The spread of ideas across a social network can be studied using complex contagion models, in which agents are activated by contact with multiple activated neighbors. The investigation of complex contagions can provide crucial insights into social influence and behavior-adoption cascades on networks. In this paper, we introduce a model of a multi-stage complex contagion on networks. Agents at different stages -which could, for example, represent differing levels of support for a social movement or differing levels of commitment to a certain product or idea -exert different amounts of influence on their neighbors. We demonstrate that the presence of even one additional stage introduces novel dynamical behavior, including interplay between multiple cascades, that cannot occur in single-stage contagion models. We find, for example, that cascades -and hence collective action -can be driven not only by higher-stage influencers but also by lower-stage influencers. Social movements and other forms of collective action, which often arise spontaneously, require an ensemble of supporters with different levels of commitment. Social influence and its potential to yield a critical mass of supporters can make a crucial difference as to whether or not movements succeed [1] [2] [3] [4] . More generally, the effect that other people's opinions and actions have on the decisions that we make is a crucial sociological consideration [1, 5, 6] , and the impact that individual influence can have on the large-scale spread of rumors, fads, beliefs, and norms via social networks is of particular interest . A closely related societal concern is that the mechanisms rooted in social interaction can give rise to financial crashes [31] , political revolutions [32] , successful technologies [33] , and cultural market sensations [34] . The sudden changes in state exhibited in these examples are known as cascades: Initially local behavior becomes widespread through collective action. The perceived similarity between social and biological epidemics [9, 35] has led to the use of the term contagion for the spread of social influence [36] . Specifically, contagion refers to cases in which -much like with a virus or a disease -exposure to some source is enough to initiate propagation. Importantly, sociological contagions need not just spread from one specific source to another. In many situations, the chance of a node becoming active (e.g., adopting a new technology or joining a political revolution) depends on several other people who are active -and this is particularly true of people who are "close" or perceived as close in a social network. These types of contagions have been called complex contagions [10] [11] [12] . Key investigations of complex contagions have examined the diffusion of applications on the social networking site Facebook [37] , memes (short textual phrases) on news websites [38] and other social media [39] , information on blogs [40] and on the micro-blogging service Twitter [41] . Although large data sets have the potential to help provide a better picture of social contagion, analyzing them without accompanying dynamical models offers little hope of distinguishing between underlying causes of individual behavior (social influence versus homophily versus covariates) [42] . Statistical methods have been developed to approach the data side of this problem [43] , but mathematical modelling is an underappreciated and crucial component of these efforts. In particular, simple models make it possible to isolate effects (e.g., social contagion) and develop and test quantitative diagnostics that characterize macroscopic dynamics. Early studies in the sociology of behavioral cascades considered threshold models of binary decisions [21] . In these models, agents can switch from an initially inactive state to an active state if a sufficient proportion of other agents are active. These models capture two important features [1]: interdependence (an agent's behavior depends on the behavior of other agents) and heterogeneity (differences in behavior are reflected in the distribution of thresholds). In recent studies [7, [44] [45] [46] , threshold models of social influence have been examined on networks in which the nodes correspond to agents and the edges between nodes indicate who influences whom. In a network setting, models with simple threshold dynamics also capture fundamental mechanisms that can be related (at least by analogy) to a large range of phenomena that include failures in power grids and the transmission of infectious diseases [47, 48] . Tractable models of threshold dynamics have also sparked a great deal of interest in the physics and applied mathematics communities, because many results from graph theory, statistical physics, and dynamical systems can be directly applied in this setting. For example, such techniques allow one to identify critical cascade thresholds [7] , mean cascade sizes [49] , and the effects of seed size [44] and network topology [48] . Motivated