PHYSICAL REVIEW RESEARCH 1, 033125 (2019)

Susceptible individuals drive active social contagion

N. N. Chung,1 L. Y. Chew,2,3,* W. Chen,4,5,6,† R. M. D’Souza,7,‡ and C. H. Lai8 1Centre for University Core, Singapore University of Social Sciences, Singapore 599494 2School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371 3Complexity Institute, Nanyang Technological University, Singapore 637723 4LMIB and School of Mathematical Sciences, Beihang University, Beijing 100191, China 5Peng Cheng Laboratory, Shenzhen, Guangdong 518055, China 6Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100191, China 7University of California, Davis, Davis, California 95616, USA 8Department of Physics, National University of Singapore, Singapore 117542

(Received 17 May 2019; published 22 November 2019)

The catalyzing role that influential individuals play in driving large-scale social contagion has been the focus of many recent empirical and theoretical studies. Nevertheless, complementary studies suggest that the success of social contagion depends largely on having a group of highly susceptible individuals. Here we show that being influential is not due solely to having susceptible peers and that the relative importance of influential versus susceptible individuals depends on the underlying mechanism driving the contagion. Our mathematical models show that for passive social influence both influential and susceptible individuals catalyze the spread of the contagion progressively and equitably. In contrast, for active social influence, susceptible individuals facilitate rapid initial contagion, but then influential individuals are necessary for sustaining the growth of the contagion. We show that active influence characterizes certain online social networks, like retweeting behavior on Twitter, while passive influence characterizes other online social networks, like adoption of whom to follow on the Digg network.

DOI: 10.1103/PhysRevResearch.1.033125

I. INTRODUCTION studies have focused on the susceptibility of the individuals to social influence [25–28]. Social influence has long been a major topic in social The debate on whether it is the influentials or the sus- psychology, which examines how innovations, behaviors, and ceptibles that drive social change is still actively ongoing. opinions diffuse through a society [1]. Stylized mathematical Thus, there is a crucial need to determine whether or not models and direct observation of social networks attempt to it is necessary to distinguish between being influential and elucidate these mechanisms and take forms as diverse as the being susceptible in the modeling of social contagion. In spread of rumors [2,3], the outbreaks of political or social other words, while influence and susceptibility to influence unrest [4], the development of [5,6], marketing are often studied separately, are their roles substitutional? If of products [7], adoption of new practices [8], and syn- not, we may be missing half of the picture by incorporating chrony of emotion [9,10]. Researchers are investigating not only influence or only susceptibility to influence in models of only whether and how various contagious phenomena spread contagion. within social networks with different topologies [11,12], but In an empirical study [29], Aral and Walker conducted also the controllability of the diffusion process [13–15]. Much a randomized experiment to gauge the influence and sus- research effort has been put into identifying the influential ceptibility to influence among 1.3 × 106 Facebook users in individuals that are the key drivers of large-scale social con- the adoption of recommended movies. Using this large sta- tagions [16–24]. Along these lines, various measures such as tistical sample, they showed that an individual’s influence network coreness and degree are proposed to identify the best and susceptibility to influence are dependent on his or her spreaders in complex social networks. In contrast, a variety of attributes such as age, gender, and relationship status. This implies that being influential or susceptible does not depend merely on personal network characteristics such as degree and *[email protected] coreness [20]; rather, it can be an implicit characteristic of an †[email protected] individual. In fact, some of us are most effective at pushing a ‡[email protected] message to a wider reach, while some of us are more likely to be influenced by the opinion of our peers. In addition, Published by the American Physical Society under the terms of the they establish empirically that being influential is not simply Creative Commons Attribution 4.0 International license. Further because of having susceptible peers. distribution of this work must maintain attribution to the author(s) Here we examine the roles that both susceptible and influ- and the published article’s title, journal citation, and DOI. ential individuals play in driving change using mathematical

