Diffusion and Cascading Behavior
in Random Networks
The spread of new ideas, behaviors or technologies has been extensively studied using epidemic models. Here we consider a model of diffusion where the individuals’ behavior is the result of a strategic choice. We study a simple coordination game with binary choice and give a condition for a new action to become widespread in a random network. We also analyze the possible equilibria of this game and identify conditions for the coexistence of both strategies in large connected sets. Finally we look at how can firms use social networks to promote their goals with limited information. Our results differ strongly from the one derived with epidemic models and show that connectivity plays an ambiguous role: while it allows the diffusion to spread, when the network is highly connected, the diffusion is also limited by high-degree nodes which are very stable.
Keywords: social networks, diffusion, random graphs, empirical distribution
JEL codes: C73, O33, L14
There is a vast literature on epidemics on complex networks (see (Newman, 2003) for a review). Most of the epidemic models consider a transmission mechanism which is independent of the local condition faced by the agents concerned. However, if there is a factor of coordination or persuasion involved, relative considerations tend to be important in understanding whether some new belief or behavior is adopted (Vega-Redondo, 2007). To fix ideas, it may be useful to think of the diffusion process as modeling the adoption of a new technology. In this case, when an agent is confronted with the possibility of adoption, her decision depends on the persuasion effort exerted on her by each of her neighbors in the social network. More formally, those neighborhood effects can be captured as follows: the probability that an agent adopts the new technology when out of her neighbors have already adopted can be modeled by a threshold function: this probability is one if and it is zero otherwise. For , this would correspond to a local majority rule. More generally, some simple models for the diffusion of a new behavior have been proposed in term of a basic underlying model of individual decision-making: as individuals make decisions based on the choices of their neighbors, a particular pattern of behavior can begin to spread across the links of the network (Vega-Redondo, 2007), (Easley and Kleinberg, 2010). In this work, we analyze the diffusion in the large population limit when the underlying graph is a random network with given vertex degrees. There is now a large literature on the study of complex networks across different disciplines such as physics (Newman, 2003), mathematics (Janson et al., 2000), sociology (Watts, 2002), networking (Lelarge and Bolot, 2008) or economics (Vega-Redondo, 2007). There is also a large literature on local interaction and adoption externalities (Jackson and Yariv, 2007), (López-Pintado, 2008), (Galeotti et al., 2010), (Young, 2009). Similarly to these works, we study a game where players’ payoffs depend on the actions taken by their neighbors in the network but not on the specific identities of these neighbors. Our most general framework (presented in Section 3) allows to deal with threshold games of complements, i.e. a player has an increasing incentive to take an action as more neighbors take the action.
The main contribution of our paper is a model which allows us to study rigorously semi-anonymous threshold games of complements with local interactions on a complex network. We use a very simple dynamics for the evolution of the play: each time, agents choose the strategy with best payoff, given the current behavior of their neighbors. Stochastic versions of best response dynamics have been studied extensively as a simple model for the emergence of conventions (Kandori et al., 1993), (Young, 1993). A key finding in this line of work is that local interaction may drive the system towards a particular equilibrium in which all players take the same action. Most of the literature following (Ellison, 1993), (Blume, 1993) is concerned with stochastic versions of best response dynamics on fixed networks and a simple condition known as risk dominance (Harsanyi and Selten, 1988) determines whether an innovation introduced in the network will eventually become widespread. In this case, bounds on the convergence time have been computed (Young, 2000), (Montanari and Saberi, 2010) as a function of the structure of the interaction network. In this paper, we focus on properties of deterministic (myopic) best response dynamics. In such a framework, as shown in (Blume, 1995), coordination on the risk dominant strategy depends on the network structure, the dynamics of the game and the initial configuration. In particular, this coordination might fail and equilibria with co-existent strategies can exist. This was shown in (Blume, 1995) already in the simple case where the underlying network is the two-dimensional lattice. While the results in (Blume, 1995) depend very much on the details of the particular graph considered, (Morris, 2000) derives general results that depend on local properties of the graphs.
More formally, agents are nodes of a fixed infinite graph and revise their strategies synchronously. There is an edge if agents and can interact with each other. Each node has a choice between two possible behaviors labeled and . On each edge , there is an incentive for and to have their behaviors match, which is modeled as the following coordination game parametrized by a real number : if and choose (resp. ), they each receive a payoff of (resp. ); if they choose opposite strategies, then they receive a payoff of . Then the total payoff of a player is the sum of the payoffs with each of her neighbors. If the degree of node is and is her number of neighbors playing , then the payoff to from choosing is while the payoff from choosing is . Hence the best response strategy for agent is to adopt if and if . A number of qualitative insights can be derived from this simple diffusion model (Blume, 1995; Morris, 2000). Clearly, a network where all nodes play is an equilibrium of the game as is the state where all nodes play . Consider now a network where all nodes initially play and a small number of nodes are forced to adopt strategy (constituting the seed). Other nodes in the network apply best-response updates, then these nodes will be repeatedly applying the following rule: switch to if enough of your neighbors have already adopted . There can be a cascading sequence of nodes switching to such that a network-wide equilibrium is reached in the limit. In particular, contagion is said to occur if action can spread from a finite set of players to the whole population. It is shown in (Morris, 2000) that there is a contagion threshold such that contagion occurs if and only if the parameter is less than the contagion threshold. In particular the contagion threshold is always at most so that needs to be the risk-dominant action in order to spread. However this condition is not sufficient and (Morris, 2000) gives a number of characterizations of the contagion threshold and also shows that low contagion threshold implies the existence of equilibria where both actions are played.