by the observation that not all opinions have equal weight, we introduce in this paper a model for multi-stage complex contagions in which agents can adopt several different states with variable levels of in-2 fluence. Almost all existing models assume that active agents exert equal influence over their peers. (A notable exception is Ref. [15], which recently considered networked agents with variable levels of social influence but used a very different framework from ours and also focused on different dynamics.) However, such a binary description of agents' states can be woefully inadequate. For example, it has been well-documented that supporters with varying level of commitment are crucial in social movements [1, 2] ; regular users of a product are more enthusiastic recommenders than casual users; supporters of a political party can vary significantly in their desire and ability to recruit new members; there are substantial behavioral differences between fans and fanatics of sports teams; and experience with free trial versions of software can greatly influence a user's decision to purchase a costly commercial version. We show schematics of single-stage and two-stage complex contagion models in Fig. 1 . To focus our discussion, we consider the simplest multi-stage progression, in which there are three possible states [see Fig. 1(b) ]: inactive (no influence; unaware of an innovation or indifferent to a social movement), active (low level of influence; testing an innovation or supporting a social movement), and hyper-active (high level of influence; full adoption of an innovation or activists in a social movement). The rest of this paper is structured as follows. In the next two sections, we define our model and illustrate some of its novel dynamics. We demonstrate that it is possible for one cascade to drive the other one even in situations in which there would be no propagation (or only small propagation) in the associated single-stage cascade model. We then use synthetic networks and analytical approximations to further explore the model's dynamics, and derive conditions for the appearance of cascades. Additional details are included in the Appendices. We consider situations in which there is an underlying social network, which we represent using an unweighted, undirected graph with N nodes. At any given point in time, there are three possible influence levels for each node: Inactive nodes exert no influence on their neighbors and are said to be in state S 0 , active nodes exert some influence and are said to be in state S 1 , and hyperactive nodes exert a higher level of influence and are said to be in state S 2 [see Fig. 1(b) ]. Importantly, nodes in state S 2 are a subset of nodes in state S 1 , because nodes that are S 2 -active are necessarily S 1 -active (i.e., we consider fanatics to be a specific type of fan), but S 1 -active nodes need not be S 2 -active. A natural generalization of the two-stage model includes further levels of influence such that an S n -active agent is S i -active for all 1 ≤ i ≤ n. To specify the model, we also need to indicate precisely how nodes influence their neighbors and how the neighbors respond to that influence. Accordingly, we define the influence response function F i (m 1 , m 2 , k) as the probability that a degree-k node (i.e., a node with k neighbors) becomes S i -active given that it has m 1 neighbors in state S 1 and m 2 neighbors in state S 2 . This macroscopic description can be derived directly from a microscopic description of response functions of individual nodes [47] . Formulating our model in these general terms allows us to capture a wide range of types of local interactions between nodes via the detailed form of the response functions F i [48] . In the following section, we use a threshold model to illustrate the dynamics of multi-stage contagion models. All nodes can be either inactive (S0) or active (S1). Nodes that are barely above the S1-threshold have the same level of influence as nodes that are strongly above the threshold. (b) Multi-stage complex contagion. A subset of active nodes (called "S1-active") can become hyper-active (called "S2-active") and have additional influence. Note that S2-active nodes are necessarily also S1-active. We define the "peer pressure" P = (m 1 + βm 2 )/k to be the total influence received by a degree-k node from its m 1 neighbors in state S 1 and m 2 neighbors in state S 2 , scaled by the node degree k. Note that our definition of states implies that a neighbor in state S 2 contributes 1 + β to the influence, so β measures the bonus influence exerted by S 2 -active nodes. It is the presence of such bonus influence that distinguishes our multi-stage contagion model from single-stage models. For threshold models of complex contagion, a node becomes S i -active if P is equal to or exceeds a certain threshold R i (which can be different for each node). Therefore, the response function is F i (m 1 , m 2 , k) = C i (P ), where C i is the cumulative distribution of