Two is a Crowd - Optimal Trend Adoption in Social Networks

Transcription

1 Two is a Crowd - Optimal Trend Adoption in Social Networs Lilin Zhang, Peter Marbach Department of Computer Science University of Toronto {llzhang, marbach}@cs.toronto.edu Abstract In this paper, we study how an individual in a social networ should decide whether or not to adopt a trend, based on how many people in his/her neighborhood in the social networ adopted the trend. In particular we are interested in the question in what adoption policy leads to an optimal trend-adoption in the sense that an individual only adopts a trend if the majority of his/her social networ will do so. We use a decision process on a Erdös-Rényi rom graph model to model study this situation. Using this model, we obtain the result that the optimal policy for an individual is to adopt a trend if two people in their neighborhood did. Interestingly, this result/behavior was experimentally observed in real social networs. We hope that this wor will help towards building applications that is able to automatically push relevant content to users in online social networs. I. INTRODUCTION Online Social Networs have experienced an explosive growth over the recent a few years, that growth has accompanied abundant interest in the study of the spread of trends in social networs. In this paper, we study how an individual in a social networ should decide on whether or not to adopt a trend. We use the term trend quite broadly where a trend could for example be a fashion trend, a new technology, an idea, or a behavior. Two paradigms for modeling trend adoption in social networs have been proposed in the literature: the threshold model the cascade model. Both models are based on the natural assumption that individuals tend to be influenced by their acquaintances with whom they communicate. Yet the two are intrinsically modeling two different types of influence in social networs. In the threshold model, each individual observes the number of adoptions among his friends, only adopts the trend if the number of adoptions among his/her friends exceeds a particular threshold value. On the other h, in the cascade model, each individual can be convinced to adopt the trend through interactions with friends in the social networ who have already adopted the trend. However, two recent studies by Bacstorm el. [4] on friendship lins community membership on LiveJournal, the co-authorship networ in the computer science bibliography hosted by the Digital Bibliography & Library Project (DBLP), showed that neither the threshold model nor the cascade model is sufficiently realistic (for more details, see in Section 2). Hence in this paper, we re-investigate the problem of trend adoption in social networs, propose a new model that better predicts the observed behavior in real-life social networs. Our first contribution in this paper is thus a new hybrid model for studying how individuals are influenced by their acquaintances in social networs. The hybrid networ model considers the social networ population as two groups: one group in which individuals follow cascade process to decide whether or not to adopt; the other group in which users obey threshold strategy to mae the decision. The basic idea behind the hybrid networ is that, with respect to trend adoption, we can distinguish between two types of individuals: ) individuals who mae informed decisions 2) individuals who imitate. We refer to individuals who mae informed decisions as informed adopters. Informed adopters have (insider) nowledge about the trend to adopt decide whether or not to adopt based on information/discussion with other insiders who have already adopted. As a result, among the insider s group, the adoption process can be modeled by the cascade process. We refer to individuals who imitate as followers. Followers do not have enough nowledge about the trend so as to mae informed decisions, as a consequence, decide whether or not to adopt depending on how many other people already have adopted the item. Hence for the latter group, the adoption process can be captured by the threshold model. In addition, we consider an initial set of individuals who adopt the trend without being influenced by others. We refer to these individuals as trendsetters (early adopters). The trendsetters are important as they serve as seeds that enable the spread of the trend. Note that followers have no special nowledge about the trend therefore need to rely on their observation of what informed adopters do in order to decide on whether or not to adopt the trend. As such, the goal of a follower is to adopt the trend only if a large fraction of informed adopters does so, since this can be taen as an indication that the trend is indeed worth adopting. However, followers do not have a global view but only a local view, i.e. they can only observe what the informed adopters who are neighbors in their social networ do, but they can not observe the adoption behavior of all informed adopters. As such, the question we want to explore is whether it is possible to choose a threshold value that guarantees that followers mae the correct decision adopt the trend only if a majority of the informed adopters

