Di usion on Social Networks Matthew O. Jackson y Caltech Leeat Yariv z Caltech Current Version: June 6, 2006 Appeare in: Économie Publique, Numéro 16, pp 3-16, 2005/1. Abstract. We analyze a moel of i usion on social networks. Agents are connecte accoring to an unirecte graph (the network) an choose one of two actions (e.g., either to aopt a new behavior or technology or not to aopt it). The return to each of the actions epens on how many neighbors an agent has, which actions the agent s neighbors choose, an some agent-speci c cost an bene t parameters. At the outset, a small portion of the population is ranomly selecte to aopt the behavior. We analyze whether the behavior spreas to a larger portion of the population. We show that there is a threshol where tipping occurs: if a large enough initial group is selecte then the behavior grows an spreas to a signi cant portion of the population, while otherwise the behavior collapses so that no one in the population chooses to aopt the behavior. We characterize the tipping threshol an the eventual portion that aopts if the threshol is surpasse. We also show how the threshol an aoption rate epen on the network structure. Applications of the techniques introuce in this paper inclue marketing, epiemiology, technological transfers, an information transmission, among others. JEL classi cation: C45, C70, C73, D85, L15. Keywors: Di usion, Social Networks, Tipping, Technology Aoption, Coorination. An earlier version of this work was presente in a lecture at the Public Economic Theory Meetings 2005 in Marseille. We thank the organizers: Nicolas Gravel, Alain Trannoy, an Myrna Wooers, for the invitation to present this work there, an Laurent Methevet for his helpful comments. We are also grateful for nancial support from the Center for Avance Stuies in the Behavioral Sciences, the Lee Center for Avance Networking, an the Guggenheim Founation. y Division of the Humanities an Social Sciences, Caltech, Pasaena, CA 91125. http://www.hss.caltech.eu/jacksonm/jackson.html e-mail: jacksonm@hss.caltech.eu z Division of the Humanities an Social Sciences, Caltech, Pasaena, CA 91125. http://www.hss.caltech.eu/~lyariv/inex.htm e-mail: lyariv@hss.caltech.eu
Diffusion on Social Networks 1 1. Introuction An iniviual s ecision to aopt a new behavior often epens on the istribution of similar choices the iniviual observes among her peers, be they friens, colleagues, or acquaintances. This may be riven by unerlying network externalities, as in a ecision to use a new technology such as a new operating system or a new language, where the bene ts of the new technology are larger when more of an agent s acquaintances have aopte the technology. It may also be an artifact of simple learning processes, where the chance that an iniviual learns about a new behavior or its bene ts is increasing in the number of neighbors who have aopte the behavior. For instance, ecisions regaring whether to go to a particular movie or restaurant, or whether to buy a new prouct, provie examples of situations in which information learne through friens an their behavior are important. Of course, there are many other potential channels by which peer ecisions may have signi cant impact on iniviual behavior. The starting point of our analysis is the observation that in all such environments, the extent to which a new behavior spreas throughout a society epens not only on its relative attractiveness or payo, but also on the unerlying social structure. In this paper, we analyze how social structure in uences the sprea of a new behavior or technology. We consier a binary choice moel with two actions: A an B: We prescribe action A to be the status quo. Agents aopt the new behavior B only if it appears worthwhile for them to o so. This epens on the costs an bene ts of the action, an how many of an agent s neighbors have aopte behavior B. The cost an bene ts of aopting the action B i er ranomly across agents. The novelty of the moel arises from the speci cation of the social interactions that each agent experiences. Here we work with a stylize moel of a social network. Each agent has some number of neighbors. These are the people that (irectly) in uence the agent s ecision. Di erent agents in the society may have i erent numbers of neighbors. This number of neighbors is terme the agent s egree. The game is therefore escribe by two istributions: one corresponing to the bene ts of the behavior B an one corresponing to
