,y) = minν,h

‖ x− νy(h) ‖‖ x ‖

, (27)

where y(h) is the time series after shifting y by h time unitsand ‖ · ‖ is the l2 norm.

Similar to K-means clustering algorithm, K-SC clusteringneeds to set the number of clusters manually. To obtainthe optimal number of clusters, we measure the quality ofclustering in different cluster numbers based on the AverageSilhouette [39] and the Hartigan’s Index [40]. Note that thedistance measures of time series in these indices are computedby Eq. 27. We find from Fig. 5 that the two measures onclustering quality in different cluster numbers do not agreewith each other since they measure the clustering quality in

12Although realistic time series exhibit burstiness and periodicity, whichinduce many noises in clustering, here we just perform clustering on thewhole time series roughly to prevent from selecting by hand.

2 4 6 8 100.04

0.06

0.08

0.1

0.12

0.14

Number of clusters

Aver

age

Silh

ouet

te

(a) Average Silhouette

2 4 6 8 10-200

0

200

400

600

Number of clusters

Har

tigan

's In

dex

(b) Hartigans Index

Fig. 5. Clustering quality versus the number of clusters. (a) The averageSilhouette measures the tightness of grouped data in the cluster and theseparability among clusters in compromise. (b) The Hartigan’s Index is arelative measure of the square error decrease as the number of clustersincreases.

different perspectives and K-SC cannot satisfy both. Hartigan[40] suggested we should choose the smallest number whoseHartigan’s Index is smaller than a number (we set 200 here) asthe best cluster number. At the same time, the number with thelargest average Silhouette is the ideal cluster number. Hencewe conclude that three is the most reasonable cluster number.The centroids of clusters C1, C2 and C3 generated after thealgorithm converges are presented in Fig. 6. Note that thecentroids are not realistic time series in the data set. They arecomputed by the K-SC algorithm and represent the generalshapes of time series curves in each cluster.

We find from Fig. 6 that three centroids express obviouspatterns of time series. Time series in cluster C1 own onespiky peak and a long tail which contains several gentle peaks.They may come from the videos that gain user’s attentionsfor a moment and then lose their attractions rapidly. As alimitation, the data set cannot record the exact whole processof each video sharing, thus some of the time series in C1 mayalso represent the ending process of video sharing. Centroid ofC2 is similar to the curve in Fig. 2b, which is the typical timeseries of video sharing. Periodic peaks and gradual decay aregeneral characteristics of dynamic patterns of video sharing inRenren. Also limited by the data set, a large number of timeseries are truncated before terminating, which results in thecluster C3. Time series in C3 could be classified into C2 ina longer period of time. Therefore, we need to fit Eq. 23 totime series of C1 and C2 to test the improved model.

We apply the LM algorithm [23], a standard algorithm tosolve non-linear least squares problems, to do curve fitting(i.e., to learn the best parameters in Eq. 23 based on realistictime series). Because centroids are not realistic time series,we first select representative time series s1 and s2 of clusterC1 and C2 by minimizing the distances defined by Eq. 27,respectively, i.e.,

s1 = arg mins∈C1

d(s, c1) and s2 = arg mins∈C2

d(s, c2),

where c1 and c2 are centroids of C1 and C2, respectively.Note that s1 and s2 are recovered from Gaussian smoothingfor fairness. The results of curve fitting on s1 and s2 arepresented in Tab. I and Fig. 7.

As there is a great difference in time measures between themodel and realistic scenarios, we should first balance two timemeasures. In fact, the spreading process in the model is much

SAI ZHANG et al.: DYNAMICS OF INFORMATION SPREADING IN ONLINE SOCIAL NETWORKS 11

0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

Time (unit interval is 0.5h)

Sha

ring

times

per

uni

t

spiky peak

long tail

(a) Centroid of C1

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

0.2

Time (unit interval is 0.5h)

Sha

ring

times

per

uni

t

periodic peaks

decaying

(b) Centroid of C2

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

Time (unit interval is 0.5h)

Sha

ring

times

per

uni

t

ascending

(c) Centroid of C3

Fig. 6. Cluster centroids generated by the K-SC algorithm. Centroids are not realistic time series in the data set but are generated by K-SC.