2 does so, even though they have only local information. Our second contribution is to characterize a optimal threshold strategy for the followers. More precisely we show that is optimal for followers to adopt a trend as soon as two friends in their social networ adopted the trend. Interestingly, this behavior was observed in the Bacstorm el. [4] on friendship lins community membership on LiveJournal, the coauthorship networ in DBLP. To the best of our nowledge, it is the first time that a mathematical model is proposed to address reassemble the real-life influence propagation pattern. The rest of the paper is organized as follows. In Section 2, we discuss related wor of empirical studies on real social networs. Section 3 presents our mathematical model, problem statement numerical results. In Section 4, we propose a tractable model. In Section 5, we perform the analysis to address the choice of threshold value. Finally we conclude discuss the future wor in Section 6. Fig.. The probability p of joining a LiveJournal community as a function of the number of friends already in the community. Error bars represent two stard errors [4]. II. RELATED WORK In this section, we provide the related wor of empirical studies concerning threshold strategy on real social networs. The subject of threshold is not new. In fact, it is well used in various fields of science [3]: as a method of segmentation in image processing, as a model in toxicology, as a phenomenon in particle physics, etc. However, the mathematical study of the threshold strategy the optimum threshold value on social networs is the first time in the paper. Bacstorm el. [4] studied two large sources of data: friendship lins community membership on LiveJournal, the co-authorship networ in DBLP. They plotted the individual s probability p of joining a LiveJournal/DBLP community as a function of the number of friends,, already in the community (Fig, Fig2). Both plots exhibit qualitatively similar shapes in which p continues increasing, but more more slowly as increases. We tae a closer loo at Fig., since the error bars are smaller here. The probability of joining LiveJournal has a close fit to a logistic function for = 0,, 2, with a rapid growth at = 2. After this point, the probability of joining the community has a close fit to a logarithmic function. For the largest jump in p at = 2, Bacstorm el. claimed that the marginal benefit of having a second friend in a community is particularly strong. Such a curve suggests the individual s strategy of a threshold pattern: individuals have a low tendency to tae an action if not many acquaintances did so; then the tendency increases very fast if a certain number of acquaintances have done so; beyond this point the tendency still increases, but does so slower slower. The action herein is an abstract notation, which could be joining a community (Fig.), or attending another conference (Fig.2). Though the empirical study of Bacstorm el. reveals the individuals threshold strategy, their research focus was on quite a different subject: the evolution of large social networs. Fig. 2. The probability p of joining a DBLP community as a function of the number of friends already in the community. Error bars represent two stard errors [4]. One variation of the above threshold strategy is in the context of voluntary vaccination [5]: once a sufficient proportion of the population is already immune, either naturally or by vaccination, people tend to sip vaccination under the voluntary policy due to morbidity riss of the vaccination itself. That is individuals tend to avoid taing an action once a threshold is reached. Bauch el. [5] performed a game theoretical analysis based on Nash equilibrium to explain the threshold strategy with respect to vaccination. However, they need to assume that all individuals are provided with the same information use this information in the same way to assess riss. Here this paper lifts the assumption of synchronized global information, considers the situations where individuals have social contacts with their friends, thus have merely local observations. And individuals mae decisions based on the limited information. To the best of our nowledge, our wor provides a unique approach to choose a threshold value so that individuals can mae correct choices based on local observations.

3 III. MODEL AND PROBLEM STATEMENT In this section, we introduce our hybrid model for trend adoption in social networs, state the question of interests, provide a numerical case study to illustrate our result. Recall that our scenario of trend adoption in the social networ: individuals can observe whether their friends have adopted the trend or not, mae the decision based on these observations. We refer to an individual who adopted the trend as active, to an individual who has not adopted the trend as non-active, or inactive. Note that individuals only go from non-active to active status, but not from non-active to active. A. Social Networ Graph Our model consider two main groups in the social networ population. One is the informed adopter group consisting of individuals who have nowledge about the trend mae informed decision based on information/discussion with other insiders who have already adopted. For this group, we model their user behaviour as cascade process. The other is the follower group consisting of individuals who don t have enough nowledge about the trend to mae informed decisions, as a consequence, decide whether or not to adopt by imitating their acquaintances who already have adopted the trend. For the second group, we presume they obey threshold strategy. We thin of a social networ as a collection of individuals who have connections with one another in social bonds such as friendship, inship, or relationships of beliefs nowledge [7]. We use a graph G to represent the