Diffusion on Social Networks 2 number of neighbors that each agent has. At the outset of the process, a fraction x 0 of agents is ranomly assigne the action B while all other players use the action A: For instance, this coul metaphorically be thought of as a free trial perio of the new technology. At each perio, each agent myopically best respons to her neighbors previous perio s actions. The goal of the paper is to characterize the evolving ynamics an its epenence on the unerlying network structure. There are three main insights that come out of our inquiry. First, we show the existence of a smallest x 0 that is su cient for such ynamics to lea to an increase in the number of B aopters over time. That is, we ientify a tipping point beyon which the action B becomes more prominent, i.e., i uses in the population. Secon, for a class of cost-bene t istributions of the action B we can escribe the shape of the i usion processes. The uniform istribution serves as a goo example. In that case, the spee of increase in the number of B aopters increases up to a certain point in time at which the spee begins to consistently ecrease. Thir, we show how the i usion of behavior changes as we change the structure of social interaction. That is, we perform comparative statics pertaining to the tipping point as well as the ultimate convergence point of the i usion ynamics, with respect to the network structure. We examine two sorts of changes to the structure of social interaction, one where agents are given more neighbors (in the sense of rst orer stochastic ominance of the egree istribution) an a secon where the heterogeneity of egrees, or connecteness, in the population increases (in the sense of secon orer stochastic ominance of the egree istribution). Our results can be taken as a metaphor for many applie problems. In marketing, the results provie a step towar unerstaning when the aoption of a new technology or prouct by only few consumers leas to a fa, as a function of the unerlying social structure (for several popular examples, see Glawell (2000)). In criminology, the results avance the theoretical founations for unerstaning how crime spreas or vanishes (Glaeser, Sacerote an Scheinkman (1996) show the importance of social structures for criminal behavior). In nancial markets, the results may be useful in unerstaning the evolution of partial bank
runs an other sorts of her behavior. Diffusion on Social Networks 3 There have been several moeling eneavors pertaining to i usion processes relate to the one evelope here. The rst prominent stran of literature that relates to our analysis comes from the el of epiemiology (e.g., see Bailey (1975)). The type of question that arises in that literature regars the sprea of isease among iniviuals connecte by a network, with some recent attention to power-law (aka scale-free) egree istributions (e.g., Pastor- Satorras an Vespignani (2000, 2001), May an Lloy (2001), an Dezso an Barbasi (2002)), but also some analysis pertaining to other classes of egree istributions (e.g., Lopez-Pintao (2004), Jackson an Rogers (2004)). The secon, an relate, stran of research comes from the computer science literature regaring the sprea of computer viruses (see, for instance, the empirical observations in Newman, Forrest, an Balthrop (2002)). 1 The moel from these two strans closest to ours is the so calle Susceptible, Infecte, Recovere (SIR) moel. In that moel, susceptible agents can catch a isease from infecte neighbors an, once infecte, eventually either recover or are remove from the system an no longer infect others. There are several stuies examining the sprea of such iseases as it relates to network structure (e.g., Newman (2002)). These i er from our moel, approach, an results in three notable ways. First, in our moel agents make strategic choices about behavior in contrast to being ranomly assigne an attribute (such as being infecte). These choices epen on relative costs an bene ts to behavior as well as on the proportion of neighbors choosing i erent behaviors. This i ers in structure from inepenent infection probabilities across links that is assume in the epiemiology literature (although it permits it as a special case). It also leas to stark i erences in propagation ynamics. Inee, in the epiemiology literature it is enough to have a single infecte neighbor for one to catch a isease, whereas our setup allows for a change in behavior to epen on the fraction of neighbors (for example, making aoption of a new behavior optimal if an only if the percentage of neighbors who have alreay one so surpasses a certain threshol). Secon, the tipping point that we ientify 1 There is also a rich literature of case stuies of the i usion of various sorts of information an behavior, such as the classic stuy by Coleman, Katz, an Menzel (1966) on the aoption of tetracycline.