2643-1564/2019/1(3)/033125(13) 033125-1 Published by the American Physical Society CHUNG, CHEW, CHEN, D’SOUZA, AND LAI PHYSICAL REVIEW RESEARCH 1, 033125 (2019) models of social contagion. From this theoretical approach, applied in a passive manner. Such an approach has been we establish that influential individuals cannot always be used widely to study the propagation of social influence and modeled by individuals with susceptible peers. More im- information [36–38]. portantly, we show that a fundamental, and somewhat over- In contrast, active social influence refers to a direct out- looked, aspect is whether the social influence is exerted in reach action by the spreader to disseminate information or a passive versus active manner. The standard mathematical share opinions in an attempt to entice others to adopt a product models (including threshold models, diffusion models, the or behavior. For instance, when we feel excited about a new voter model, etc. [30]) tend to assume passive influence, product or a movie, we may contact others directly to share where one mimics or converges to the behaviors of neighbor- our excitement and recommend the new product or movie. ing nodes. Passive influence plays an important role. However, Likewise, young adults being handed an alcoholic beverage in the era of pervasive online social network activity, such as by a peer is a form of active influence to drink alcohol [33,34]. Twitter, active social influence, when an individual performs To model contagion spread through active social influence, an action or sends a message to entice others to adopt a we propose what we call the excitation model. Here a node behavior, is increasingly important. reaches an excitation threshold and sends active messages to Here we introduce a mathematical model of active social neighbors, who may then reach their excitation threshold and influence with analogies to the classic Bak-Tang-Wiesenfield send out active messages to influence their neighbors, who sandpile model [31]. We show that the dynamics of some may in turn reach their activation level, etc. The model also social networks, like retweeting activity on Twitter, are better captures the accumulative nature of human emotion. When captured via this active influence model, whereas the dynam- something is first introduced to us, we may not have much in- ics of other social networks, like adopting new followers on terest in it. Receiving repeated recommendations of a product the Digg network, are better captured via a passive influence often increases our interest, as shown empirically [39,40] and model. For active social influence, the difference between sus- modeled in a stylized manner by complex contagion [41]. The ceptible and influential individuals is particularly pronounced accumulated interest may eventually result in our adoption of in the cases we studied. The difference is most significant for the product and active attempts to entice others to adopt. We the important class of terse messages, i.e., messages which find that several factors impact the dynamics of active social are disseminated instantly once received [32], such as news of influence including the underlying degree distribution of the a nearby catastrophe or natural disaster. For terse messages network and whether activation thresholds are constant for all spreading through active social influence, we find that the nodes or proportional to a node’s degree. susceptibles can be dramatically more effective in catalyzing contagion. For passive social influence, we find that the differ- ence in effectiveness of susceptible and influential individuals III. MATHEMATICAL MODELS OF CONTAGION is much less significant. The development is organized as follows. First, in Sec. II, we review the distinctions between A. Passive influence: The threshold model passive and active influence. Then, in Sec. III, we analyze In the threshold model [35], an individual’s decision to mathematical models of both types of influence. In Sec. IV we adopt a product or a message is determined by the extent to investigate numerically the mathematical models and also real which their peers have already adopted it, specifically by the data, focusing on several scenarios that serve to corroborate fraction τ of peers that have already adopted. More explicitly, the mathematical models and also let us establish the main an individual, labeled j, is in either the nonactivated state regimes and mechanisms of the models. In Sec. V we include with σ j = 0 or the activated state with σ j = 1. Typically, discussion of shortcomings and future work. the model starts with a small fraction α of activated agents chosen uniformly at random, with the rest of the agents in the nonactivated state. The contagion then propagates through the II. PASSIVE VERSUS ACTIVE passive influence of those early adopters. Specifically, at each Passive social influence is related to an individual’s per- discrete time step an agent is selected uniformly at random to ception and interpretation of behaviors or product adoption update its state. In case that agent is in a nonactivated state, patterns of others [33]. It can be further broken down into it becomes activated if at least a fraction τ of its neighbors two types: social modeling and perceived norms [34]. Social are already activated and then it remains activated forever. modeling refers to imitation of behaviors or product adop- (Thus, τ is the threshold for activation, which is the same for tion patterns of others. As an example, we often observe all agents.) A new agent is then selected at random to update and imitate the behavior or fashion style of our peers. In a and the process is iterated until the system reaches equilibrium complementary manner, perceived norms are related to beliefs and no further agent can be activated. The fraction of activated about what is normative within a particular reference group nodes at the last time step is denoted by fA. and individuals strive to match their attitudes, beliefs, and be- The above describes the original threshold model, with the haviors to the group norm to avoid social ostracism. Through analytical solution of the average final fraction of activated these mechanisms, our peers are applying influence on us nodes fA given in Ref. [42]. Our threshold model, however, is passively and unintentionally. To capture passive influence more detailed than the original model since we introduce three from a mathematical perspective, the classic threshold model types of agents, influential, susceptible, and normal, to reflect is often used [35]. Here an individual’s decision to adopt a the implicit characteristics of individuals. Now when agents product or a message is determined by the extent to which are selected to update their state, they consider the average their peers are already adopting it; thus social influence is behavior of their peers, weighting each peer’s contribution

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(1 for normal peers, γ for influential peers) to influence their play different roles through the manifestation of fI and fS in a own decision by 1/k. In other words, the contribution of each different part of Eq. (2). normal and susceptible activated neighbor is β = 1/k, while the contribution of each influential activated neighbor is βI = γ/k, with γ a parameter larger than one. On the other hand, B. Active influence: The excitation model the activation threshold for a susceptible agent is denoted by The key idea of the excitation model is to account for τS and it is lower than τ, the activation threshold of the other active attempts to influence neighbors, such as by sending a two types of agents. direct message endorsing a new behavior or product. Such Considering first the impact of influentials, we analyze the an activity may increase the receiver’s level of interest in the case that a fraction fI of the agents are influential while the product. Thus we associate with each agent j an excitation remainder are normal agents. Solving for the average final level E j = 0, 1, 2,...representing its current level of interest fraction of activated nodes gives or excitement about the contagious phenomenon under study. Here E j is cumulative and increments each time j receives ∞ k m k − m additional influence from an activated neighbor up until reach- = α + − α m − k m μ fA (1 ) pk q∞(1 q∞) ε = m r ing an excitation threshold j ck j , which we assume to be k=1 m=0 r=0 proportional to a power μ of agent j’s degree k j. (Note that − m r(γ − 1) μ = 1 means the threshold is proportional to the fraction of × f r (1 − f )m r F + , (1) I I k k neighbors, whereas μ = 0 means a constant threshold.) As soon as E j  ε j, three things happen. First, if agent j is not with pk denoting the degree distribution of the underlying yet activated (i.e., σ = 0), it becomes activated and transits θ j network and F ( ) the probability that an agent has a threshold to state σ = 1, adopting the contagious phenomenon and θ j less than . remaining in the activated state throughout the process. Sec- Similar to the analytical approach in Ref. [42], we must λ = ν ond, the agent emits j dkj excitation pulses to randomly assume that the underlying network is represented by a locally selected neighbors. Third, the agent’s excitation level is reset treelike structure. The conditional probability (qn+1) that a to zero, E = 0. Note that j then restarts its accumulation + j node on level n 1 of the tree is active, conditioned on its of excitation and can eventually topple again with the same + parent (on level n 2) being inactive, is solved through the threshold condition. Here users are allowed to reaccumulate following recursion relation: the excitation level such that activated users can participate ∞ − actively in the spreading process if a second excitement k 1 − k k 1 m k−1−m strikes them. Note, however, that a new and higher threshold qn+1 = α + (1 − α) pk q (1 − qn ) z m n condition can be set for cases where second excitements are k=1 m=0 unlikely [44]. In this context, the process of the excitation m m − m r(γ − 1) model is similar to the toppling process in models of self- × f r (1 − f )m r F + . (2) r I I k k organized criticality [31,45]. r=0 Note that above we defined two exponents μ and ν related, Here z denotes the average degree of the network, while q∞ respectively, to the excitation threshold of a node and the gives the fixed point of the recursion relation. With q0 = α, number of activation pulses sent out upon toppling. We might Eq. (2) is iterated a sufficient number of times to get a steady- naively expect the excitation threshold to be proportional to a state value for q∞ which is then substituted into Eq. (1)togive node’s degree, and thus μ = 1. Nevertheless, it was recently fA. Note that established that the activation threshold of social media users depends largely on the content of the message to be spread θ<τ θ = 0for [46,47]. While a casual message is more likely to be circulated F ( ) θ  τ, (3) 1for only after a significant fraction of the receiver’s peers have μ = as the activation threshold for all the agents is the same. done so ( 1), in contrast, a terse message is likely to be disseminated instantly once received [32], such as news of a Now considering the impact of susceptibles, we analyze μ = the case that a fraction f of the agents are susceptible while nearby catastrophe or natural disaster. In the latter case 0 S and the activation threshold can require only one neighbor. the remainder are normal agents. The distribution of the ν activation threshold is given by Likewise, we can consider the exponent governing the number of activation pulses sent out upon toppling. Most so- fS for φ = τS cial media platforms allow their users to send messages to all g(φ) = (4) ν = 1 − fS for φ = τ. peers with just a click. This corresponds to 1 and results in high-degree users emitting more excitation pulses when In this case, in analogy to the analysis in [43], the probability compared to low-degree users. However, there are situations θ θ = θ φ φ that an agent has a threshold less than is F ( ) 0 g( )d , when there is a finite budget and the number of messages to whichisusedinEq.(2) to determine q∞ and in turn in be disseminated is limited by the social media platform. As an Eq. (1) to determine the final fraction of activated agents example, a Twitter user is limited to send no more than 1000 when a fraction fS of the population are susceptibles. Note direct messages per day. Likewise, several popular games on that applicability of the analytical results is limited as it can Facebook impose limits on the number of user interactions only be applied to a sparse treelike network. It is thus shown each day [48]. Such limitations are modeled by setting ν = 0. only to highlight that influential and susceptible individuals As the overall influence imparted by a sender is proportional