entire social networ. As part of the social networ graph G, we define a undirected subgraph G of G representing the informed adopters directed subgraph G 2 representing the followers. In the rest of the paper, we will use the terms individuals nodes/vertices interchangeably. More precisely, we use the following Erdös-Rényi [] rom graph model. We denote with V (G ) the vertex set of G given by V (G ) = {, 2, 3,, }, where denotes the total number of informed adopters in the social graph G. Each node in V (G ) represents an informed adopter in the social networ. Between any pair of nodes i, j V (G ), with probability p (0, ] there exists an undirected edge between two informed adopters i j in G, independent of everything else. Let E(G ) be the resulting edge set of G given by E(G ) = {e i,j i, j V (G )}. Each edge in E(G ) sts for the social connection within G. The average node degree λ in G is then given by λ = p we have the relation p = λ. () Note that λ represents the average number of informed adopters a node in G communicates with. Let V (G 2 ) the vertex set of G 2 given by V (G 2 ) = {, 2, 3,, n 2 }, where n 2 denotes the total number of followers in the social graph G. Each node in V (G 2 ) represents a follower in the social networ. We assume that with probability p 2 (0.] there exists a directed edge e(i, j) between a follower i in G 2 an informed adopter j in G, independently of everything else. Let E(G 2 ) be the edge set of nodes in G 2. Each edge in E(G 2 ) sts for the social connection between a follower in G 2 an informed adopter in G, i.e. we have E(G 2 ) = {e i,j i V (G ), j V (G 2 )}. The average node degree λ 2 in G 2, i.e. the average number of informed adopters that a follower in G 2 nows, is then given by λ 2 = p 2 we have the relation p 2 = λ 2. Using the above notation, the graphs G G 2 are characterized by the parameters (, λ ) (n 2, λ 2 ), respectively. Accordingly, we use the notation G (, λ ) G 2 (n 2, λ 2 ) to characterize the two graphs. Aggregated, let e(g) be the rom variable denoting the number of edges in G. e(g) = E(G) And let N(v) be the set of immediate neighbours of v, which sts for the set of friends, relatives or acquaintances of v on the social networ. N(v) = {w w V (G), e v,w E(G)} B. Asymptotic Behavior In the following we are interested in the asymptotic behavior as the social graph becomes large. As such, we consider an infinite sequence of social graphs denoted by n =, 2,..., let G (, λ ) G 2 (n 2, λ 2 ) be functions of n; i.e. the parameters (, λ ) (n 2, λ 2 ) of G G 2, respectively, depend on n. We assume that n 2 become large (approach infinity) as n increases. Assumptio. We have lim (n) = lim n 2(n) =.

4 To eep the notation light, in the following we typically do not explicitly state the dependency of parameters such as (, λ ) (n 2, λ 2 ) on n. Note that the above implies that the edge probabilities p p 2 are also functions of n. For our analysis, we mae the following assumption on p 2 (n). Assumption 2. We have lim p 2(n) = λ 2(n) nto (n) = 0. The above assumption states that the followers can only observe a vanishingly small fraction of the informed adopters. C. Trendsetters In addition to the informed adopters followers, we consider a third group of individuals, namely the trendsetters or early adopters. Trendsetters adopt the trend without being influenced by others in the social networ. Trendsetters serve as seeds that enable the spread of the trend. In our model, the trendsetters are given by a subset A of the informed adopters G, i.e. A is the the initial active set within the insiders group, G. This size of the set A can depend on n we mae the following technical assumption on the size of the set A (n), i.e. the total number of trendsetters as a function of n. Assumption 3. There exist positive constants ɛ, c [0, 2 ) such that lim A (n) > ɛ lim A (n) (n c ). (2) We restrict the size of A (n) because we want to focus on the case where the numbers of trendsetters does not dominate the total number of informed adopters. D. Trend Adoption We consider the following policies for informed adopters followers to adopt a trend. Informed Adopters Influence Cascade: Informed adopters in G are influenced by the decisions made by other informed adopters who are within one hop friendship on G (, λ ). We model this process using the independent information cascade model that was introduced to study the problem of the maximum influenced set in social networs [6], [8], [9]. More precisely, we divide the set E(G ) into two disjoint subsets of open bloced edges. An edge e(i, j) E(G ) is open with probability ρ [0, ], independent of everything else; otherwise it is bloced. The trend adoption then only propagates over open edges, i.e. if node i has adopted the trend node e(i, j) is an open edge, then node j will also adopt the trend [6], [8], [9]. Followers Threshold Strategy: Here we assume that followers use a threshold policy in order to adopt the trend, i.e. a follower will adopt the trend only if the number of informed adopters that adopted the trend exceeds a given threshold value. More precisely, let t a be a threshold value used by the followers in G 2 (n 2, λ 2 ) in order to decide whether or not to adopt the trend. A follower then adopts the trend only if he observes t a number of adoptions among his one-hop friends. E. Adoption Process The trend adoption process then proceeds as a discrete time. At time t = 0, only the trendsetters A in G have adopted the trend, i.e. are active in G. The trend then spreads from