thresholds for state S i across network nodes. In this paper, we focus mostly on uniform-threshold cases in which S iactivation thresholds R i are the same for all nodes. In the uniform-threshold cases, the response functions are step functions: Observe that the response functions F i (m 1 , m 2 , 1) can be used to model cases in which the received influence 3 is not scaled by a node's degree k, so an active node influences all of its neighbors equally (regardless of their degrees). We require that R 2 ≥ R 1 in order to satisfy , thereby guaranteeing that all S 2 -active nodes are also S 1 -active. When β = 0, the S 1 -state dynamics reduce to a single-stage case because the S 2 -active nodes are indistinguishable from S 1 nodes. As an initial condition, we select a small fraction φ (i) of nodes to be initially S i -active, and these are not allowed to become less active. At each subsequent time step (of size ∆t = 1/N ), we update a randomly chosen node according to the threshold rules (1). This implies monotone dynamics-i.e., nodes can never become less active than they are currently. In single-stage threshold models, it is usually the case that if a node needs a number m of active neighbors to become active, then any subset of its neighbors with at least m active nodes is sufficient to make it active. In our multi-stage model, however, there is a heterogeneity in the subset of a node's neighbors needed for activation. For example, the subset might consist of 4 active neighbors, 2 active and 1 hyper-active neighbor, or just 2 hyper-active neighbors. Moreover, depending on a node's threshold values, various possible subsets of neighbors can make it active or even hyper-active. We now compare single-stage and multi-stage cascade dynamics, highlighting cascades that occur in the multistage models that do not occur in the corresponding single-stage models. Because these dynamics cannot arise if there are only two types of nodes (inactive and active), we see that the presence of hyper-active nodes plays a crucial role in driving cascades on networks. We start by simulating these dynamics on the Facebook network of students at the University of Oklahoma (recorded in September 2005 as a single-time snapshot) [50, 51] . This example illustrates that a small amount of additional influence can trigger cascades. This may model, for example, the role of a charismatic leader in a social movement, or the potential effect of customer product reviews on retail websites such as Amazon. In the context of our model, the presence of S 2 -active nodes triggers a cascade of S 1 -active nodes that would not have occurred otherwise. In particular, this effect arises specifically due to the extra influence attributed to S 2 -active nodes, who are significantly above the S 1 -activation threshold. In (1) with the following parameter values: R 1 = 0.15 and R 2 = 0.3. In both cases, we show an average over 100 realizations with the same initially active nodes [52] . In Fig. 2(a) , S 2 -active nodes have no influence on the activation of S 1 -active nodes (because β = 0) and there is no cascade: the fraction of S 1 nodes remains small. Note, however that some nodes are well above the activation threshold (purple region). Specifically, these nodes surpass the threshold R 2 = 0.3 but have no additional influence. In Fig. 2(b) , S 2 -active nodes that surpass the higher threshold R 2 = 0.3 have 1.5 times as much influence as S 1 -active nodes. This additional influence is enough to trigger a global (system-wide) cascade. This example illustrates that a small number of additional low-level influencers that are S 1 -active can also trigger a cascade. This could represent situations in which a company gives out free trials of a product to potential customers with the aim of boosting full product sales. Once again, we use the Facebook network to compare two simulations. In the first, we set R 1 = R 2 = 0.2; in the second, we set R 1 = 0.15 and R 2 = 0.2. The first case is essentially a single-stage process because the S 1 dynamics are slaved to the S 2 dynamics. The other parameters are the same in both simulations: β = 0.3 and φ (1) = φ (2) ≈ 0.02, and the response function is again given by Eq. (1). We again average over 100 realizations using the same initially active nodes in both cases [53] . There is no cascade in Fig. 3(a) , in which the activation thresholds are equal. In Fig. 3(b) , however, the S 1 -activation threshold is slightly lower. This results in a small number of additional S 1 -active nodes, which is enough to trigger a cascade. Fig. 2 , the time t is on the horizontal axis, and the fractions of nodes in each state are indicated on the vertical axis. Light blue, blue, and red regions represent S0-, S1-, and S2-active nodes, respectively. We use 348 S2-active seed nodes (corresponding to φ (2) = φ (1) ≈ 0.02) and β = 0.3. In panel (a), R1 = R2 = 0.2, so the S1 dynamics are slaved to the S2 dynamics. A small change in the threshold parameter (R1 = 0.15) in panel (b) yields a small