TABLE IPARAMETER VALUES LEARNED BY THE LM ALGORITHM

Θ p η z ϑ Cp C

s1 0.4677 10.0662 0.8443 1.4631 0.0395 0.1586s2 0.5157 11.5924 -0.9050 1.3159 0.0493 0.2382

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0

50

100

-10Time (after scaling)

s(t)

s1sfitsfit+1.96σssfit-1.96σs

(a) Curve fitting of time series s1 in C1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

5

10

15

20

Time (after scaling)

s(t)

s2sfitsfit+1.96σssfit-1.96σs

(b) Curve fitting of time series s2 in C2

Fig. 7. Curve fitting on representative time series of C1 and C2. The scalebetween real time and model time is adjusted to 500:1 (in hour). We alsoplot the 95% confidence interval (red and purple curves) of the fit, whereσs is the asymptotic standard parameter error that measures how unexplainedvariability in the data propagates to variability in the parameters.

faster than that in real world. In our experiments, we set thescale between the real time and model time as 500:1 (in hour),which means that one step in model time equals to 500 hoursin real world. We find from Fig. 7 that Eq. 23 derived fromthe time-varying model fits the time series of video sharing inRenren very well. Specifically, the model can characterize twomajor features of information spreading in OSNs, i.e., gradualdecay and periodic spiky peaks, via two parameters α(t) andλ(t) that depict information popularity and user behavior,respectively. Note that limited by somewhat coarseness of thevon Mises distribution, Eq. 23 cannot model more delicatefeatures in time series, such as two local maximum alwaysappear in one big peak (see Fig. 6b and Fig. 6c). We believe

0 50 100 150 200 250 300 350

0

5

10

15

20

Time (unit interval is 0.5h)

Shar

ing

times

per

uni

t

dataAR(6)AR(39)fittingpredicted

Fig. 8. Comparison of prediction performance of AR and the improvedmodel on time series s2. The first 1/3 part (about 56 hours) of s2 is given totrain models. Prediction performances of the remaining 2/3 (about 112 hours)are shown. Note that there are totally 6 parameters in our model. For fairness,we set the order (i.e., the number of regression coefficients) of AR as 6 aswell. We also increase the order to 39 for a better prediction performance.

that if λ(t) is defined as other periodic distributions13 withtwo or more local maximums, the time-varying model cancharacterize more detailed features of information spreadingin OSNs.

C. Predicting Temporal Dynamics of Video Sharing in Renren

Other than characterizing dynamic patterns of video sharingin Renren, our improved model with time-varying parameterscan be used to carry out more realistic task —- to predictfuture temporal dynamics of time series whose head part isgiven. Here is our idea: given the first part of time series, welearn parameters in Eq. 23 using the LM algorithm the same aswhat we do in Section VI-B. Then Eq. 23 determines how thetime series evolve in the future and we can use this equation topredict the last part of time series. We take the representativetime series s2 as an example. We first truncate the first 1/3part of s2 and learn Θ by the LM curve fitting. This givesp = 0.8822, η = 0.2461, z = −0.7737, ϑ = 1.5292,Cp = 0.0498, C = 0.0586. Then we use Eq. 23 equippedwith these parameters to generate time series of the remaining2/3 part. The traditional time series analysis approach, theAutoregressive (AR) model [41], is used to compare with ourmethod. Similarly, we estimate parameters of AR based on thefirst 1/3 of time series and predict the remaining part with thelearned AR. Detailed results are shown in Fig. 8.

From Fig. 8, we find that our model is very good atpredicting time series in a long range (see the blue curve inFig. 8). While AR with the same number of parameters fails to

13However, to our knowledge, there are few periodic distributions of thiskind.

12 TECHNICAL REPORT

predict future dynamics and degrades to zero immediately (seethe purple curve in Fig. 8). In fact, if we increase the order ofAR to 39, more than 6 times of the parameter number of ourmodel, AR can characterize future dynamics as well. But theperformance is also not as good as ours. The relative errors [4],which are defined as

√∑t(s(t)− s(t))2

/√∑t s(t)

2 wheres(t) is the predicted value, are 0.9828, 0.7156, 0.7046 forAR(6), AR(39) and our model, respectively.

VII. RELATED WORK

Representative studies on epidemic spreading [6–13] laythe foundations for studying spreading process in complexnetworks. Classic epidemic models, such as susceptible-infected-susceptible (SIS) and susceptible-infected-removed(SIR) models, have been studied extensively. Similar to epi-demic spreading, rumor diffusion [14–17] can be investigatedin the same principle. The main difference between epidemicand rumor spreading is that vertices do not always act activelyor react to received messages in rumor diffusion. This leadsto related modifications of rumor models [17].