033125-3 CHUNG, CHEW, CHEN, D’SOUZA, AND LAI PHYSICAL REVIEW RESEARCH 1, 033125 (2019) to the size of the audience for a message, this results in the C. Threshold versus excitation model number of excitation pulses sent by all agents being the same We compare the two models through the activation deci- regardless of the agent’s degree. (Note, as shown herein, that sion of an agent which is defined by μ = the excitation threshold, e.g., whether 0 or 1, has a much ⎧ ⎫ bigger impact than the pulses sent, e.g., whether ν = 0or1.) ⎨ ⎬ We start the simulation of the excitation model with a small σ =  ϒ > , (7) α i ⎩ j i⎭ fraction of activated agents sending pulses while the rest ∈ A j Pi are inactive and initialized with E j = 0. The social contagion is thus initially propagated through the active influence sent where ϒ j parametrizes the influence of peer j, i A out from the early adopters. At each time step, the excitation parametrizes the susceptibility of agent i, Pi is the set of peers levels of all the agents are scanned. Those with excitation level of agent i who are activated,  is the step function, and σi is an ε higher than or equal to their threshold level j topple, sending adoption indicator. For the threshold model, i = τiki. Thus, out excitation pulses. For simplicity, we assume that the pulses the number of an agent’s peers who must activate before the are sent uniformly at random to all the agent’s peers. agent will is dependent on the agent’s degree. On the other To bring in influential and susceptible agents, we define a hand, i = εi/p for the excitation model. The event of getting scenario analogous to that in the threshold model of passive excited by an activated peer is stochastic with a probability p. influence. We want an influential agent to increase the excita- With μ = 1, the number of an agent’s peers who must activate tion level of its peers with a higher probability than a normal before the agent will is dependent on the agent’s degree. agent. Thus, upon toppling, an influential agent j sends out λI >λ λ j j excitation pulses, where j is the number of pulses sent if j is either a normal or a susceptible agent. On the other IV. RESULTS hand, a susceptible agent is associated with a lower excitation Here we will now investigate numerically several distinct εS <ε ε threshold of j j, with j being the threshold if the jth scenarios that let us establish the main regimes and mech- agent is either an influential or normal agent. anisms of the models. We will simulate the threshold and Next we evaluate analytically the fraction of agents who excitation models under four different settings. The settings emit pulses at time step n + 1, i.e., Qn+1. For the sake of are (i) an equal fraction of either influential or susceptible analytic tractability, we restrict our solution to the case of agents distributed randomly throughout the social networks, z-regular random networks where by definition all nodes j studied in Sec. III A; (ii) an equal fraction of either influential = λI = λI λ = λ εS = εS have the same degree k j z. Thus, j , j , j , or susceptible agents forming clusters within the networks, and ε j = ε. The probability for a randomly selected node to in Sec. III B; (iii) an influential agent versus an agent with receive s pulses at time step n is given by purely susceptible peers, in Sec. III C; and (iv) the influential and susceptible agents in the network are linked by real-world z m z − m connections, in Sec. III D. (s) = Qm(1 − Q )z m f r n m n n r I = = m 1 r 0 A. Effectiveness at catalyzing contagion − λ λ − ×(1 − f )m r M (1/z)s(1 − 1/z) M s, (5) I s While both influential and susceptible individuals play important roles in promoting the spread of social contagion, I apriori with Q0 = α and λM = (m − r)λ + rλ . The probability for a it is not clear how different or similar they are in their randomly selected node to emit pulses at time step n + 1is effectiveness. It is thus essential that we study separately the then given by impact of the influential and of the susceptible individuals. We label the two scenarios I and S, where in scenario I a fraction λ ε− M 1 fI of agents are selected randomly as influential agents, with Qn+1 = n(s) ρn(E )F (E + s), (6) the rest being normal agents. Likewise, for scenario S,a s=1 E=0 fraction fS of agents are selected randomly as susceptible agents, with the rest being normal agents. We then compare θ with F ( ) again being the probability that an agent has a the extent of contagion reached for different values of fI = fS, θ φ threshold less than . Given the threshold distribution g( ), as shown in Fig. 1, as measured by fA, the expected fraction of θ = θ φ φ ρ we have F ( ) 0 g( )d .Here n(E ) denotes the distribu- activated nodes, and the time taken for the contagion to reach tion of excitation levels for the neighborhood of an agent who half of the population, denoted by T1/2. is sending out pulses at time step n. As activation activities are The results for Erdos-Rényi˝ (ER) networks are shown in originated locally, pulse emission events at each time step are Fig. 1(a) for the passive influence threshold model and in dependent on the local distribution of excitation level. For in- Figs. 1(b) and 1(c) for the active influence excitation model. stance, since all the agents start with E = 0, we have ρ0(0) = All numerical results are obtained for a network of size N = −z k 1 at time step n = 0. With this, the fraction of activated 8000, with ER degree distribution pk = e z /k! and average agents at the next time step Q1 can be determined. In other degree set to z = 8. Each numerical result is averaged over words, Qn+1, defined in Eq. (6), can be solved numerically 500 independent realizations, where 10 random instances of by substituting the values of Qn and ρn into Eqs. (4) and (5). networks are generated and 50 simulations are done for each Results are show in Fig. 2 and discussed in detail in the next instance. In all cases error bars are smaller than the marker’s section. size.