Diffusion on Social Networks 4 relates to the percentage of the population that nees to be seee as initial aopters in orer to have the new behavior persist. This i ers from the threshols usually investigate in the epiemiology literature, where it is the probability of transmission that must pass a threshol. This i erence is a natural consequence of the type of questions explore in the epiemiology literature. Inee, in the context of epiemics, a single iniviual is often the rst source of a isease an can generate an epiemic epening on (exogenous) infection probabilities. 2 In contrast, with behavior there can be some nontrivial portion of the population that are initial aopters (inepenent of neighbors behavior), such as those who gain utility from experimenting with new behaviors or proucts, or those expose to a trial run or free sample. Furthermore, probabilities of aoption may epen on the istribution of aopters at each point in time. Thus, the focus of our analysis is on the volume of initial aopters (that enogenously generate transmission probabilities). Thir, using techniques erive from Jackson an Rogers (2004) base on stochastic ominance arguments, we are able to make comparisons across general network structures, whereas the previous literature has ha to resort either to simulations or speci c egree istributions in orer to make comparisons. In the economics literature, Young (2000) approaches a similar set of questions to ours with a i erent moeling setup. In Young s analysis, neighbors e ects on an agent s utility are separable. Young stuies a process reminiscent of the one use here in which at each point in time, agents upate with a logistic istribution that is a function of payo i erences arising from the i erent actions playe against current play (rather than a simple best response). Young s main result shows that for su ciently ense networks, there is an upper boun on the time span it takes the entire population to switch actions with arbitrarily high probability. There is also a literature that examines the equilibrium outcomes of a variety of games playe on networks (e.g., Chwe (2000), Morris (2000) an Galeotti, Goyal, Jackson, Vega, an Yariv (2005)). Those analyses have a i erent structure as to how neighbors 2 A classical example is that of AIDS, in which one person, patient O, has been ienti e as the trigger to the sprea of the isease in the westernize worl - see Auerbach, Darrow, Ja e, an Curran (1984).
Diffusion on Social Networks 5 actions matter. In aition, they focus on the overall equilibrium structure rather than the tipping point an i usion of behavior that we analyze here. The paper is structure as follows. Section 2 contains the escription of the moel an the results. We rst present results characterizing the i usion ynamics. We then present some comparative statics of the analyze ynamics. Section 3 conclues. 2. Diffusion Dynamics an Tipping 2.1. The Moel. We consier a society of iniviuals who each start out taking an action A. The possibility arises of switching to a new action B (a metaphor for a new technology, for example). We consier a countable set of agents an capture the social structure by its unerlying network. The way in which we moel the network is through the istribution of the number of irect neighbors, or egree, that each agent has. Agent i s egree is enote i : The fraction of agents in the population with neighbors is given by P () > 0; for = 1; ::::; D; an X D =1 P () = 1: Behavior A is the efault behavior (for example, the status-quo technology) an its payo to an agent is normalize to 0. An agent i has a cost of choosing B, enote c i > 0. An agent also has some bene t from B, enote v i 0. These are ranomly an inepenently istribute across the society, accoring to a istribution that we specify shortly. Agent i s payo from aopting behavior B when i has i neighbors is: v i g( i ) i c i where an i is the fraction of i s neighbors who have chosen B an g( i ) is a function capturing how the number of neighbors that i has a ects the bene ts to i from aopting B. So, i will switch to B if the corresponing cost-bene t analysis is favorable, that is, if v i c i g( i ) i 1: (1)
Diffusion on Social Networks 6 Thus, the primitives of the moel are the istribution of i s in the population (P ), the speci cation of g, an the istribution of v i =c i. Let F be the cumulative istribution function of v i =c i. For ease of exposition we assume that F is twice i erentiable an has a ensity f: To get some feeling for behavior as a function of the number of neighbors that an agent has, let us examine a case where g() =. If > 0, then agents with higher egrees (i.e., more neighbors) are more likely to aopt the new technology or behavior for any given fraction of neighbors who have aopte i, while if < 0, then agents with higher egrees are less likely to aopt the new technology or behavior. The case where > 0 is one where bene ts epen not only on the fraction, but also on the number of an agent s neighbors who have aopte the behavior. For instance, if = 1, then g( i ) i is simply proportional to the number of neighbors that an agent has who have aopte the behavior (which is a stanar case in the epiemiology literature, where infection rates are proportional to the number of contacts with infecte iniviuals). If = 0, then an agent cares only about the fraction of neighbors who have aopte the action B an not on their absolute number (which is a stanar case stuie in coorination games, where players are often thought of to be ranomly matche with a neighbor to play a game). In that case, an agent s egree plays less of a role than in cases where 6= 0. At t = 0; a fraction x 0 of the population is exogenously an ranomly switche to the action B. At each stage t > 0; each agent, incluing the fraction of x 0 agents who are assigne the action B at the outset, best respons to the istribution of agents choosing the action B in perio t 1: As we shall show below, convergence of behavior from the starting point is monotone, either upwars or ownwars. So, once an agent (voluntarily) switches behaviors, the agent will not want to switch back at a later ate. Thus, although these best responses are myopic, any changes in behavior are equivalently forwar-looking. The eventual rest point of the system is an equilibrium of the system.