Many other works model the information spreading processin OSNs in different assumptions. Two standard models ininnovation diffusion [42], i.e., Independent Cascades (IC) andLinear Threshold (LT), have been widely employed to modelinformation spreading in OSNs. Galba et al. [2] proposed amodel based on the LT model to predict information cascadesin Twitter14. Similarly, Guille et al. [3] proposed an IC-based model with time-varying parameters to predict temporaldynamics of information spreading in Twitter. Also to predicttemporal dynamics of information spreading, Yang et al. [4]proposed the so-called Linear Influence Model (LIM) igno-rant with the underlying network topology. As LIM mainlyfocuses on the temporal dynamics of spreading, the influenceof vertices is assumed to be the only factor impacting thespreading process. In addition, Leskovec et al. [5] used SIS tomodel blog citing in the blogosphere. All these models eithercharacterize information spreading with finer granularity, ortreat the spreading process in different perspectives. However,some of them are too naive to recover realistic temporaldynamics, let alone model dynamic patterns explicitly. Someothers do not take network topology into consideration, whichindeed highly influences information spreading in OSNs. Evenif some do this, they cannot give definite analytic resultsto show how the network topology influences the spreadingprocess. To understand more related work on informationspreading in OSNs, readers can refer to [43].

VIII. CONCLUSION AND FUTURE WORK

In this paper, we first propose a probabilistic spreadingmodel based on IMCs, which features realistic process ofinformation spreading in OSNs. These IMCs also take theunderlying network topology (e.g., degree distributions anddegree-degree correlations) into account. Then we derive adynamical system from the original IMCs based on themean-field principle. With these coupled differential equations

14Twitter. http://www.twitter.com.

(Eqs. 1a to 1d), we analyze the impact of network topology oninformation spreading in OSNs. Our theoretical and numericalresults show that the network heterogeneity promotes thespreading process, while the positive degree-degree correla-tions inhibit information spreading in general.

To characterize the temporal dynamics of informationspreading in OSNs, we further propose an improved modelwith time-varying parameters that depict user behavior evolu-tion and information popularity. We derive an explicit function(Eq. 23) to model dynamic patterns of the spreading process.Our experiments show that by learning relevant parameters inthe function using the curve fitting algorithm, our improvedmodel can reproduce temporal dynamics of video sharing inRenren precisely. Given historical measurements, the improvedmodel is also able to accurately predict future spreadingdynamics in the long range.

Several questions remain to be answered. In the aspectof spreading strategy, we identify vertex influence with eachother at present. While in reality the influence of verticesvaries a lot and is unnecessarily related to vertex degree[44]. How to differentiate those influential vertices in theprobabilistic model needs further study. In network topology,we currently only consider two orders of correlations in thenetwork. Although these correlations have been found inrealistic OSNs widely [24], they are not the only factorsinfluencing information spreading. Some other topologicalfeatures, e.g., the largest degree [8], k-core [44], have beenfound to impact the spreading process largely. The modelneeds to be adjusted to express these influences. In modelparameters, the spreading rate α is not only related to time butalso the influence of vertices as stated above. Moreover, it isrecommended [45] to consider indirect links when inspectingambient influences on one vertex. In this case, α should becalculated by both time and pairwise distances. Note that allthe above model improvements should be investigated basedon empirical studies.

APPENDIX AFROM PROBABILISTIC MODEL TO DYNAMICAL SYSTEM

In this section, we derive the dynamical system of Eqs. 1a to1d from the probabilistic model described in Section II basedon the mean-field principle.

Consider a vertex nj that is ignorant at time t. We denotepjii as the probability that nj stays ignorant in the time interval[t, t+ ∆t]. Thus we have pj

ii= pjia+pjir = 1−pjii, where pj

ii,

pjia, pjir are probabilities of nj changing its state, becomingactive, and becoming indifferent, respectively. Let g = g(t)denotes the number of active vertices among nj’s neighborsat time t, it follows that

pjii = (1−∆tλ)g.