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FIG. 2. Analytical results for the excitation model on z-regular random networks [Eq. (6)] show the importance of susceptible FIG. 1. Role that influential versus susceptible individuals play individuals in catalyzing social contagion especially at its early in catalyzing social contagion under (a) passive influence and (b) and phase when the excitation level across the population is low. Here (c) active influence. Plotted is fA, the expected fraction of the popu- the populations consist of either a fraction fI of influential agents lation adopting the new behavior, versus the fraction of susceptible (solid line) or a fraction fS of susceptible agents (dashed line). The (influential) individuals. Numerical results are for ER random graphs parameters used are Qn = 0.02, z = 4, c = d = 1, εS = ε/2, and = = of size N 8000, average degree z 8, and initial activated fraction λI = 2λ. The distributions of excitation levels are (a) ρn(0) = 1and α = . 0 02. Each data point is the average over 500 realizations. (b) ρn(1) = ρn(3) = 0.5. (a) For the threshold model, τ = 0.4, τS = 0.2, and γ = 2, meaning that susceptibles need only half the influence to activate and influ- entials are twice as effective. (b) and (c) Analogously, for active individuals when a larger fraction of these individuals are present. In Fig. 1(b), an obvious difference is observed when influence, εS = ε/2andλI = 2λ, meaning that susceptibles need only half the influence and influentials exert twice as many activation the contagion is driven by influential versus susceptible pulses. For (b), μ = 1, ν = 1, c = 0.65, and d = 1, while (c) is for individuals. In addition, numerical results for the excitation terse messages (μ = 0). In (c) each newly activated node emits on model passing terse messages, i.e., constant activation average 5.5 pulses (ν = 0 with d = 5.5). With c = 4, the excitation threshold μ = 0 [Fig. 1(c)], show the dramatic role played threshold is set to ε = 4. Analytical results from Eq. (1)areshown by susceptible agents in catalyzing the social contagion. in (a) with solid and dashed lines. Note that choose parameter values Figure 1(c) is for ν = 0, corresponding to activation pulses that best highlight the interesting phenomena. The insets show the sent to a fixed number of neighbors. time for contagion to reach half the population. Analytic results for the excitation model for z-regular ran- dom networks are derived in Eqs. (5) and (6). Figure 2 shows the probability of activation at the next time step given the For the threshold model [Fig. 1(a)] the average number of probability of activation at the current time step, as calculated peers that have to be activated before an agent will is given by Eq. (6). In Fig. 2(a), the initial excitation level of all agents by  = zτ = 3.2. There will be no contagion without the is set to zero. In Fig. 2(b), the initial overall excitation level presence of influential or susceptible individuals as τ>1/z. is set to 2. The probability is significantly higher for the S Large-scale adoptions are observed when fI > 0.6 and fS > over the I scenarios, with the susceptible agents being more 0.6. The analytic results (1) agree well with the numerical effective in catalyzing social contagion, particularly when the simulations and show overall little difference when driven by overall excitation level of the population is low [Fig. 2(a)]. influential versus susceptible individuals. Rather than a homogenous Erdos-Rényi˝ network structure, Analogously, we set μ = 1, ν = 1, c = 0.4, and d = 1for real-world social networks typically possess an approximately the excitation model. With ν = 1 and d = 1, the probability scale-free (SF) structure, with a small number of nodes having of receiving an excitation pulse from an activated peer is very high degrees compared to a vast majority of others. approximately 1. Therefore,  = 3.2. In this case, contagion It is thus imperative to examine the roles of the influential is induced when fI > 0.05 and fS > 0.05 [44]. We next and susceptible agents in the more realistic setting of a SF compare the roles played by influential and susceptible network topology. Figure 3 shows the fraction of activated

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influence on catalyzing effectiveness of influential individuals and susceptible individuals.