Most of the results on this model are restricted to deterministic infinite graphs and the results in (Morris, 2000) depend on local properties of graphs that need to be checked node-by-node. In networks exhibiting heterogeneity among the nodes, this approach will be limited in its application. In this work, we analyze the diffusion in the large population limit when the underlying graph is a random network with vertices and where is a given degree sequence (see Section 2.1 for a detailed definition). Although random graphs are not considered to be highly realistic models of most real-world networks, they are often used as first approximation and are a natural first choice for a sparse interaction network in the absence of any known geometry of the problem (Jackson, 2008). Our analysis yields several insights into how the diffusion propagates and we believe that they could be used in real-world situations. As an example, we show some implications of our results for the design of optimal advertising strategies. Our results extend the previous analysis of global cascades made in (Watts, 2002) using a threshold model. They differ greatly from the study of standard epidemics models used for the analysis of the spread of viruses (Bailey, 1975) where an entity begin as ’susceptible’ and may become ’infected’ and infectious through contacts with her neighbors with a given probability. Already in the simple coordination game presented above, we show that connectivity (i.e. the average number of neighbors of a typical agent) plays an ambiguous role: while it allows the diffusion to spread, when the network is highly connected, the diffusion is also limited by high-degree nodes, which are very stable. These nodes require a lot of their neighbors to switch to in order to play themselves.
In the case of a sparse random network of interacting agents, we compute the contagion threshold in Section 2.2 and relate it to the important notion of pivotal players. We also show the existence of (continuous and discontinuous) phase transitions. In Section 2.3, we analyze the possible equilibria of the game for low values of . In particular, we give conditions under which an equilibrium with coexistence of large (i.e. containing a positive fraction of the total population) connected sets of players and is possible. In Section 2.4, we also compute the minimal size of a seed of new adopters in order to trigger a global cascade in the case where these adopters can only be sampled without any information on the graph. We show that this minimal size has a non-trivial behavior as a function of the connectivity. In Section 2.5, we give a heuristic argument allowing to recover the technical results which gives some intuition behind our formulas. Our results also explain why social networks can display a great stability in the presence of continual small shocks that are as large as the shocks that ultimately generate a global cascade. Cascades can therefore be regarded as a specific manifestation of the robust yet fragile nature of many complex systems (Carlson and Doyle, 2002): a system may appear stable for long periods of time and withstand many external shocks (robustness), then suddenly and apparently inexplicably exhibit a large scale cascade (fragility).
Our paper is divided into two parts. The first part is contained in the next section. We apply our technical results to the particular case of the coordination game presented above. We describe our main findings and provide heuristics and intuitions for them. These results are direct consequences of our main theorems stated and proved in the second part of the paper. This second part starts with Section 3, where we present in details our most general model of diffusion (where initial activation and thresholds can vary among the population and also be random). This general model allows to recover and extend results in the random graphs literature and is used in (Coupechoux and Lelarge, 2011) for the analysis of the impact of clustering. We also state our main technical results: Theorem 10 and Theorem 11. Their proofs can be found in Sections 4 and 5 respectively. Section 7 contains our concluding remarks.
Probability asymptotics: in this paper, we consider sequences of (random) graphs and asymptotics as the number of vertices tends to infinity. For notational simplicity we will usually not show the dependency on explicitly. All unspecified limits and other asymptotics statement are for . For example, w.h.p. (with high probability) means with probability tending to 1 as and means convergence in probability as . Similarly, we use , and in a standard way, see Janson et al. (2000). For example, if is a parameter of the random graph, means that as for every , equivalently , or for every , w.h.p.
2 Analysis of a simple model of cascades
In this section, we first present the model for the underlying graph. This is the only part of this section required to understand our main technical results in Section 3. The remaining parts of this section present and discuss applications and special cases exploring the significance of the technical results derived in the sequel.
2.1 Graphs: the configuration model
We consider a set of agents interacting over a social network. Let be a sequence of non-negative integers such that is even. For notational simplicity we will usually not show the dependency on explicitly. This sequence is the degree sequence of the graph: agent has degree , i.e. has neighbors. We define a random multigraph (allowing for self-loop and multiple links) with given degree sequence , denoted by by the configuration model (Bollobás, 2001): take a set of half-edges for each vertex and combine the half-edges into pairs by a uniformly random matching of the set of all half-edges. Conditioned on the multigraph being a simple graph, we obtain a uniformly distributed random graph with the given degree sequence, which we denote by (Janson, 2009b).