Assume that the degree of nj is k, and g can be consideredas a random variable that has the following binomial distribu-tion,

Π(g, t) =

(k

g

)ψ(k, t)g (1− ψ(k, t))

k−g,

SAI ZHANG et al.: DYNAMICS OF INFORMATION SPREADING IN ONLINE SOCIAL NETWORKS 13

where ψ(k, t) is the probability at time t that an edge connectsan ignorant vertex of degree k with an active vertex. Thusψ(k, t) can be written as

ψ(k, t) =∑k′

P (k′ | k)P (ak′ | ik) ≈∑k′

P (k′ | k)ak′(t),

where we approximate P (ak′ | ik) with ak′(t) by ignoring thecorrelations between neighboring vertices in different states,

The transition probability pii(k, t) averaged over all possiblevalues of g is given by

pii(k, t) =

k∑g=0

(k

g

)(1−∆tλ)gψ(k, t)g(1− ψ(k, t))k−g

=

(1−∆tλ

∑k′

P (k′ | k)ak′(t)

)k. (A.1)

Based on Eq. A.1 we get

pii(k, t) = 1− pii(k, t), (A.2)pia(k, t) = α (1− pii(k, t)) , (A.3)pir(k, t) = (1− α) (1− pii(k, t)) . (A.4)

From Fig. 1, we have

paq(k, t) = ∆tβ, (A.5)paa(k, t) = 1−∆tβ. (A.6)

Denote Ik(t), Ak(t), Rk(t), Qk(t) as the expected pop-ulations of vertices with degree k that are ignorant, active,indifferent and quiet at time t, respectively. The event that anignorant vertex with degree k becomes active during [t, t+∆t]is a Bernoulli random variable with probability 1 − pii(k, t)of success. Since the sum of Bernoulli variables follows thebinomial distribution with expectation Ik(t)(1− pii(k, t)), thechange of the expected population of ignorant vertices withdegree k is

Ik(t + ∆t)− Ik(t)

= −Ik(t)

1−

(1−∆tλ

∑k′

P (k′ | k)ak′(t)

)k .

(A.7)

Similarly, we can get the change of populations of active,indifferent and quiet vertices as follows.

Ak(t+ ∆t)−Ak(t)

= α Ik(t)

1−

(1−∆tλ

∑k′

P (k′ | k)ak′(t)

)k− Ak(t)∆tβ. (A.8)

Rk(t+ ∆t)−Rk(t)

= (1−α)Ik(t)

1−

(1−∆tλ

∑k′

P (k′ | k)ak′(t)

)k .

(A.9)Qk(t+ ∆t)−Qk(t) = Ak(t)∆tβ. (A.10)

For Eqs. A.7 to A.10, we let ∆t → 0, omit the higherorder infinitesimal and divide each equation by correspondingNk (the number of vertices with degree k), then finally getEqs. 1a to 1d.

APPENDIX BPROOF OF THEOREM 1

In uncorrelated heterogeneous networks, the conditionalprobability P (k′ | k) satisfies

P (k′ | k) = q(k′) =k′P (k′)

〈k〉.

Thus the system of Eqs. 1a to 1d degrades into

dik(t)

dt= −λkik(t)

∑k′

q(k′)ak′(t), (B.1)

dak(t)

dt= αλkik(t)

∑k′

q(k′)ak′(t)− βak(t), (B.2)

drk(t)

dt= (1− α)λkik(t)

∑k′

q(k′)ak′(t), (B.3)

dqk(t)

dt= βak(t). (B.4)

By integrating Eq. B.1, we get

ik(t) = ik(0)e−λkφ(t), (B.5)

where ik(0) ≈ 1 and

φ(t) =∑k

q(k)

∫ t

0

ak(x)dx =

∫ t

0

〈〈ak(x)〉〉dx.

Note that we define 〈〈?〉〉 ,∑k q(k)?.

By multiplying Eq. B.2 with q(k), summing over k andintegrating over t, we get

dφ(t)

dt= 〈〈ak(t)〉〉 = α− α〈〈e−λkφ(t)〉〉 − βφ(t). (B.6)

Let t→∞, we have dφ/dt→ 0, and Eq. B.6 becomes

φ∞ =α

β− α

β

∑k

q(k)e−λkφ∞ , (B.7)

where φ∞ = limt→∞ φ(t).It is obvious that φ∞ = 0 is a trivial solution of Eq. B.7.