B. Effects of clustering Based on empirical data on movie adoption activities in Facebook, influential individuals are found to cluster together in the network, while susceptible users have a tendency not to cluster [29]. Interestingly, the clustering of influential indi- viduals generates a multiplier effect which causes a cascade enhancing the spread of the social contagion. It is however unclear whether similar effects happen when susceptible in- dividuals cluster together in the network. Hence, we examine the effects of clustering of the susceptible agents in scenario S and the influential agents in I separately. In other words, our studies investigate the consequences of assortativity of the influentials as well as assortativity of the susceptibles on the propagation of the social contagion. To focus on the impact of clustering, we consider an ER network topology with average degree z = 4 and set either fS = 0.5or fI = 0.5 and the initial fraction of activated agents α = 0.005. Increasing the clustering between the susceptible agents (or the influential agents) is done by rewiring the links connecting agents of different types. The clustering is quantified by the coefficient R, whose definition is given in Sec. VI. Figure 4 shows the extent of the final contagion fA as a function of R for the passive influence model [Fig. 4(a)] and FIG. 3. Impact of scale-free topology on the relative importance the active influence model [Fig. 4(b)] with μ = ν = 0. of influential versus susceptible individuals. The figures here are The results in Fig. 4 clearly demonstrate that increasing analogous to Fig. 1, except that the underlying network topology is clustering of either the influentials or the susceptibles leads γ =− . generated using the Chung-Lu model [49] with exponent 2 5. in general to an increase in effectiveness of their roles in Shown is the asymptotic fraction of activated agents fA for (a) the catalyzing the transmission of social contagions. Here again threshold model, (b) the excitation model with μ = ν = 1, and we see that for active influence [Fig. 4(b)] the susceptibles (c) the excitation model with μ = ν = 0. All parameter choices are play a more important role. For passive influence [Fig. 4(a)], identical to those in Fig. 1. The insets show the time for the contagion to reach half of the population. initially increasing clustering among susceptible or among influential agents is equally effective at driving adoption. Nevertheless, ultimately, the susceptible individuals fail to maintain the improved rate of contagion transmission beyond agents for a SF network for the passive influence threshold a certain high level of clustering. As we see in Fig. 4(a), model [Fig. 3(a)] and the active influence excitation model fA drops abruptly as R increases beyond a critical value. [Figs. 3(b) and 3(c)]. As with Erdos-Rényi˝ networks, we again This results from the appearance of one or a few highly see that under passive influence [Fig. 3(a)] both influential clustered groups of susceptible individuals which are sparsely and susceptible agents drive change in an equitable manner; connected to the rest of the normal agents. While these suscep- however, for active influence [Figs. 3(b) and 3(c)], depending tible agents are all easily activated, their relative isolation has on whether the excitation threshold is dependent on degree, prevented them from activating the rest of the normal agents, catalyzing effectiveness of susceptible and influential agents thus leading to the observed drop in performance. can be different. It is interesting that for active influence the increasing In summary, the effectiveness of influential individuals and clustering of susceptibles does not encounter a similar drop in susceptible individuals at catalyzing social contagion is not spreading effectiveness. This results from the key difference always the same. The difference between the two can be between active susceptibles and passive susceptibles. Unlike obvious or insignificant, depending on mechanisms (active passive susceptibles who affect the normal individual through or passive influence and dependence of threshold on degree) their presence, active susceptibles impart excitement through underlying the spreading. Thus, we show here a need to excitation pulses. This makes it possible for the relatively distinguish between being influential and being susceptible isolated active susceptibles to transmit a sufficiently high level in the modeling of social contagion as their effectiveness of excitement to the normal agents and activate them, which is at catalyzing contagion is not always the same. We show impossible for the passive susceptibles to achieve in the same also a need to distinguish between active and passive in- clustered configuration. fluence, which would give a different extent of adoptions under similar settings. Note that other factors, such as average C. Being influential versus having susceptible peers degree, community structure, assortativity of the network, Let us now consider a crucial question posed by Aral and and heterogeneity of the threshold, may also have a different Walker on whether the role of an influential individual is

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the data were collected. We label this individual user A and examine the subset consisting of all users who begin to follow user A at some point during the time of the data collection. There is a large proportion of spontaneous adoption of user A, where the followers’ activation happened in the ab- sence of quantified influence (of course others have noted the confounding effects of [51,52]). Nonetheless, a significant number (around 30%) of those who chose to begin following user A did so after one or more of their peers had done so. Social media users rarely send active messages to others when they adopt a new followee. More often, they imitate the behavior of the individuals whom they already follow [53]. In other words, there is evidence suggesting that who to follow is spread through the mechanism of passive social influence. The other data set we investigate, the Twitter data set, covers the spreading of news on the discovery of the Higgs boson at CERN between 1 July 2012 and 7 July 2012 [54]. We analyzed a subset of these data excluding tweets with no interactions among the users, resulting in a data set of 563 069 tweeting activities and a network of 304 691 users who have either tweeted, mentioned, or replied to a tweet related to the discovery of the Higgs boson. Here we consider that a user adopts the behavior of another user if they retweet, mention, FIG. 4. Clustering among influential or among susceptible in- or reply to that user. Specifically, the data set contains iden- dividuals in general enhances the extent of adoption. Shown is tifications of the spreader and the adopter, the time when the message was created, and the type of adoption. the fraction of activated agents fA as a function of the clustering coefficient R for (a) the threshold model and (b) the excitation model In contrast to the Digg data set, spontaneous adoption, i.e., with μ = ν = 0 when half the population is susceptible, fS = 0.5, or a spontaneous tweet on the subject, contributes to only around when half the population is influential, fI = 0.5. Although clustering 10% of the total number of activities in this data set. Hence, among influential and among susceptible individuals has similar a large proportion of adoption activities here result from impact initially for passive influence, ultimately too much clustering social interaction, suggesting that the spread of news through of susceptible agents leads to a steep drop in the extent of adoption, the Twitter network is driven by active social influence. The as shown for the last two data points in (a). empirical details of spreading on Twitter, including the effects of divided attention, are explored in Ref. [55]. equivalent to an agent having susceptible peers [29]. Their The spreading processes on these two real-world networks empirical results showed that they are not equivalent. Here, exhibit very different dynamics as shown in Fig. 5.Inthe using our theoretical models, we arrive at the same conclusion Twitter network [Fig. 5(b)], we observe a slow increase in the [44]. beginning until a critical point at which the spread increases abruptly, characteristic of active influence and that the pent-up excitement in the contagion is suddenly released. However, D. Catalyzing social contagions in real-world settings the Digg network [Fig. 5(a)] shows a consistent and steady We have thus far studied the impact of influential and rise in adoption behavior of the followers, characteristic of of susceptible agents separately. In real-world settings it is passive influence. likely that all types of agents will be present simultaneously. To understand the specific roles of influential and suscepti- This is the case for real-world social media networks [29], ble individuals in catalyzing the contagion, we first identify including two that we investigate in detail: the Digg and the influential individuals, the susceptible individuals, and Twitter networks. the spontaneous and early adopters in both networks. (See In the Digg network users vote for interesting stories to be Sec. VI for the identification methods.) In Fig. 5, in addition promoted to the front page of Digg.com. Users also follow one to the aggregate behavior discussed above, we also show the another and the data set contains information on the formation adoption dynamics specifically for influential agents, suscep- of follower links among 71 367 Digg users. Specifically, it tible agents, and neighbors of the influential agents. We are contains pairs of anonymized user’s identifications, the time interested in the neighbors of the influential agents because when the follower’s link was created, and whether the link their activations will serve as a direct measure of the spread of represents a mutual following relationship. This data set was the contagion due to the influentials. used originally in Ref. [50] to study the voting dynamics of To discern the major underlying mechanism that drives the Digg’s users over a period of one month in 2009. Our purpose social contagion, we have simulated the spreading processes here is instead to understand the dynamics of adopting new on both the Digg and the Twitter networks with both the followers, especially as we observe that one of the users in this threshold and the excitation model. Note that the spreading data set gained 12 097 followers during the time range that processes in real-world social networks can be complex and