We will let and assume that we are given satisfying the following regularity conditions, see (Molloy and Reed, 1995):
For each , is a sequence of non-negative integers such that is even and, for some probability distribution independent of ,
for every as ;
In words, we assume that the empirical distribution of the degree sequence converges to a fixed probability distribution with a finite mean . Condition 1 (iii) ensures that has a finite second moment and is technically required. It implies that the empirical mean of the degrees converges to .
In some cases, we will need the following additional condition which implies that the asymptotic distribution of the degrees has a finite third moment:
For each , is a sequence of non-negative integers such that .
The results of this work can be applied to some other random graphs models too by conditioning on the vertex degrees (see (Janson and Luczak, 2007, 2009)). For example, for the Erdős-Rényi graph with , our results will hold with the limiting probability distribution being a Poisson distribution with mean .
2.2 Contagion threshold for random networks
An interesting perspective is to understand how different network structures are more or less hospitable to cascades. Going back to the diffusion model described in the Introduction, we see that the lower is, the easiest the diffusion spreads. In (Morris, 2000), the contagion threshold of a connected infinite network (called the cascade capacity in (Easley and Kleinberg, 2010)) is defined as the maximum threshold at which a finite set of initial adopters can cause a complete cascade, i.e. the resulting cascade of adoptions of eventually causes every node to switch from to . In this section, we consider a model where the initial adopters are forced to play forever. In this case, the diffusion is monotone and the number of nodes playing is non-decreasing. We say that this case corresponds to the permanent adoption model: a player playing will never play again. We will discuss a model where the initial adopters also apply best-response update in Section 2.3.
To illustrate the notion of contagion threshold, consider the infinite network depicted on Figure 1. It is easy to see that the contagion threshold for such a network is , the same as for the -dimensional lattice. Note that in the network of Figure 1 the density of nodes with degree is and the density of nodes of degree is , whereas in the -dimensional lattice, all vertices have degree .
We now compute the contagion threshold for a sequence of random networks. Since a random network is finite and not necessarily connected, we first need to adapt the definition of contagion threshold to our context. We start by recalling a basic result on the size of the largest component of a random graph with a given degree sequence (Molloy and Reed, 1995; Janson and Luczak, 2009), namely a sufficient condition for the existence of a giant component.
Consider the random graph satisfying Conditions 1 with asymptotic degree distribution . If then the size of the largest component of is . In this case, we say that there exists a giant component.
For a graph and a parameter , we consider the largest connected component of the induced subgraph in which we keep only vertices of degree strictly less than . We call the vertices in this component pivotal players: if only one pivotal player switches from to then the whole set of pivotal players will eventually switch to in the permanent adoption model. For a player , we denote by the final number of players in the permanent adoption model with parameter , when the initial state consists of only playing , all other players playing . Informally, we say that is the size of the cascade induced by player .
Let be the set of pivotal players in .
For , there are constants such that w.h.p. and for any , .
For , for an uniformly chosen player , we have . The same result holds if players are chosen uniformly at random.
Note that Proposition 3 is valid only for graphs with a giant component (for graphs with all components , we have trivially that for all values of ). We can rewrite (1) as follows: let be a random variable with distribution , i.e. , then
We can restate Proposition 3 as follows: let be the size of a cascade induced by switching a random player from to . Proposition 3 implies that for , we have as for every , whereas for there exists constants depending only on and the parameters of the graph such that for every , . Informally, we will say that global cascades (i.e. reaching a positive fraction of the population) occur when and do not occur when . We call defined by (1) the contagion threshold for the sequence of random networks with degree distribution . This result is in accordance with the heuristic result of (Watts, 2002) (see in particular the cascade condition Eq. 5 in (Watts, 2002)). Proposition 3 follows from Theorem 11 below which also gives estimates for the parameters and when . Under additional technical conditions, Theorem 11 shows that in case (i).
Note that the condition implies that so that for , we have, . Hence we have the following elementary corollary:
Recall that for general graphs, it is shown in (Morris, 2000) that . Note also that our bound in Corollary 4 is tight: for a -regular network with chosen uniformly at random (i.e. such that for all ) , Proposition 3 implies that corresponding to the contagion threshold of a -regular tree, see (Morris, 2000). For , a -regular random graph is connected w.h.p, so that for any , an initial adopter will cause a complete cascade (i.e. reaching all players). In particular, in this case, the only possible equilibria of the game are all players playing or all players playing . We will come back to the analysis of equilibria of the game in Section 2.3.