The non-zero solution exists if only if the condition

d

dφ∞

(α

β− α

β

∑k

q(k)e−λkφ∞

)∣∣∣∣∣φ∞=0

> 1

is satisfied, i.e.,αλ

β>〈k〉〈k2〉

. (B.8)

Thus we get the threshold ρc = 〈k〉/〈k2〉 in uncorrelatedheterogeneous networks.

In SF networks, we can calculate 〈k〉 and 〈k2〉 as follows.

〈k〉 =

∞∑k=1

kP (k) = Z

∞∑k=1

k−γ+1

≈ Z∫ ∞

1

k−γ+1dk =γ − 1

2− γk−γ+2

∣∣∣∣∞1

, (B.9)

14 TECHNICAL REPORT

〈k2〉 =

∞∑k=1

k2P (k) = Z

∞∑k=1

k−γ+2

≈ Z∫ ∞

1

k−γ+2dk =γ − 1

3− γk−γ+3

∣∣∣∣∞1

. (B.10)

To get reasonable result, the networks we consider shouldhave finite average degree. Thus according to Eq. B.9, wehave γ > 2. In addition, 〈k2〉 < ∞ if γ > 3, and 〈k2〉 = ∞otherwise. Hence the spreading threshold of uncorrelated SFnetworks is given by Eq. 4.

APPENDIX CPROOF OF THEOREM 2

We consider uncorrelated SF networks in this theorem.Denote the spreading prevalence of each degree k as Pk =rk(∞) + qk(∞), and we have P =

∑k P (k)Pk. Given

Eq. B.5, we have

P =∑k

Zk−γ(1− e−λkφ∞

)≈ 1− (γ − 1)

∫ ∞1

k−γe−λkφ∞dk

= 1− (γ − 1)zγ−1

∫ ∞z

x−γe−xdx

= 1− (γ − 1)zγ−1Γ(−γ + 1, z), (C.1)

where z = λφ∞, x = λkφ∞ and Γ(s, z) is the incompleteGamma function.

As Γ(s, z) can be written as

Γ(s, z) = Γ(s)−∫ z

0

xs−1e−xdx, (C.2)

we perform the Taylor expansion on the integrand of the right-hand side of Eq. C.2 for small z and get

Γ(s, z) = Γ(s)− zs

s− zs

∞∑n=1

(−z)n

(s+ n)n!, (C.3)

where Γ(s) is the standard Gamma function. Note that thisexpansion makes sense only if s 6= 0,−1,−2, . . . . The casesof γ ∈ N should be discussed case by case.

By substituting Eq. C.3 into Eq. C.1, we get

P = Γ(−γ + 2)zγ−1 + (γ − 1)

∞∑n=1

(−z)n

(n− γ + 1)n!

≈ γ − 1

γ − 2z +O(zγ−1).

Thus for any γ > 2, we have

P ≈ γ − 1

γ − 2z =

γ − 1

γ − 2λφ∞. (C.4)

We will get the expression of P once φ∞ is obtained. ForEq. B.7, we perform the same calculations as Eq. C.1 and get

φ∞ =α

β− α

β(γ − 2)zγ−2Γ(−γ + 2, z),

which gives

φ∞ =α

βzγ−2Γ(3−γ)+

α

β(γ−2)

∞∑n=1

(−z)n

(n+ 2− γ)n!. (C.5)

The leading behavior of φ∞ depends on particular values ofγ. We consider the following cases.

(1) 2 < γ < 3:In this case, we have

φ∞ ≈α

β(λφ∞)γ−2Γ(3− γ),

which gives

φ∞ ≈ (Γ(3− γ))1/(3−γ)

(α/β)1/(3−γ)

λ(γ−2)/(3−γ). (C.6)

Combining Eq. C.6 with Eq. C.4, we get Eq. 5.(2) 3 < γ < 4:According to Eq. C.5, we have

φ∞ ≈α

β(λφ∞)γ−2Γ(3− γ)− α

β

γ − 2

3− γλφ∞,

which gives

φ∞ ≈(

β

αλγ−2

γ − 2

Γ(4− γ)

(αλ

β− γ − 3

γ − 2

))1/(γ−3)

. (C.7)

Combining Eq. C.7 with Eq. C.4, we get Eq. 6.(3) γ > 4:The dominant terms in the expansion of φ∞ are now