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FIG. 5. Empirical data of growth dynamics for (a) the Digg follower network and (b) the Twitter network, simulation results of the threshold model on the (c) Digg and (d) Twitter topologies, and simulation results of the excitation model on the (e) Digg and (f) Twitter topologies. The four curves in each plot are the fraction of activated agents (thick solid line), the fraction of activated agents who are connected to at least one influential neighbor (dashed line), the fraction of activated susceptible agents (solid line), and the fraction of activated influential agents (dotted line). For the Digg network, τ = 0.12, τS = 0.06, γ = 2.0, c = 2, d = 6, μ = ν = 1, εS = ε/2, and λI = 2.0λ. For the Twitter network, τ = 0.4, τS = 0.1, γ = 1.6, c = 1.0, d = 1, μ = ν = 1, εS = ε/4, and λI = 1.6λ. Results are obtained by averaging over ten realizations. consist of mixtures of the two types of contagions. Nonethe- the simulated fA(t ) curve and the fA(t ) curve from the data. less, we expect either one of the models to give a good (See Sec. V C for details.) A smaller value of the L2 error approximation when the contagion is driven mainly by active indicates a better fit between the model and the data. Here or passive influence. we show only the results simulated using a homogeneous One may think that it is not fair to compare the following threshold for simplicity. A more comprehensive study of the behavior in Digg data and adoption behavior on Twitter as modeling can be done with a heterogeneous threshold and there is one and only one user to follow throughout the period optimization over parameters [44]. while there are more and more tweets about the discovery For the Digg network, the dynamics predicted by the of the Higgs boson to react to over time. Thus, instead of threshold model [Fig. 5(c)] corresponds better to the real randomly choosing a seed as initial adopters and having the data [Fig. 5(a)], with an L2 error of 0.0376. (Note that influence to propagate through them, we determine the seed the L2 error for dynamics simulated using the excitation as well as spontaneous adoptions through the data. The same model is 0.5307 [Fig. 5(e)].) The growth of the activated information is then used in both the threshold and excitation fraction of the population shows a steady rise, which we models to obtain two fits for comparison. The results for the conjecture to be characteristic of spreading due to passive threshold model are shown in Fig. 5(c) for the Digg network influence. Interestingly, the feature of a steady rise in ac- and in Fig. 5(d) for the Twitter network. The results for the tivation is not unique to the follower dynamics. In fact, excitation model are shown in Fig. 5(e) for the Digg network scientific papers also receive their citation counts in steady and in Fig. 5(f) for the Twitter network. We then quantify the paces [56,57]. Although scientists do broadcast and promote difference in the adoption dynamics predicted by the models their research in conferences, there seems to be a greater with the dynamics derived from the actual data using the L2 tendency for researchers to cite works that are more often error. This is given by the L2-norm of the difference between cited by others. Thus, we believe that the spread of scientific