A as follows. When a node i V (G ) first becomes active in step t, it is given a single chance to activate each currently nonactive neighbour j N(i), it succeeds with probability ρ. If i succeeds, then j will become active in step t +. The cascade terminates when no node has become newly active at a given time step. Followers V (G 2 ) in G 2 observe whether their one-hop friends in G adopt the trend as soon as t a of their friends have adopted the trend, i.e.have become active. F. Problem Statement In the above scenario of the individual adoptions on social networs, we are interested in the question of an optimal threshold decision policy, which allows followers to mae the correct decision in the sense that they only adopts a trend if the majority of informed adopters in G will adopt the trend. Thus we are interested in which threshold value t a to choose such that followers will almost always mae the correct decision to conform with the majority of individuals in G. Note that the threshold decision policy merely maes use of individuals local observations. It is well nown [] that (asymptotically) a majority of informed adopters in G will adopt the trend only if lim ρ λ > only a small fraction of informed adopters in G will adopt the trend if lim ρ λ <. Therefore, the question we want to study is whether there exists a threshold value t a that ensures that (asymptotically) no follower will adopt the trend in the case where lim ρ λ <, each follower will adopt the trend with positive probability if lim ρ λ >. G. Main Result Our main result is to prove that choosing the threshold value t a = 2 is optimal for followers in the sense that the threshold value t a = 2 ensures that a follower never maes a mistae by adopting a trend that is adopted only by a small fraction of the informed adopters, while maximizing the probability that a follower adopts the trend when a majority of the informed adopters does so. More precisely, we obtain the following result. Consider a follower i in G 2 let the rom variable K i denote the number of active neighbours of i after the cascade process

5 the fraction of threshold nodes that adopt O000 nodes( ) λ ρ Fig. 3. Double jump at λ ρ =. has finished, i.e. the numbers of active informed adopters in G to whom follower i is connected in the social graph G. Conditional on λ ρ, let F () be the probability there exists at least one node in G 2 who has at least number of active neighbours in G, i.e. F () is given by F () = P r( i V (G 2 ), such that K i λ ρ < ). (3) In our main result in Section 5 we show that lim F () > 0 lim F () = 0, for = 2, 3,. Note that this result implies that under a any threshold value t a 2, no follower will mae the mistae to adopt a trend if only a small fraction of the informed adopters does so. This is not true for the threshold value t a =, i.e. under the threshold value t a = some followers will mae a the mistae adopt the trend even though only a small fraction of the informed adopters does so. Hence, using a threshold t a 2 is safe as ensures that followers never mae a mistae. Moreover, among all value t a 2 the value t 2 = 2 is optimal in the sense that it maximizes the probability of adopting the trend when a majority of the informed adopters does so. In particular, one can show that under the threshold t a = 2 followers will adopt the trend with a strictly positive probability when a majority of the informed adopters adopt the trend. H. Numerical Results To illustrate our result, we provide a numerical case study. First, we plot the fraction of nodes in G 2 that adopts the trend as the product of λ ρ increases (Fig. 3). This leads to an important observation, nown as the double jump [2]. Furthermore, in order to examine closely the relation between individual adoption behaviour their neighbourhood adoption status, we plot Fig. 4, in which we find that the fraction of nodes who adopt the trend has a sudden rise with two active neighbours presenting. Interestingly, this observation was experimentally observed in real social networs [4]. This is one important feature of our model. As fraction of nodes that adopt probability to adopt at x number of active neighbours presenting on 2000 nodes( ) λ =0, ρ = 0.5, A = 8, λ 2 = 4, threshold = 2 aggregated on G insiders on G followers on G x: number of active neighbours on 2000 nodes( ) λ =0, ρ = 0.5, A = 8, λ 2 = 4, threshold = 2 aggreated on G insiders on G followers on G number of active neighbours Fig. 4. Second-adoption phenomenon. before, no propagation models can reproduce this secondadoption phenomenon (Fig., 2) that appears in real life. IV. EQUIVALENT MODEL In this section, we propose the equivalent model of G in Section 3, G (, n 2, λ ρ, A, λ 2, t a ). In particular, we shall prove that the size of the influenced set produced by ρ -cascade process over G (, λ, A ) has the same distribution as that of the influenced set on G (, λ ρ, A ). Before proceeding, we need to define the influenced set I on G. Let the influenced set be the set of vertices connected to A via open edges in G, i.e., I = {v v V (G ), v connects to A via open edges} Hence by the end of the influence cascade process, all the nodes in I become active. The equivalent model G is a Erdös-Rényi graph. G has the same vertex set as G, but with a different edge preserving probability p = p ρ, where p is defined in (3.), the edge preserving probability of G, the success probability of the influence cascade process. G have the same initial active set as G, A, subjecting to the same size constraint.