φ∞ ≈α

β

γ − 2

γ − 3λφ∞ −

α

2β

γ − 2

γ − 4(λφ∞)2,

which gives

φ∞ ≈2(γ − 4)

γ − 3

β

αλ2

(αλ

β− γ − 3

γ − 2

). (C.8)

Combining Eq. C.8 with Eq. C.4, we get Eq. 7.(4) γ = 3:By carrying out similar approximation techniques as those

in cases (1) to (3) except making use of the expansion ofΓ(0, z) as z → 0, i.e.,

Γ(0, z) = −(γE + ln(z)) + z +O(z2), (C.9)

where γE is the Euler’s constant, instead of the expansionEq. C.3, we can get Eq. 8. For brevity, we omit the detailedproof here.

APPENDIX DPROOF OF THEOREM 3

By omitting terms of O(a2), we can write Eq. B.2 as

dak(t)

dt≈ αλk

∑k′

q(k′)ak′(t)− βak(t)

= αλk〈〈ak(t)〉〉 − βak(t), (D.1)

where 〈〈ak(t)〉〉 is a function of t with 〈〈ak(0)〉〉 ≈ a(0) ≈ 1.Then the change rate equation of 〈〈ak(t)〉〉 can be written as

d〈〈ak(t)〉〉dt

=∑k

q(k)dak(t)

dt

=∑k

q(k) (αλk〈〈ak(t)〉〉 − βak(t))

= αλ

(〈k2〉〈k〉− β

αλ

)〈〈ak(t)〉〉. (D.2)

SAI ZHANG et al.: DYNAMICS OF INFORMATION SPREADING IN ONLINE SOCIAL NETWORKS 15

By solving Eqs. D.2 and D.1, we have

〈〈ak(t)〉〉 = et/τ (D.3)

and

ak(t) = e−βt(

1 +k〈k〉〈k2〉

(e(

1τ +β)t − 1

)), (D.4)

where τ is given by Eq. 9.For the total number of active vertices a(t) which equals to∑k P (k)ak(t), we have

a(t) = e−βt

(1 +〈k〉2

〈k2〉

(e(

1τ +β)t − 1

)). (D.5)

Note that et/τ is the dominant term of Eqs. D.4 and D.5.Furthermore, for uncorrelated SF networks, Eq. 10 can be geteasily from Eq. 9 given Eqs. B.9 and B.10. This completesthe proof.

APPENDIX EPROOF OF THEOREM 4

We consider correlated heterogeneous networks (i.e.,Markovian networks) in this theorem. Consider the Jacobianmatrix L defined as Eq. 13, the solution a = 0 is unstable ifthere exists at least one positive eigenvalue of L.

Based on Eq. 3, we have the connectivity detailed balancecondition [6]

kP (k′|k)P (k) = k′P (k|k′)P (k′) = 〈k〉P (k, k′). (E.1)

As to the connectivity matrix C, if (vk) is an eigenvector of Cwith eigenvalue Λ, then by Eq. E.1, (P (k)vk) is an eigenvectorof CT with the same eigenvalue. Hence all eigenvalues ofC are real. Let Λm be the largest eigenvalue of C. SinceL = αλC − βI , the largest eigenvalue of L is αλΛm − β.Therefore, L has at least one positive eigenvalue wheneverαλΛm − β > 0. This gives the threshold ρc = 1/Λm.

APPENDIX FPROOF OF THEOREM 5

The proof techniques used here are similar to those in theproof of Theorem 2.

By integrating Eq. 17a, we get

ik(t) = ik(0)e−λkφk(t), (F.1)

where

φk(t) = (1− θ)φ(t) + θ

∫ t

0

ak(x)dx (F.2)

and φ(t) is defined in Appendix B. Given ak(0) ≈ 0, weintegrate Eq. 17b and have

ak(t) = α− αik(t)− β∫ t

0

ak(x)dx.