033125-8 SUSCEPTIBLE INDIVIDUALS DRIVE ACTIVE SOCIAL … PHYSICAL REVIEW RESEARCH 1, 033125 (2019) information in this context is also predominantly driven by passive influence. In the case of the Twitter network, the excitation model (a) [Fig. 5(f)] clearly provides a better prediction of the real data [Fig. 5(b)] with respect to the aggregate behavior as well as the behavior of the different types of individuals. In particular, a similar slow spread in the beginning, as well as a sudden increase in the fraction of activated agents, the fraction of activated susceptibles, and the fraction of activated influentials’ neighbors at the same critical point, is observed. The threshold model [Fig. 5(d) also does not correspond to the real data. First, it overestimates the number of activated agents at the initial stage by giving a prediction almost twice as large. Second, its spread has the feature of a steady rise instead of the observed abrupt increase at the critical stage of the contagion. More importantly, the L2 error for the excitation model (0.3543) is found to be smaller than that of the threshold model (0.6218). It is interesting to note that although the Digg follower network seems to obey passive influences, other aspects of the Digg network seem to obey active social influence, in partic- (b) ular the voting behavior among fans. In this case, whenever a Digg user votes for a story, short messages revealing the vote will be sent to the news feeds of the user’s followers. These messages act like active influences sent to excite the fol- lowers’ sentiments on the story. Notably, activation dynamics similar to that in Figs. 5(b) and 5(f) is observed in the spread of votes within this Digg network [50]. To further quantify the roles of the different types of agents, we wish to identify the time at which they were first activated. Figure 6 shows a randomly chosen subset of the actual inter- action networks for both the Digg and Twitter networks. The colors of the nodes depict the different roles, and the size of the node is proportional to how quickly in the process a node was activated, i.e., a bigger diameter indicates that a node was FIG. 6. Interaction networks for different types of agents: normal activated earlier in the process. In the Digg’s follower network (blue), susceptibles (green), influentials (red), influentials’ neighbors [Fig. 6(a)] we observe a steady rate of activation of agents of (orange), and susceptibles linked to influentials (pink), obtained from all categories as shown by the occurrence of nodes of all sizes sampling (a) the Digg network and (b) the Twitter network. The size for the influentials, purely susceptibles, susceptibles linked to of the node indicates activation time, with larger nodes activating influentials, influentials’ neighbors, and the normal agents. In earlier in the process, plotted using [58]. contrast, the active spreading process in the Twitter network [Fig. 6(b)] is observed to occur in two stages. Here the nodes and the other with μ>0.5, we show the adoption dynamics of susceptibles (green) and susceptibles linked to influentials for terse and nonterse messages in Fig. 7. The growth of both (pink) appear only with a large diameter, implying that these contagions is observed to occur in two stages. Terse messages, nodes are almost all activated in an early stage. The nodes of however, gain a wider reach during the first stage. This implies the influentials’ neighbors (orange) appear, however, predom- that Twitter users are more susceptible to terse messages since inately in very small sizes. Thus, the adoption activities in the the susceptibles drive early-stage contagions. second stage are driven mainly by the influential spreaders. Interestingly, similar dynamics is observed in the spreading of two terse messages on Twitter, one related to Supertyphoon E. Terse messages Haiyan [59] and the other to Hurricane Sandy [60]. (See Fig. 7.) To physicists, a tweet sharing the news on the discovery of the Higgs boson is likely a terse message, but probably not to other users in Twitter. Here we examine the number of a user’s F. Summary of the simulations neighbors who have tweeted the message before the user does In summary, we have simulated the threshold and exci- so and refer to this as the preceding tweeting neighbors n.For tation models under four different scenarios. The scenarios the spreading of terse messages, the dependence of n on the are (i) an equal fraction of either influential or susceptible user degree k given by μ = ln n/ ln k is minimal. Indeed, users agents distributed randomly throughout the social networks, in the Twitter data set show values of μ ranging from 0 to 1. (ii) an equal fraction of either influential or susceptible agents By separating the users into two groups, one with μ  0.5 forming clusters within the networks, (iii) an influential agent

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increase in activated individuals as the contagion mechanism, the excitation model shows an abrupt rise in activation. We surmise that the main social mechanisms that drive contagion in the threshold model and the excitation model correspond to those of the real-world Digg and Twitter networks, respec- tively. Our conjecture is supported by the fact that simulation and empirical observations give the same conclusion in the respective networks: (a) Influential and susceptible individ- uals play equitable roles in both the Digg network and its simulated threshold model version and (b) for the Twitter network and its simulated version from the excitation model, susceptible individuals are more active in the early phase of the activation before the influential individuals take over the role of spreading.

V. DISCUSSION In general, influential and susceptible individuals catalyze the spread of social contagion through different mechanisms. Influential individuals act by directly influencing their peers in an impactful way. On the other hand, susceptible individuals have the characteristic of being easily activated. They are especially effective in influencing individuals that are sur- rounded by them in sufficient numbers. The catalyzing effec- tiveness of a single activated susceptible individual, however, is limited. In addition, being influential is not equivalent to having susceptible peers and we cannot replace an influential spreader with an agent who has susceptible peers in the modeling of social contagion via either passive or active social contagion. Another essential aspect of the propagation of real-world contagion concerns its underlying spreading mechanisms. In this paper, we have demonstrated that the spreading of social contagions can be mediated by two distinct mechanisms: passive and active social influence. We have exemplified this in the Digg follower network and the spreading of scientific news in Twitter networks, where the threshold and excitation models have been employed, respectively, to affirm the under- lying spreading mechanisms through a close correspondence between empirical data and model predictions. Note, however, FIG. 7. Growth dynamics observed in the empirical Twitter data that the spreading processes in real-world social networks of (a) the discovery of the Higgs boson, (b) Hurricane Sandy, and can be complex and may even consist of both spreading (c) Supertyphoon Haiyan. The growth dynamics predicted by direct mechanisms simultaneously. In certain cases, it could be hard simulation of the excitation model is shown in (b) and (c) for Chung- to reduce the underlying mechanism into solely an active or Lu networks [49] with size N = 8000 and average degree z = 20. In a passive influence. For instance, it has been shown that both = . = . μ = ν = ε = ε = λ = λ = (b) fS 0 7, fI 0 1, 0, 4, S 2, 4, I 8, and active and passive social influences have strong effects on the α = . 0 08. In addition, an excitation threshold of 15% of the users is spreading of heavy drinking behaviors among college students = . = . μ = ν = ε = ε = λ = set to 13. In (c) fS 0 2, fI 0 2, 0, 4, S 1, 3, [61,62]. Likewise, successful modeling of the adoption of λ = α = . I 6, and 0 06. In addition, an excitation threshold of 39% of microfinance practices in rural villages considers the mech- the users is set to 8. anisms of information passing (passive) and endorsement (active) [8]. In the Supplemental Material we extend our versus an agent with purely susceptible peers, and (iv) the model to incorporate when both active and passive social influential and susceptible agents in the network are linked influences are simultaneously at play [44]. by real-world connections. Results from (i) and (ii) show Distinguishing the role of influential and susceptible indi- the distinct effect of the catalyzing effectiveness of influ- viduals is of fundamental importance when the social con- ential and susceptible agents on the contagion. Moreover, tagion is driven by active social influence. This is clearly scenario (iii) tells us that being influential is not equivalent to demonstrated in the Twitter network, where we observe how having susceptible peers. Finally, scenario (iv) demonstrates active social influence has led the influential and susceptible that the threshold and excitation models have very different individuals to assert themselves at different periods of the growth dynamics. While the threshold model displays a steady contagion: In the early stage, the susceptible individuals are