Given a degree distribution , Proposition 3 gives the typical value of under the uniform distribution on graphs with the prescribed degree distribution. A question of interest is to find among all graphs with a given degree sequence, which ones achieve the lowest or highest value of the contagion threshold. Indeed, the contagion threshold of a graph can be seen as a proxy for its susceptibility to cascades: the higher the contagion threshold is, the more susceptible the graph is to cascading behavior. Clearly if is the maximal degree in the graph, we have . In the case of regular graphs, we saw that the typical contagion threshold is equal to this lower bound: . Note however that there are examples of regular graphs in (Morris, 2000) with a strictly higher threshold (see Figure 3 there: a -regular graph with contagion threshold ). Hence among all -regular graphs, picking a graph at random will give w.h.p. a graph which is the least susceptible to cascades among those -regular graphs. However it is not always the case that picking a graph at random among all graphs with a given degree sequence will result in the least susceptible one. There are situations where the typical value of the contagion threshold exceeds the lowest possible value: this is the case in the example presented above in Figure 1 for which the contagion threshold is whereas the sequence of random graphs with asymptotic degree distribution such that and has a contagion threshold of . The recent work (Blume et al., 2011) has a similar focus.
We now consider some examples. First the case of Erdős-Rényi random graphs where each of the edges is present with probability for a fixed parameter . In this case, we can apply our results with for all , i.e. the Poisson distribution with mean . As shown in Figure 2 (Left) for the case of Erdős-Rényi random graphs , is well below , indeed we have for any value of . As shown in Figure 2 (Right), we see that is a non-decreasing function of the average degree in the graph for . Clearly on Figure 2 (Left), is a non-increasing function of , for . The second curve in Figure 2 corresponds to the contagion threshold for a scale-free random network whose degree distribution (with ) is parametrized by the decay parameter . We see that in this case we have . In other words, in an Erdős-Rényi random graph, in order to have a global cascade, the parameter must be such that any node with no more than four neighbors must be able to adopt even if it has a single adopting neighbor. In the case of the scale free random network considered, the parameter must be much lower and any node with no more than nine neighbors must be able to adopt with a single adopting neighbor. This simply reflects the intuitive idea that for widespread diffusion to occur there must be a sufficient high frequency of nodes that are certain to propagate the adoption.
We also observe that in both cases, for sufficiently low, there are two critical values for the parameter , such that a global cascade for a fixed is only possible for . The heuristic reason for these two thresholds is that a cascade can be prematurely stopped at high-degree nodes. For Erdős-Rényi random graphs, when , there exists a giant component, i.e. a connected component containing a positive fraction of the nodes. The high-degree nodes are quite infrequent so that the diffusion should spread easily. However, for close to one, the diffusion does not branch much and progresses along a very thin tree, “almost a line”, so that its progression is stopped as soon as it encounters a high-degree node. Due to the variability of the Poisson distribution, this happens before the diffusion becomes too big for . Nevertheless the condition is not sufficient for a global cascade. Global diffusion also requires that the network not be too highly connected. This is reflected by the existence of the second threshold where a further transition occurs, now in the opposite direction. For , the diffusion will not reach a positive fraction of the population. The intuition here is clear: the frequency of high-degree nodes is so large that diffusion cannot avoid them and typically stops there since it is unlikely that a high enough fraction of their many neighbors eventually adopts. Following (Watts, 2002), we say that these nodes are locally stable.
The proof of our Theorem 11 makes previous intuition rigorous for a more general model of diffusion and gives also more insights on the nature of the phase transitions. We describe it now. The lower curve in Figure 3 represents the number of pivotal players in an Erdős-Rényi random graph as a function of the average connectivity for : hence we keep only the largest connected component of an Erdős-Rényi random graph where we removed all vertices of degree greater than . By the same heuristic argument as above, we expect two phase transitions for the size of the set of pivotal players. In the proof of Theorem 11, we show that it is indeed the case and moreover that the phase transitions occur at the same values and as can be seen on Figure 3 where the normalized size (i.e. fraction) of the set of pivotal players is positive only for . Hence a cascade is possible if and only if there is a ’giant’ component of pivotal players. Note also that both phase transitions for the pivotal players are continuous, in the sense that the function is continuous. This is not the case for the second phase transition for the normalized size of the cascade given by in Proposition 3: the function is continuous in but not in as depicted on Figure 3. This has important consequences: around the propagation of cascades is limited by the connectivity of the network as in standard epidemic models. But around , the propagation of cascades is not limited by the connectivity but by the high-degree nodes which are locally stable.
To better understand this second phase transition, consider a dynamical model where at each step a player is chosen at random and switches to . Once the corresponding cascade is triggered, we come back to the initial state where every node play before going to the next step. Then for less than but very close to , most cascades die out before spreading very far. However, a set of pivotal players still exists, so very rarely a cascade will be triggered by such a player in which case the high connectivity of the network ensures that it will be extremely large. As approaches from below, global cascades become larger but increasingly rare until they disappear implying a discontinuous phase transition. For low values of , the cascades are infrequent and small. As increases, their frequencies and sizes also increase until a point where the cascade reaches almost all vertices of the giant component of the graph. Then as increases, their sizes remain almost constant but their frequencies are decreasing.