Since ak(∞) = a∞ = 0, we have∫ ∞0

ak(x)dx =α

β(1− ik(∞)) . (F.3)

By combining Eqs. F.1, F.2 and F.3, we have

φk(∞) = (1− θ)φ∞ + θ

∫ ∞0

ak(x)dx

= (1− θ)φ∞ +αθ

β

(1− e−λkφk(∞)

). (F.4)

By performing the Taylor expansion on e−λkφk(∞) of Eq. F.4and omitting higher-order terms, we get

φk(∞) =1− θ

1− θρkφ∞. (F.5)

Given Eqs. F.1 and F.5, we have

P =∑k

P (k)Pk =∑k

Zk−γ(

1− e−λkφk(∞))

= 1− (γ − 1)

∫ ∞1

k−γe−(1−θ)λk1−θρk φ∞dk, (F.6)

where 1/(1− θρk) can be expanded as

1

1− θρk=∞∑n=0

(θρk)n

when θρk < 1. At the same time, part of the integrand inEq. F.6, exp

(− (1−θ)λk

1−θρk φ∞

), can be written as

e−(1−θ)λk1−θρk φ∞ = e(

(1−θ)λθρ φ∞)

/(1− 1

θρk ),

where 1/(

1− 1θρk

)has the expansion

1

/(1− 1

θρk

)=

∞∑n=0

(1

θρk

)nwhen θρk > 1.

Now we analyze relations between P and θ on a case-by-case basis.

(1) γ > 2 and γ /∈ N:In this case, we have the following Eq. F.7, where x1 =

(1 − θ)λφ∞, x2 = x1/θρ, A1 =((θρ)γ−2 − 1

)γ−1γ−2 , A2 =

− (θρ)γ−1, and θρk1 = 1. Note that if k < bk1c, then θρk <

1, and if k > dk1e, then θρk > 1.Moreover, we write Eq. F.7 as

P =(A1 +A2x

−1)

(1− θ)λφ∞, (F.8)

where x = θρ. Note that in the calculation of Eq. F.8 and thefollowing Eq. F.9, we use the Taylor expansion of Γ(s, z), i.e.,Eq. C.3. Thus Eqs. F.8 and F.9 make sense only if γ /∈ N. Thecases of γ ∈ N should be discussed case by case.

By multiplying Eq. 17b with q(k), summing over k andintegrating over t, we get

dφ(t)

dt= α− α〈〈e−λkφk(t)〉〉 − βφ(t).

As t→∞, we have dφ/dt→ 0. Then we get

φ∞ ≈α

β

(1− (γ − 2)

∫ ∞1

k−γ+1e−λkφk(∞)dk

),

which is similar to Eq. F.6. Hence we use the same techniqueand obtain

φ∞ ≈α

β

(B1x1 +B2x

21 +B3x2

)

16 TECHNICAL REPORT

P = 1− (γ − 1)

(∫ bk1c1

k−γe−(1−θ)λkφ∞∑∞n=0 (θρk)ndk +

∫ ∞dk1e

k−γe(1−θ)λθρ φ∞·

∑∞n=0 ( 1

θρk )n

dk

)

≈ 1− (γ − 1)

(∫ bk1c1

k−γe−(1−θ)λkφ∞dk +

∫ ∞dk1e

k−γe(1−θ)λθρ φ∞dk

)

= 1− (γ − 1)

(xγ−1

1 (Γ(−γ + 1, x1)− Γ(−γ + 1, x1bk1c)) +dk1e−γ+1

γ − 1ex2

)≈ A1x1 +A2x2. (F.7)

=(B1 +B2(1− θ)λφ∞ +B3x

−1)

(1− θ)ρφ∞,

where x1, x2 and x have the same meaning as those in Eq. F.7,and B1 =

((θρ)γ−3 − 1

)γ−2γ−3 , B2 =

((θρ)γ−4 − 1

)γ−2

2(γ−4) ,B3 = − (θρ)

γ−2. Then we have

(1− θ)λφ∞ =1− (1− θ)ρB1 − (1− θ)ρB3x

−1

(1− θ)ρB2. (F.9)

Given Eqs. F.8 and F.9, we get

P =p1(θ)

p2(θ), (F.10)

where γ is fixed, p1(θ) and p2(θ) are polynomials of θ, andtheir coefficients are polynomials of ρ. This implies that, givenγ, the impact of θ on P is related to ρ, i.e., there exists a realnumber c0(γ) such that the degree-degree correlations promotethe spreading prevalence if ρ < c0(γ), whereas inhibit theprevalence if ρ > c0(γ). Note that the opposite case is alsopossible.

(2) γ = 3:The approximation techniques used here are similar to those

in case (1) except making use of Eq. C.9 instead of Eq. C.3.The conclusion of this case is the same as that of case (1).For brevity, we omit the detailed proof here.

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