6 Now we define the influenced set I on G. Let the influenced set on G be the set of vertices connected to A via edges in G, i.e., I = {v v V (G ), v connects to A via edges e E(G )} The following theorem guarantees that on both models the distribution in the size of the influenced set is the same. To prove that unfolding the influence cascade by determining the open/bloc status of edges gradually at each time step is the same as unfolding the influence cascade by determining the status of all the edges at the beginning during the construction of G, the idea is to argue that the distribution for a rom node joining the influenced set is the same over time steps in both models. Theorem. Given an initial active set A, the distribution of the influenced sets obtained by cascade process on G starting from A, is the same as the distribution of sets reachable from A via edges on G. Proof: First we argue the iterations of the influence cascade process over G. Define A (t) to be the set of active nodes at the end of iteration t, for t = 0,, 2, 3,. If a rom node v has not become active by the end of iteration t, then under the cascade process, the probability that v will become active in iteration t + is equal to the chance that one of its active neighbours in A (t) but not in A (t ) succeeds in activating v. This probability is P (v A (t+) v / A (t) ) = ( w A (t) ρλ ). \A(t ) Second we construct G gradually as follows. We start with the initial set A. For each node v with at least one edge stub, we determine whether it connects to A. If so, then v is reachable; if not, we eep the source of v s edge unnown subject to the condition that it comes from outside of A. Having now exposed a new set of reachable nodes A () in the first time step, we proceed to reveal further reachable nodes by performing the same process on edges from A, in this way produce sets A (2), A(3), A(4),. If node v has not been determined to be reachable by the end of time step t, then the probability that it is determined to be reachable in time t + is equal to the chance that its ; this probability is edge comes from A (t) but not in A (t ) A (t+) v / A (t) ) = w A (t) \A(t ) ( ρλ ). Thus, by induction over the iterations, we see that the cascade process produces the same distribution over influenced sets as the construction of G. Since we have established the equivalence between the G (, λ, ρ, A ) in Section 3 G (, λ ρ, A ) with respect to the distribution in the size of the influenced set, G is essentially a Erdös-Rényi graph. V. ANALYSIS In this section, we derive in detail the answer to the question of the optimal trend-adoption policy on G (, n 2, λ ρ, A, λ 2, t a). We shall specifically consider the adoption policy in subcritical phase, obtain the optimal policy. A. Decision Policy in Subcritical Phase When p = p ρ [0, ), G is in subcritical phase. We first derive the probability, u(), for a rom vertex in G 2 (n 2, λ 2 ) possessing active neighbours. Then by maing use of this formula, we derive the conditional probability, F (), for existing any node in G 2 (n 2, λ 2 ) with at least active neighbours. And we thus prove that in the subcritical phase the correct decision policies are threshold strategies with threshold value: t a = 2, 3,. Let P a be the probability for a rom node in G being active after the influence cascade terminates. Then P a is given by: P a = I (4) In order to quantify P a, we mae use of the expectation of I, which can be expressed in closed form using percolation theory on rom graphs. Percolation property of rom graphs has been extensively studied in the field of epidemics spread [5], [6], [7]. It studies whether a disease will dominate the whole population. If the disease does not percolate the whole networ, it quantifies the