Diffusion on Social Networks 7 2.2. Di usion. Let x t enote the fraction of those agents with egree who have aopte the behavior B by time t, an let x t enote the link-weighte fraction of agents who have aopte by time t. That is, x t = X x t P () ; where is the average egree uner P. ~P () P () The reason for weighting by links is stanar: is the probability that any given neighbor of some agent is of egree (uner the presumption that there is no correlation in egrees of linke agents). We analyze a simple ynamic that leas to an overall equilibrium of the system. We begin with some ranom perturbation where x 0 of the agents of egree have aopte. Given this, we then check each agent s best response to the system. This leas to a new x 1 for each. Iterating on this process, we show that the system will eventually converge to a steay state. The convergence point is an equilibrium in the sense that given the state of the system, no aitional agents wish to aopt, an none of the agents who have aopte woul like to change their mins. Given the complexity of the system, we use a stanar technique for estimating the solutions. Namely, we use a mean- el analysis to estimate the proportion of the population that will have aopte at each time. This is escribe as follows. We start with the assumption that each i has the same initial fraction of neighbors aopting B, x 0 (an ignore the constraint that this be an integer). We also ignore the ranom istribution of initial aopters throughout the population. Each agent is matche with the actual istribution of the population. 3 So, i will aopt B in the rst perio if v i =c i > 1=(g()x 0 ). Base on this, the fraction of 3 Another way to think about this approximation is as follows. Contemplate a two stage process such that at the rst stage, each agent has a probability of x 0 of being assigne the new behavior B, an at the secon stage, each agent is ranomly matche to neighbors accoring to P (): The expecte fraction of neighbors of each iniviual choosing B is then x 0 ; an our approximation assumes that agents place a probability of 1 on the mean.
egree types who will aopt B in the rst perio is Diffusion on Social Networks 8 x 1 = 1 F [1=(g()x 0 )]: We now have a new probability that a given link points to an aopter, which is x 1 = P P ()x1 =. Iterating on this, at time t we get x t = 1 F [1=(g()xt 1 )]. This gives us an equation: or x t = 1 X P () 1 F [1=(g()x t 1 )] ; x t = 1 1 X P ()F 1 g()x t 1 : (2) Let us note a few things about this system. The right han sie is non-ecreasing in x t 1, an when starting with x t 1 = 0 the generate next perio level of aoption is x t = 0 (noting that F (1) = 1). Provie x 1 x 0, this system converges upwars to some point above x 0. Note that this happens even if we allow the initial aopters to only stay aopters if they prefer to. Once we have gotten to x 1, this inclues exactly those who prefer to have aopte given the initial shock of x 0, an now the level is either above or below x 0, epening on the speci cs of the system. So we can ask what minimal x 0 is neee in orer to have the action B i use throughout the population; that is, to have x t converge to a point above the initial point. We call this minimal x 0 the tipping point of the system: 4 We can then also ask what x t converges to. In orer to gain some insights regaring how the network structure (as capture through P ) an how preferences vary with egree (as capture through g), we examine a case where F is the uniform istribution on some interval [0; b]. 4 In general, it is possible to have multiple convergence points epening on the initial seeing. Here we look for the smallest seeing that will lea to some upwars convergence, an consequently analyze its corresponing convergence point. In many cases, there will be a unique point that we coul converge to from below.