2.3 Equilibria of the game and coexistence
We considered so far the permanent adoption model: the only possible transitions are from playing to . There is another possible model in which the initial adopters playing also apply best-response update. We call this model the non-monotonic model (by opposition to the permanent adoption model). In this model, if the dynamic converges, the final state will be an equilibrium of the game. An equilibrium of the game is a fixed point of the best response dynamics. For example, the states in which all players play or all players play are trivial equilibria of the game. Note that the permanent adoption model does not necessarily yield to an equilibrium of the game as the initial seed does not apply best response dynamics. To illustrate the differences between the two models consider the graph of Figure 4 for a value of : if the circled player switches to , the whole network will eventually switch to in the permanent adoption model, whereas the dynamic for the non-monotonic model will oscillate between the state where only the circled player plays and the state where only his two neighbors of degree two play (agents still revise their strategies synchronously).
Clearly if a player induces a global cascade for the non-monotonic model, it will also induce a global cascade in the permanent adoption model. As illustrated by the case of Figure 4, the converse is not true in general. Hence, a priori, a pivotal player as defined in previous section might not induce a global cascade in the non-monotonic model. In (Morris, 2000), it is shown that if one can find a finite set of initial adopters causing a complete cascade for the permanent adoption model, it is also possible to find another, possibly larger but still finite, set of initial adopters leading to a complete cascade for the non-monotonic model. Hence the contagion threshold as defined in (Morris, 2000) for infinite graphs is the same for both models. In our case, we see that if we switch from to two pivotal players who are neighbors, then the whole set of pivotal players will eventually switch to in the non-monotonic model. Hence in our analysis on random networks, we also say that the contagion threshold is the same in both models. Moreover, both models will have exactly the same dynamics if started with the set of pivotal players playing and all other players playing . In particular, it shows that the dynamic converges and reaches an equilibrium of the game. Hence we have the following corollary:
Consider the random graph satisfying Conditions 1 and 2 with asymptotic degree distribution . For , there exists w.h.p. an equilibrium of the game in which the number of players is more than (defined in Proposition 3) and it can be obtained from the trivial equilibrium with all players playing by switching only two neighboring pivotal players.
Hence for , the trivial equilibrium all , is rather ’weak’ since two pivotal players can induce a global cascade and there are such players so that switching two neighbors at random will after a finite number of trials (i.e. not increasing with ) induce such a global cascade. We call this equilibrium the pivotal equilibrium. Figure 5 shows the average number of trials required. It goes to infinity at both extremes and . Moreover, we see that for most values of inside this interval, the average number of trials is less than . If , then by definition if there are pivotal players, their number must be . Indeed, in the case of -regular graphs, there are no pivotal players for . Hence, it is either impossible or very difficult (by sampling) to find a set of players with cardinality bounded in leading to a global cascade since in all cases, their number is .
We now show that equilibria with co-existent strategies exist. Here by co-existent, we mean that a connected fraction of players and are present in the equilibrium. More formally, we call a giant component: a subset of vertices containing a positive fraction (as tends to infinity) of the total size of the graph such that the corresponding induced graph is connected. We ask if there exists equilibrium with a giant component of players and ? In the case , for values of close to , the global cascade reaches almost all nodes of the giant component leaving only very small of connected component of players . However for close to , this is not the case and co-existence is possible as shown by the following proposition:
In an Erdős-Rényi random graph , for , there exists such that:
for , in the pivotal equilibrium, there is coexistence of a giant component of players and a giant component of players .
for , in the pivotal equilibrium, there is no giant component of players , although there might be a positive fraction of players .
Figure 6 illustrates this proposition in the case of Erdős-Rényi random graphs . The difference between the upper (red) and lower (green) curve is exactly the fractions of players in the pivotal equilibrium while the brown curve represent the size (in percentage of the total population) of the largest connected component of players in the pivotal equilibrium. This curve reaches zero exactly at . Hence for , we see that there is still a positive fraction of the population playing , but the set of players is divided in small (i.e. of size ) connected components, like ’islands’ of players in a ’sea’ of players . Note also that for Erdős-Rényi random graphs, the value of is close to . Proposition 6 follows from the following proposition whose proof is given in Section 6.