expected value of the size of the infected population. The main approach used is generating functions [8]. By the same approach, we are now deriving E( I ) on Erdös-Rényi rom graph, G n,p, with percolation probability ρ. Theorem 2. On Erdös-Rényi rom graph G(n, p = λ n ), the expectation of the size of the influenced set with one vertex percolated initially is ρ ρ λρ. Proof: Let p be the probability for a rom vertex having degree. p is given by: ( n p = )p ( p) n λ e λ. (5)! Let G 0 (x) be the generating function of {p }, which is given by: G 0 (x) = p x 0 Follow a rom edge reach a vertex v. Let {q } be the degree distribution of so-reached vertex v. Let G (x) be the generating function of {q }, which is given by: G (x) = 0 q x A rom edge has probability e(g) to be chosen by the uniform choosing policy, where e(g) denotes the total number of edges in G. Conditional on v having degree, the chosen edge has probability being incident to v, i.e., e(g) P ( the chosen edge incident to v deg(v) = ) = e(g) (6)

7 By Total law of probability, from (5, (6) we have: P (the chosen edge incident to v) = p e(g). (7) Thus by Bayes rule, from (5), (7) we have q which is given by: q = P (deg(v) = the chosen edge incident to v) = p p = p (8) z where z is the average vertex degree. And in the case of Erdös-Rényi rom graph z = λ on our model. Let p m be the probability for a rom vertex having m open edges. Let G 0 (x; ρ) be the generating function of { p m }, which is given by: Thus P (m edges of v are open deg(v) = ) ( ) = ρ m ( ρ) m binomial distribution m p m = P (m edges of v are open) = ( ) p ρ m ( ρ) m m G 0 (x; ρ) = ( ) p ρ m ( ρ) m x m m m 0 0 = ( ) p (xρ) m ( ρ) m m 0 m 0 = p ( ρ + xρ) 0 G 0 (x; ρ) = G 0 ( ρ + xρ) (9) Now follow a rom edge reach a vertex v. Let q m be the probability for so-reached v having m open edges. Let G (x; ρ) be the generating function of { q m }. Similar with the derivation of (9), G (x; ρ) is given by: G (x; ρ) = G ( ρ + xρ) (0) Let Z be a rom variable denoting the size of influenced set connected to a rom vertex. Define H 0 (x; ρ) be generating function for the distribution of Z. Let Z be a rom variable denoting the size of influenced set connected to a rom edge. Define H (x; ρ) be generating function for the distribution of Z. The influenced components generated by H 0 (x; ρ) consist of the initial infected node, plus any number of tree-lie clusters, joined to it by single edges. Each tree-lie cluster has the size distribution generated by H (x; ρ). Thus we have: H 0 (x; ρ) = xq 0 + xq H (x; ρ) + xq 2 [H (x; ρ)] 2 + xq 3 [H (x; ρ)] i.e., i.e., H 0 (x; ρ) = xg 0 (H (x; ρ); ρ) H (x; ρ) = x q 0 + x q H (x; ρ) + x q 2 [H (x; ρ)] H (x; ρ) = xg (H (x; ρ); ρ) () The expected value of Z is given by differentiating H 0 (x; ρ), i.e., E[Z] = H 0(; ρ) = + G 0(; ρ)h (; ρ) (2) Differentiating equation (), we have: H (; ρ) = + G (; ρ)h (; ρ) = G (; ρ) (3) substitute (3) into (2), mae use of (8),(9), (0), we have: E[Z] = + ρg 0() ρg () (4) We notice G 0() = 0 With (8), we also have p x x= = 0 p = λ (5) G () = 0 q x x= = 0 2 p z = E(K2 ) z = V ar(k) + E2 (K) z = + λ = λ + λ2 λ (6) where K in (6) is a r.v. denoting the vertex degree of a rom node in Erdös-Rényi rom graph G(n, p = λ n ). Thus, K has approximately Poisson distribution with E(K) = λ V ar(k) = λ. Substituting (5) (6) into (4), we have E[Z] = ρ ρ( + λ) By Theorem 2, we have the expectation of the influenced set given by: E( I ) = ρ ρ (λ + ) for A = Further for A n c for some, we have E( I ) ρ ρ (λ + ) nc = n c, for some. If the influenced sets associated with each node in A are independent, the sum of the influenced sets has the ρ expectation of ρ (λ +) nc. The inequality comes from the possibility that those influenced sets overlap with one another.