In that case, (2) becomes Diffusion on Social Networks 9 x t = 1 X P () 1 min[1; ]: (3) bg()xt 1 In a case where x t 1 is large enough so that bg()x t 1 1 for each, then we can rewrite this as x t 1 (1 x t ) = X P () b g() : (4) Let = P P () b g(). From (4) we euce the following proposition. Proposition 1. Suppose that F is uniform on [0; b] an bg()(1 p 1 4)=2 1 for all. If x 0 < (1 If x 0 (1 p 1 4)=2 then the system converges to x = 0. p 1 4)=2 then the system converges (upwars) to x = (1+ p 1 4)=2. Proposition 1 tells us that (1 p 1 4)=2 is the tipping point of the system, beyon which there is convergence upwars. If the initial number of aopters is pushe above this level, then the ynamics converge upwars to an eventual point of x = (1 + p 1 4)=2. If the threshol is not reache, then the system collapses back to 0. Figure 1 illustrates the ynamics of the system by showing the epenence of x t+1 on x t. The gures are for a bene t/cost istribution which is uniform on [0; 5] (F U[0; 5]) an a scale-free network with power 2:5: That is, P () / 2:5 for 6 D = 1000: 5 The relationship between x t+1 an x t are rawn for g() = 1; g() = ; an g() = 2 : As is clearly seen, up to a certain x t ; the resulting x t+1 = 0. Beyon this point there is a range where x t+1 > 0, but still x t > x t+1. The tipping point is the rst point where x t+1 = x t. 5 Scale-free networks have been claime to approximate the egree istributions of some social networks, ranging from the Worl Wie Web links to phone lines (see Newman (2003) for an overview), an have been ienti e by a power parameter which falls in between 2 an 3. Jackson an Rogers (2004) provie empirical ts illustrating the iversity of egree istributions that real-worl social networks exhibit. In particular, some networks previously claime to be scale-free are, in fact, not so. Nevertheless, the scale-free istributions are a class that has been extensively use in parts of the literature to moel social networks an are thus of some interest, an they o capture some features of observe networks.
Diffusion on Social Networks 10 Above that point, we see that x t+1 > x t, up to the secon point where x t+1 = x t. This secon point is where the system converges to if the initial tipping threshol is surpasse. If the tipping point is not initially surpasse, then the system converges back to 0. Figure 1: Tipping Dynamics When we look above the tipping point, we see that the population of those choosing B increases, with increasing spee at rst, an then ecreasing spee later on. For higher values of g(); the returns from a marginal increase in the probability of a neighbor choosing the action B is higher an hence the tipping point is lower an the response to any xe fraction of the population choosing B is higher in terms of the new fraction of agents choosing B. These sorts of changes in the rate of convergence are characteristic of a wie variety of settings, as we now show. Let G(x) = 1 X P () (1 F [1=(g()x)]) (5)
Diffusion on Social Networks 11 so that x t+1 = G(x t ): Note that if F (y) is a strictly increasing function then G(x) is strictly increasing as well. In particular, if one starts with any x 0 such that G(x 0 ) > x 0, then the resulting x t s will form an increasing sequence an converge upwars to some limit. The shape of the ynamic process epens on the shape of the function G: As we show below, if the initial threshol is passe, then the spee with which the fraction of B aopters increases is increasing at rst, an ecreasing after some threshol point in time. Proposition 2. If F (y) is strictly increasing an yf (y) is a convex function of y, then there exists T 2 f0; 1; : : : ; 1g such that if 0 t < T; then x t x t 1 > xt+1 x t (where x 1 = G 1 (x 0 ) provie x 0 > 0). xt < xt+1 x t 1 x t an if t > T; then Proof of Proposition 2: Using (5), we write Now, G(x) x x t+1 = G(x t ) an xt+1 x t = G(xt ) x t : P h 0 P () 1 f g()x = 1 g()x x 2 + F 1 g()x Notice that yf(y) + F (y) = (yf (y)) 0. If (yf (y)) 00 > 0; then as x increases, the numerator ecreases. Suppose we start with su ciently high x 0 so that x 1 > x 0 : In that case, x t+1 > x t 0 for all t; an G(x t ) ecreases with time, either reaching 0 at which case T < 1; or not. x t 0 Alternatively, if x 0 is so low so that x 1 < x 0 then x t+1 < x t for all t; an G(x t ) x increases t 0 0 0 with time. If G(x 0 ) > 0 then T = 0. If G(x 0 ) < 0; then T > 0; (in fact, if G(x t ) x 0 x 0 x t converges below 0 then T = 1). If x 1 = x 0 ; then the steay state is achieve immeiately an T = 0: i 1 2.3. Comparisons across Networks. We can also euce how the tipping threshol an eventual aoption fraction change as the network structure is varie. This is an important issue in many contexts. In marketing, the tipping points for the initiations of fashions (in proucts, in the use of a new technology, etc.) may i er across emographics if those are