We first introduce some notations: for a random variable, we define for , the thinning of obtained by taking points and then randomly and independently keeping each of them with probability , i.e. given , . We define (corresponding to the function defined in previous section for and ), and
Consider the random graph satisfying Conditions 1 and 2 with asymptotic degree distribution . Let be a random variable with distribution . If , then . Assume moreover that there exists such that for . There is coexistence of a giant component of players and a giant component of players in the pivotal equilibrium if
2.4 Advertising with word of mouth communication
We consider now scenarios where and the initial set of adopters grows linearly with the total population . More precisely, consider now a firm advertising to a group of consumers, who share product information among themselves: potential buyers are not aware of the existence of the product and the firm undertakes costly informative advertising. The firm chooses the fraction of individuals who receive advertisements. Individuals are located in a social network modeled by a random network with given vertex degrees as in previous section. However contrary to most work on viral marketing (Richardson and Domingos, 2002), (Kempe et al., 2003), we assume that the advertiser has limited knowledge about the network: the firm only knows the proportions of individuals having different degrees in the social network. One possibility for the firm is to sample individuals randomly and to decide the costly advertising for this individual based on her degree (i.e. her number of neighbors). The action of the firm is then encoded in a vector , where represents the fraction of individuals with degree which are directly targeted by the firm. These individuals will constitute the seed (in a permanent adoption model) and we call them the early adopters. Note that the case for all corresponds to a case where the firm samples individuals uniformly. This might be one possibility if it is unable to observe their degrees. In order to optimize its strategy, the firm needs to compute the expected payoff of its marketing strategy as a function of . Our results allows to estimate this function in terms of and the degree distribution in the social network.
We assume that a buyer might buy either if she receives advertisement from the firm or if she receives information via word of mouth communication (Ellison and Fudenberg, 1995). More precisely, following (Galeotti and Goyal, 2009), we consider the following general model for the diffusion of information: a buyer obtains information as soon as one of her neighbors buys the product but she decides to buy the product when where is her number of neighbors (i.e. degree) and the number of neighbors having the product, is a Binomial random variable with parameters and and is a general random variable (depending on ). In words, is the probability that a particular neighbor does influence the possible buyer. This possible buyer does actually buy when the number of influential neighbors having bought the product exceeds a threshold . Thus, the thresholds represent the different propensity of nodes to buy the new product when their neighbors do. The fact that these are possibly randomly selected is intended to model our lack of knowledge of their values and a possibly heterogeneous population. Note that for and , our model of diffusion is exactly a contact process with probability of contagion between neighbors equals to . This model is also called the SI (susceptible-infected) model in mathematical epidemiology (Bailey, 1975). Also for and , we recover the model of (Morris, 2000) described previously where players are non-buyers and players are buyers.
We now give the asymptotic for the final number of buyers for the case and , with (the general case with a random threshold is given in Theorem 10). We first need to introduce some notations. For integers and let denote the binomial probabilities . Given a distribution , we define the functions:
where . We define
We refer to Section 2.5 for an intuition behind the definitions of the functions and in terms of a branching process approximation of the local structure of the graph.
Consider the random graph for a sequence satisfying Condition 1. If the strategy of the firm is given by , then the final number of buyers is given by provided , or , and further for any in some interval .
To illustrate this result, we consider the simple case of Erdős-Rényi random graphs with , for all , and where is the contagion threshold for this network (defined in previous section). For simplicity, we omit to write explicitly the dependence in and replace the dependence in by the parameter .
As we will see, for some values of there is a phase transition in . For a certain value , we have: if , the size of the diffusion is rather small whereas it becomes very large for . Clearly the advertising firm has a high interest in reaching the ’critical mass’ of early adopters. This phase transition is reminiscent of the one described in previous section. Recall that we are now in a setting where so that a global cascade triggered by a single (or a small number of) individual(s) uniformly sampled is not possible. Hence, the firm will have to advertise to a positive fraction of individuals constituting the seed for the diffusion. The intuition is the following: if the seed is too small, then each early adopter starts a small cascade which is stopped by high-degree nodes. When the seed reaches the critical mass, then the cascades ’coalesce’ and are able to overcome ’barriers’ constituted by high-degree nodes so that a large fraction of the nodes in the ’giant component’ of the graph adopt. Then increasing the size of the seed has very little effect since most of the time, the new early adopters (chosen at random by the firm) are already reached by the global diffusion. Our Theorem 10 makes this heuristic rigorous for the general model of diffusion described above (with random thresholds).
We now give some more insights by exploring different scenarios for different values of .
Consider first a case where , then thanks to the results of previous section, we know that there exists such that for any , a global cascade is possible with positive probability if only one random player switches to . In particular, if a fraction of individuals uniformly sampled are playing , then for any , such a global cascade will occur with high probability. For , we know that a single player cannot trigger a global cascade. More precisely, if players are chosen at random without any information on the underlying network, any set of initial adopters with size cannot trigger a global cascade, as shown by Theorem 11. However the final size of the set of players playing is a discontinuous function of the size of the initial seed. If is defined as the point at which this function is discontinuous, we have: for , the final set of players will be only slightly larger than but if the final set of players will be very large. Hence we will say that there is a global cascade when and that there is no global cascade when . As shown in Figure 7, for , we have for and for . In Figure 7, we also see that our definition of global cascade when is consistent with our previous definition since the function giving the size of the final set of players when the seed has normalized size is continuous in and agrees with the previous curve for the size of the final set of players when and with only one early adopter.