8 By Marov inequality Theorem 2, we have an upper bound of I given by: P r( I n 2c ) E( I ) n 2c nc n 2c = n c 0 for large Since we have restricted the size of A by assuming that 0 c < 2 in (2), then lim P ( I n c ) = 0, where c [0, ). By Theorem, we have the same bound of I : lim P ( I n c ) = 0, where c [0, ) (7) Alternatively, (7) can be written as: lim P ( I < n c ) = (8) Now we bound P a by substituting (8) into (4): P a = I < nc = 0 with probability for large (9) Equation (9) has an intuitive interpretation. In the subcritical phase of G (, λ ρ, A ), we romly pic a node when the influence cascade terminates. Such a node is non-active with probability for large networs. The following theorem provides the distribution for a rom node in G 2 (n 2, λ 2 ) having n a number of active neighbours. Theorem 3. On our model G (, n 2, λρ, A, λ 2, t a ), we have P (K v = n a ) = (λ2pa)na n a!e λ 2 Pa, where P a is given by (4). Proof: First, we derive the degree distribution of a rom node in G 2 (n 2, λ 2 ). Let p be the probability for a rom node v G 2 having neighbours. p satisfies binomial distribution is given by: ( ) n p = p 2( p 2 ) n where p 2 = λ2. We apply Poisson approximation as goes to infinity: ( ) n p = p 2( p 2 ) n λ 2e λ2 (20)! Next we derive the distribution of the number of active neighbours a rom node in G 2 possess. On the condition that a rom node v has degree, the probability for v having n a active neighbours also satisfies binomial distribution: ( ) P (K v = n a deg(v) = ) = Pa na ( P a ) na (2) where P a is the probability for a rom node being active, as defined in (4). By Total law of probability, we remove the condition in (2), have the probability for v having n a active neighbours given by: P (K v = n a ) = By substituting p n a p ( n a )P na a ( P a ) na (22) in (20) into (22) applying Taylor expansion [20], we have P (K v = n a ) = = λ 2e λ2!! ( n a )!n a! P a na ( P a ) na = e λ2 Pa na λ 2( P a ) n a!( P a ) na ( n a )! = e λ2 Pa na n a!( P a ) [λ 2( P a )] na e λ2( Pa) na = (λ 2P a ) na n a!e λ2pa Next, we use Theorem 3 to determine threshold decision policy. In particular, we define F () to be a conditional probability. It denotes the probability that there exists one G 2 node having at least number of active neighbours, conditional in subcritical phase. The next theorem quantifies F (), which in turn suggests that if choosing t a 2, with probability, no nodes in G 2 mae a wrong decision. Theorem 4. We have lim F () > 0 lim F () = 0, for = 2, 3,. Proof: To see whether there exists a follower node in G 2 having at least one active insider-neighbour in subcritical phase, we compute lim F (): lim F () = lim [ P r(k v )] n2 = lim [P r(k v = 0)] n2 = lim [(λ 2P a ) 0 0!e = lim = λ2pa ]n2 I [(λ 2 ) 0 ] n2 0!e λ2 I e λ2 I n 2 > 0 To see whether there exists a node in G 2 having at least two active neighbours in subcritical phase, we compute lim F (2): lim F (2) = lim [ P r(k v 2)] n2 = lim [P r(k v = 0) + P r(k v = )] n2 = lim [(λ 2P a ) 0 0!e + (λ 2P a ) λ2pa!e = lim I [(λ 2 ) 0 0!e λ2 I = lim ( + λ2 I + (λ 2 λ2pa ]n2 I ) ] n2!e λ2 I ) n e λ2 I (23)

9 To further reduce (23), let m = n λ. Note that lim 2 I m =. Thus we reduce the nominator in (23) as follows: lim ( + λ 2 I ) = lim {( + λ 2 I ) n λ 2 I } λ2 I = lim m [( + m )m ] λ2 I = e λ2 I (24) In the derivation of (24), we mae use of the common limit [2]: lim m ( + m )m = e Substitute (24) into (23). We have: lim F (2) = lim = eλ2 I e λ2 I = 0. ( + λ2 I ) n2 e λ2 I (25) (25) says that there exists no node having 2 or more active neighbours in subcritical phase. It generalizes that there exists no node having 3 or more active neighbours in subcritical phase, so forth. Thus we have lim F () = 0, for = 2, 3,. Theorem 4 suggests the decision policy for followers in subcritical phase. The first part of the theorem says that the threshold value should not be t a =. Since lim F () > 0 suggests that the probability is positive for existing at least one node having active neighbour by the end of influence cascade process. If we choose t a = as the threshold value, then by the strategy the nodes with one active neighbour shall adopt thus mae the wrong decision in subcritical phase. The second part says that the threshold value could be t a = 2, 3,. Since lim F () = 0 