Diffusion on Social Networks 12 characterize by i erent social structures. In epiemiology, the likelihoo of the eruption of an epiemic may epen on the unerlying social network. These are but two of many possible examples. The network shifts we consier are characterize by statistical shifts of the relevant egree istributions. In particular, we consier shifts that raise the fraction of agents with many neighbors (First Orer Stochastic Dominance, or FOSD, shifts), an shifts that raise the heterogeneity of connecteness in the population (Secon Orer Stochastic Dominance, or SOSD, shifts). Note that from Proposition 1 we see that any change that leas = P P () b g() to increase will lea to a higher threshol an lower eventual convergence point. A ecrease in will o the reverse. Since shifts in the egree istribution P a ect in very particular ways, we can euce the implications of a variety of network shifts. Recall that ~ P () = P () enote the egree istribution of an arbitrary neighbor. The rst proposition aresses rst orer stochastic ominance shifts in the inuce neighbor egree istribution. Proposition 3. Suppose that F is uniform on [0; b], that bg()(1 p 1 4)=2 1 for all, an that P an P 0 are such that the inuce ~ P rst orer stochastically ominates ~ P 0. (1) If g() is an increasing function of, then the tipping point is lower an the upper convergence point is higher uner P. (2) If g() is a ecreasing function of, then the tipping point is higher an the upper convergence point is lower uner P. (3) If g() is constant, then the tipping point an the upper convergence point uner P are the same as uner P 0. Proposition 3 follows irectly from noting that the change in ue to a rst orer stochastic ominance shift in the istribution epens on whether =g() is an increasing or
Diffusion on Social Networks 13 ecreasing function of. 6 Proposition 3 tells us something about how aing links to the network changes the convergence behavior. In cases where g() is an increasing function of we see that this leas to lower threshols an higher convergence points. Thus, larger egree noes become more sensitive to neighbors aopting the behavior. In such a situation, increasing average neighbor egree (in the sense of FOSD) increases overall sensitivity of the population to the behavior of others, leaing to lower threshols an higher convergence. The reverse is true if g() is ecreasing. Aressing shifts in the istribution of links, we use the notion of mean preserving sprea (MPS) an a similar logic to euce the following proposition. Proposition 4. Suppose that F is uniform on [0; b] an suppose that bg()(1 p 1 4)=2 1 for all. Consier P that is a mean preserving sprea of P 0. (1) If =g() is strictly concave, then the tipping point is lower an the upper convergence point is higher uner P 0. (2) If =g() is strictly convex, then the tipping point is higher an the upper convergence point is lower uner P 0. (3) If g() is either linear or constant, then the tipping point an the upper convergence point are the same. Again, the proof is achieve irectly from examining the changes in ue to the MPS shift in istributions. 7 This proposition provies a look at how changing the sprea in egrees throughout the population changes the behavior of i usion. 6 First orer stochastic ominance of P over P 0 is equivalent to having the expectation of all increasing functions be larger uner P than uner P 0 (an ecreasing functions be smaller). 7 If P secon orer stochastically ominates P 0, then it leas to larger expectations of all strictly concave functions, an smaller expectations of strictly convex functions.
Diffusion on Social Networks 14 To illustrate the conitions in Propositions 3 an 4, consier g() =, where 0. In that case, =g() = 1 =. This is concave an increasing if 0 < < 1 an is convex an ecreasing if > 1. Note that g() is constant if = 0 an =g() is constant if = 1. 3. Conclusions We introuce a simple moel of behavioral shifts in the presence of network externalities an network structure. There are three main insights that come out of the paper. First, the ynamics are characterize by a threshol level of initial aopters: a tipping point. If that point is surpasse, then there is an increase in the eventual number of aopters of the behavior. If the initial number of aopters falls below this threshol, then the behavior will eventually ie out. Secon, if the tipping point is surpasse, then the i usion ynamics are characterize by increasing spees of aoption initially an slower spees of aoption later on. Thir, uner some assumptions on the primitives of the moel, we can escribe how the tipping point an eventual convergence point epen on the network structure. First orer an secon orer stochastic ominance shifts in the egree istributions a ect the tipping point as well as the convergence point in ways that epen on the returns to each agent from a xe fraction of her neighbors choosing to aopt the action in question.
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