We now consider the case where . In this case, thanks to the results of previous section, we know that for any value of , a single initial player (sampled uniformly) cannot trigger a global cascade. But our definition of still makes sense and we now have for all . Figure 8 gives a plot of the function . We see that again there are two regimes associated with the low/high connectivity of the graph. For low values of , the function is non-increasing in . This situation corresponds to the intuition that is correct for standard epidemic models according to which an increase in the connectivity makes the diffusion easier and hence the size of the critical initial seed will decrease.
Figure 9 shows the size of the final set of players as a function of the size of the initial seed for various small values of . We see that for the smallest values of , there is no discontinuity for the function . In this case is not defined. We also see that there is a natural monotonicity in : as the connectivity increases, the diffusion also increases. However Figure 8 shows that for , the function becomes non-decreasing in . Hence even if the connectivity increases, in order to trigger a global cascade the size of the initial seed has to increase too. The intuition for this phenomenon is now clear: in order to trigger a global cascade, the seed has to overcome the local stability of high-degree nodes which now dominates the effect of the increase in connectivity.
Figure 10 shows the size of the final set of players as a function of the size of the initial seed for various value of . Here we see that connectivity hurts the start of the diffusion: for small value of , an increase in results in a lower size for the final set of players ! However when reaches the critical value , then a global cascade occurs and its size is an increasing function of . In other words, high connectivity inhibits the global cascade but once it occurs, it facilitates its spread.
2.5 Local Mean Field approximation
In this subsection, we say that a player is inactive and a player is active. We describe an approximation to the local structure of the graph and present a heuristic argument which leads easily to a prediction for the asymptotic probability of being active. This heuristic allows us to recover the formulas for the functions . Results from this section are not needed for the rigorous proofs given in Sections 4 and 5 and are only presented to give some intuition about our technical formulas. This branching process approximation is standard in the random graphs literature (Durrett, 2007) and is related to the generating function approach in physics Watts (2002). It is called Local Mean Field (LMF) in (Lelarge and Bolot, 2008; Lelarge, 2009) and in the particular case of an epidemics with independent contagion on each edge, the LMF approximation was turned into a rigorous argument. For our model of diffusion with threshold, this seems unlikely to be straightforward and our proof will not use the LMF approximation.
The LMF model is characterized by its connectivity distribution defined in Condition 1. We now construct a tree-indexed process. Let be a Galton-Watson branching process (Athreya and Ney, 1972) with a root which has offspring distribution and all other nodes have offspring distribution given by for all . Recall that corresponds to the probability that an edge points to a node with degree , see (Durrett, 2007). The tree describes the local structure of the graph (as tends to infinity): the exploration of the successive neighborhoods of a given vertex is approximated by the branching process as long as the exploration is local (typically restricted to a finite neighborhood independent of ).
We denote by the root of the tree and for a node , we denote by the generation of , i.e. the length of the minimal path from to . Also we denote if belongs to the children of , i.e. and is on the minimal path from to . For an edge with , we denote by the sub-tree of with root obtained by the deletion of edge from .
We now consider the diffusion model described in Section 2.4 (with threshold for a vertex of degree given by and probability for a neighbor to be influential given by ) on the tree , where the initial set of active nodes is given by a vector : if then vertex is in the initial set of active vertices, otherwise . In our model the ’s are independent Bernoulli random variables with parameter where is the degree of node in the tree. Thanks to the tree structure, it is possible to compute the probability of being active recursively as follows: for any node , let if node is active on the sub-graph with initial set of active nodes given by the restriction of to individuals in and with individual held fix in the inactive state. Then for any node of degree , becomes active if the number of active influential children exceeds . Hence, we get
where the ’s are independent Bernoulli random variables with parameter . Then the state of the root is given by
In order to compute the distribution of , we first solve the Recursive Distributional Equation (RDE) associated to the ’s: thanks to the tree structure, the random variables in (2) are i.i.d. and have the same distribution as . Hence their distribution solve the RDE given by
where for a given , the random variable is Bernoulli with parameter , ’s are independent Bernoulli with parameter , has distribution , and the ’s are i.i.d. copies (with unknown distribution). To solve the RDE (4), we need to compute only the mean of the Bernoulli random variable . Hence taking expectation in (4) directly gives a fixed-point equation for this mean and the following lemma follows (its proof is deferred to Appendix A.1):
By the change of variable , we see that Lemma 9 is consistent with Proposition 8 and allow to recover the functions . However note that, the RDE (4) has in general several solutions one being the trivial one with corresponding to the trivial equilibrium where every node are active. The fact that this RDE has several solutions is a major difficulty in order to turn this LMF approximation into a rigorous argument. Finally, the crucial point allowing previous computation is the fact that in recursion (2) the can be computed “bottom-up” so that the ’s of a given generation (from the root) are independent. The ’s in (3) encode the information that is activated by a node in the subtree of “below” (and not by the root). If one considers a node in the original graph and runs a directed contagion model on a local neighborhood of this node where only ’directed’ contagion toward this node are allowed, then the state of the graph seen from this node is well approximated by the ’s.