for = 2, 3, says the probability tends to zero for existing any nodes having or more active neighbours. In words, with probability one there is no node with or more active neighbours for 2. Hence these choices of can guarantee all the follower nodes in G 2 not to adopt thus mae the correct decision in the subcritical phase. Moreover, one can show that under the threshold t a = 2 followers will adopt the trend with a strictly positive probability when a majority of the informed adopters adopt the trend (i.e. when lim λ ρ > ); the larger the fraction of informed adopters who adopted the trend, the higher the probability that a follower will do so. In this sense, the threshold value t a = 2 the optimal value. VI. CONCLUSIONS In this paper, we study how an individual in a social networ should decide whether or not to adopt a trend, based on how many people in their neighborhood in the social networ adopted the trend. In particular we investigated the question in what adoption policy leads to an optimal trend-adoption in the sense that an individual only adopts a trend if the majority of their social networ will do so. we obtain the result that the optimal policy for an individual is to adopt a trend if two people in their neighborhood did. Our analysis is mainly done on the Erdös-Rényi rom graph model. Though our wor is based on an oversimplified model, it indeed shed some light on how individuals mae choices in real life. this result/behavior was experimentally observed in real social networs. The wor of Bacstrom el. [4] noticed the special effect of the second friend claimed that the marginal benefit of having a second friend in the community is particularly strong. Here in our paper, we prove the optimality of the threshold value of 2, which coincides with the discovery of the special second adoption found in [4].We hope that this wor will help towards building applications that is able to automatically push relevant content to users in online social networs. REFERENCES [] access on March. 5th, 200. [2] Aronson, E., Wilson, T.D., Aert, A.M. Social Psychology(6th edition). Upper Saddle River, NJ: Pearson Prentice Hall [3] access on March. 8th, 200. [4] L. Bacstrom, D. Huttenlocher, J. Kleinberg, X. Lan. Group Formation in Large Social Networs: Membership, Growth, Evolution. Proc. 2th ACM SIGKDD Intl. Conf. on Knowledge Discovery Data Mining, [5] C.T. Bauch D.J.D. Earn. Vaccination the theory of games. Proceedings of the National Academy of Sciences 0: [6] D. Kempe, J. Kleinberg, E. Tardos. Maximizing the Spread of Influence through a Social Networ. Proc. 9th ACM SIGKDD Intl. Conf. on Knowledge Discovery Data Mining, [7] D. Easley, J. Kleinberg. Networs, Crowds, Marets: Reasoning About a Highly Connected World. To be published by Cambridge University Press, 200. [8] D. Kempe, J. Kleinberg, E. Tardos. Influential Nodes in a Diffusion Model for Social Networs. Proc. 32nd International Colloquium on Automata, Languages Programming (ICALP), [9] J. Kleinberg. Cascading behaviour in Networs: Algorithmic Economic Issues. In Algorithmic Game Theory (N. Nisan, T. Roughgarden, E. Tardos, V. Vazirani, eds.), Cambridge University Press, [0] Hacing, Ian. Logic of statistical inference. Cambridge, University Press, 965. [] B. Bollobs. Rom graphs(2nd edition). Cambridge University Press, 200. [2] P. Erdos, A. Renyi. On the Evolution of Rom Graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5(960), 7-6. [3] R. Durrett. Rom graph dynamics. Cambridge University Press, [4] J. Spencer. The strange logic of rom graphs. Springer, c200. [5] M. E. J. Newman. The spread of epidemic disease on networs. Phys. Rev. E 66, [6] M. E. J. Newman, S. H. Strogatz, D. J. Watts. Rom graphs with arbitrary degree distributions their applications. Phys. Rev. E 64, [7] M. E. J. Newman. Rom graphs with clustering. Phys. Rev. Lett. 03, [8] Herbert S. Wilf. Generating functionology. Wellesley, Mass. : A K Peters, 3rd ed [9] NIST/SEMATECH, Counts Control Charts, e-hboo of Statistical Methods, March 200. [20] Greenberg, Michael, Advanced Engineering Mathematics (2nd ed.), Prentice Hall, ISBN , 998. [2] Catherine Roberts, Ray McLenaghan. Continuous Mathematics in Stard Mathematical Tables Formulae. ed. Daniel Zwillinger. Boca Raton: CRC Press (996): 333, 5. Differential Calculus