033125-10 SUSCEPTIBLE INDIVIDUALS DRIVE ACTIVE SOCIAL … PHYSICAL REVIEW RESEARCH 1, 033125 (2019) the key drivers of the contagion; when the interest in the immunity to infectious disease attack, while the influential contagion reaches a critical point, the influential individuals individuals are those that have a greater chance of spreading take over. the disease to others perhaps through bad habits or through In contrast, it is hard to distinguish the role played by behaviors. influential agents and susceptible agents when they operate together in the same social network through passive so- VI. METHODS cial influence. Indeed, the fraction of activated agents rises steadily in follower dynamics for the passive Digg network. A. Clustering coefficient Likewise, in our mathematical and simulation models, there An influential agent, a normal agent, and a suscepti- is no significant difference when we examine their catalyzing ble agent are marked with agent types  = 1,  = 0, and effectiveness in scenarios I and S, respectively. It is only when  =−1, respectively. We quantify the clustering among the the susceptible agents are highly clustered that their catalyz- agents by calculating the Pearson correlation between agent ing effectiveness differs significantly from that of influential type of the connecting pairs: agents. Nonetheless, susceptible users are found to have a ( − ¯ )( − ¯ ) tendency not to cluster in real-world social media networks R = x x y y . δ δ (8) [29]. x y A final remark is that we have ignored exogenous as- Here x and y denote agent types of the connected agents, sumptions with regard to our simulations on the spread of ¯ x is the mean of x, and δx is the standard deviation of x. contagion in the Digg and Twitter networks. For example, Note that · represents the operation of expectation. we have neglected the possibility that the activation threshold can change during the simulation, like a sudden change in the B. Identifying influential and susceptible members overall susceptibility of the population at the critical point of in social networks the Twitter dynamics, which might allow the threshold model to account for the observed Twitter dynamics. In addition, Truly identifying influence and susceptibility is con- targeted activations or instantaneous activation nuclei [63] founded by effects like homophily [51,52], which are beyond can also have significant impact on the speed of spreading. the scope of our data. In addition, our focus here is on However, without evidence of their presence in the contagion, contrasting results from mathematical models of active versus we have not considered them in our modeling. Note also that passive influence; thus we implement a basic operational the current model involves the setting of a large number of definition. For our analysis on the spreading process in the parameters, and we choose the values to either best highlight Digg network, an agent is identified as influential if more than the regimes of interest or match empirical data. It is an open 15% of its followers follow user A after the agent has followed question if the parameters correlate and thus if there is a way user A. With this, influential agents constitute around 12% to reduce their numbers. of the followers. An agent is identified as susceptible if it An outstanding challenge is to understand the microscopic follows user A after 10% or less of its peers do so. With interaction dynamics. For example, one may consider study- this, around 15% of the activated agents are identified as sus- ing specifically the detailed dynamics of how agents in the the ceptible agents. On the other hand, for the spreading process distinct classes infect one another, including potential back in the Twitter network, an agent is identified as influential if and forth between classes. At the moment, the data in our more than 10% of its peers retweet its message. This gives empirical networks are insufficient for such a study because rise to around 4% of influential agents among the activated there is no specific details on who infects who. Nonetheless, agents. In addition, an agent is identified as susceptible if it it allows us to determine the activation dynamics subject tweets the news after 40% or less of its peers do so. With to the collective influence of the agent’s neighbors, taking this, around 25% of the activated agents are identified as into account the agent’s susceptibility and also the degree susceptible agents. of influence of its neighbors. Future work would require a detailed data set yielding the precise causal influence between C. The L2 error individuals in order to determine the actual dynamics of the To compare the results of simulations with the data, we spreading. calculate the L2 error, which is defined as In conclusion, the relative importance of influential ver- sus susceptible individuals depends fundamentally on the = M − D 2, underlying mechanism driving the contagion. These are in eL2 fa (t ) fa (t ) (9) fact the necessary precursors to the development of a more t comprehensive approach to the controllability of complex M where fa (t ) is the fraction of activated agents at time t diffusion dynamics in social systems. Although our approach D simulated from the model and fa (t ) is the fraction of activated has mainly focused on the spreading of social influence, it agents at time t given by the data [60]. may have broad applications to other areas and contexts. One possible application is the spreading of infectious dis- ACKNOWLEDGMENTS eases under active influence with the spread of the epidemics being catalyzed by both influential and susceptible individ- The authors thank Keith Burghardt for a critical reading of uals. In this case, one could identify the children and the the manuscript. C.H.L. would like to thank the Complexity elderly to be the susceptible individuals due to their weaker Sciences Center at UC Davis for hospitality and acknowledge

033125-11 CHUNG, CHEW, CHEN, D’SOUZA, AND LAI PHYSICAL REVIEW RESEARCH 1, 033125 (2019) partial support from the Lai Teik Fatt Fund. N.N.C. would like tive, Grant No. W911NF-15-1-0502; NSF Grant No. CNS- to thank Andrew Johnathan Schauf for helpful discussions. 1302691; and DARPA Grant No. W911NF-17-1-0077. R.M.D. is grateful for support from the U.S. Army Research All authors contributed equally to all aspects of the re- Office MURI Award No. W911NF-13-1-0340 and Coopera- search. tive Agreement No. W911NF-09-2-0053; the Minerva Initia- The authors declare no competing financial interests.

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