3 General model and main results
We first present the model for the diffusion process and then our main results for the spread of the diffusion.
3.1 Diffusion: percolated threshold model
In this section, we describe the diffusion for any given finite graph with vertex set . We still denote by the degree of node . From now on, a vertex is either active or inactive. In our model, the initial set of active nodes (the seed) will remain active during the whole process of the diffusion (results for the non-monotonic model are derived in Section 6). We will consider single activation where the seed contains only one vertex and degree based activation where the vertex is in the seed with probability , where is the degree of the vertex independently of the others. In other words, each node draws independently of each other a Bernoulli random variable with parameter and is considered as initially active if and not initially active otherwise. In the case of degree based activation, we denote the parameters of this activation. In particular, if for all , then a fraction chosen uniformly at random among the population is activated before the diffusion takes place.
Symmetric threshold model: We first present the symmetric threshold model which generalizes the bootstrap percolation (Balogh and Pittel, 2007): a node becomes active when a certain threshold fraction of neighbors are already active. We allow the threshold fraction to be a random variable with distribution depending on the degree of the node and such that thresholds are independent among nodes. Formally, we define for each node a sequence of random variables in denoted by . The threshold associated to node is where is the degree of node . We assume that for any two vertices and , the sequences and are independent and have the same law as a sequence denoted by . For , we denote the probability distribution of the threshold for a node of degree . For example in the model of (Morris, 2000) described in the introduction, we take so that . We will use the notation and denotes the distribution of thresholds.
Now the progressive dynamic of the diffusion on the finite graph operates as follows: some set of nodes starts out being active; all other nodes are inactive. Time operates in discrete steps . At a given time , any inactive node becomes active if its number of active neighbors is at least . This in turn may cause other nodes to become active. It is easy to see that the final set of active nodes (after time steps if the network is of size ) only depends on the initial set (and not on the order of the activations) and can be obtained as follows: set for all . Then as long as there exists such that , set , where means that and share an edge in . When this algorithm finishes, the final state of node is represented by : if node is active and otherwise.
Note that we allow for the possibility in which case, node is never activated unless it belongs to the initial set . Note also that the condition is actually not restrictive. If we wish to consider a model where is possible, we just need to modify the initial seed so as to put node in if . Hence this model is equivalent to ours if we increase accordingly.
Percolated threshold model: this model depends on a parameter and a distribution of random thresholds given by as described above. Given any graph and initial set , we now proceed in two phases.
bond percolation: randomly delete each edge with probability independently of all other edges. We denote the resulting random graph by ;
apply the symmetric threshold model with thresholds : set and then as long as there is such that , set , where means that and share an edge in and is the degree of node in the original graph .
Clearly if , this is exactly the symmetric threshold model. If in addition , then this model is known as bootstrap percolation (Balogh and Pittel, 2007). On the other hand if and for any , then this is the contact process with probability of activation on each edge. Note that the percolated threshold model is not equivalent to the symmetric threshold model on the (bond) percolated graph since threshold depends on the degree in the original graph (and not in the percolated graph).
3.2 Diffusion with degree based activation
Recall that for integers and , denotes the binomial probabilities .
For a graph , let and denote the numbers of vertices and edges in respectively. In this section, we assume that the followings are given: a sequence of independent thresholds drawn according to a distribution , a parameter and an activation set drawn according to the distribution . The subgraph of induced by the activated (resp. inactive) nodes at the end of the diffusion is denoted by (resp. ). For , we denote by the number of vertices in with degree in and in , i.e. the number of vertices with degree in which are not activated and with neighbors which are not activated either. We denote by the number of activated vertices of degree in (and with possibly lower degree in ).
We define the functions:
where . We define
Consider the random graph (or ) for a sequence satisfying Condition 1. We are given with a sequence of independent thresholds drawn according to a distribution , a parameter and an activation set drawn according to the distribution . For the percolated threshold diffusion on the graph , we have: if , or if , and further for any in some interval , then
If we condition the induced graph of inactive nodes in on its degree sequence and let be the number of its vertices, then has the distribution of .
3.3 Diffusion with a single activation
In this section, we look at the diffusion with one (or a small number of) initial active node(s) in .
For , let (resp. ) be the subgraph induced by the final active nodes with initial active node (resp. initial active nodes ). We also define as the subgraph induced by the inactive nodes with initial active node . The set of vertices of and is a partition of the vertices of the original graph. To ease notation, we denote , and . We define
We also define