number of additional S1-active nodes, which are nevertheless enough to trigger a cascade. To better understand the cascade dynamics, it is instructive to consider well-chosen synthetic networks that allow us to control the increase in the final cascade size. We design synthetic scenarios in which some nodes do not become active in the single-stage case, but the cascade in the multi-stage case occupies the entire system. To this end, we construct random networks consisting of nodes of degrees 4 and 24 in proportion 1:1 with positive degree-degree correlations. (We dub such networks "(4,24)-correlated random networks"; degree assortativity is positive when degree-degree correlations are positive.) We then use the response function defined in Eq. (1) that scales the threshold by node degree. It is thus harder for high-degree nodes to become active because they need a larger number of active neighbors (most of which are also of high degree). We construct the network using the method described in Ref. [54] , where the assortativity is captured by the joint distribution P (k, k ′ ), which gives the probability that a randomly chosen edge connects nodes of degrees k and k ′ . For our example, we choose P (k, k ′ ) to be a (symmetric) matrix whose non-zero entries satisfy P (4, 4)/P (4, 24) = 3 and P (24, 24)/P (24, 4) = 23. This gives a network with a specified amount of (positive) assortativity. An additional consequence of the multi-stage dynamics on such networks is that trajectories spend a significant amount of time around a value that is not the final steady state. In the first example, illustrated in Fig. 4 , the extra influence exerted by S 2 -active nodes is needed to drive a system-wide cascade. To demonstrate this effect, we compare the dynamics resulting from a single-stage case, captured by the S 1 -dynamics of the multi-stage case with upper threshold R 2 = ∞, with those resulting from a (true) multi-stage case with R 2 = 0.7. (See the caption of Fig. 4 for the values of the other parameters.) In Fig. 4 , we show numerical results using symbols and analytical results given by Eqs. (2)-(3) using curves. The analytical results qualitatively reproduce the numerical behavior; in some cases, we also observe good quantitative agreement. However, as discussed in Appendix B, some novel effects arising in the multi-stage model are not captured by the analytical approximation and can potentially lead to incorrect estimates. We show the aggregate fractions of S 1 -and S 2 -active nodes for the single-stage case in Fig. 4(a) and for the multi-stage case in Fig. 4(b) . We show the separate temporal evolutions for the two degree classes in Figs. 4(c,d) and the temporal evolution for nodes in each degree class that are S 1 -but not S 2 -active in Fig. 4(e) . In the single-stage case, only the low-degree nodes ultimately become S 1 -active, which results in the aggregate active fraction of 0.5 illustrated in Fig. 4(a) . Observe that none of the high-degree nodes become S 1 -active [see Fig. 4 (c)] and that no nodes become S 2 -active because R 2 = ∞. In the multi-stage case, which we show in Fig. 4(b) , the low-degree nodes that in the single-stage case were significantly above the S 1 -activation threshold eventually become S 2 -active and hence exert more influence on their neighbors. As we show in Fig. 4(d) , this initiates a gradual increase in the number of S 1 -active high-degree nodes (starting around t = 10) until there are sufficiently many to trigger a delayed rapid transition or a secondary cascade (around t = 13) in which all nodes become S 2 active. In Fig. 4 (e), we show for each degree class the temporal evolution of the fraction of the S 1 -active nodes that are not S 2 -active. The number of such nodes is given by the difference between S 1 -active and S 2 -active nodes. The peaks in Fig. 4 (e) imply that nodes in each degree class first become S 1 active and that there is some delay for their S 2 -activation. Now imagine that only the S 1 -dynamics in Fig. 4 (b) can be observed. For example, suppose that a publisher knows which nodes purchased a particular book but has no idea about the level of excitement of any of the nodes. In this case, at time t = 8, the publisher might mistakenly conclude that the product is not going to be sold anymore and could perhaps discontinue it. However, the heterogeneity of excitement and influence levels among book purchasers suggests instead that the publisher should continue printing more copies of the book. In this toy example, there would be a sharp rise in sales at time t = 13 caused by the S 2 -active agents (who, e.g., could represent customers who write book reviews on Amazon). We now consider another example, illustrated in Fig. 5 , in which the addition of an S 1 -active state enhances the propagation of an S 2 -cascade, which (in this case) eventually reaches the entire network. In Fig. 5(a) , we illustrate (b) show the aggregate fractions of S1-and S2-active nodes, panels (c) and (d) show these fractions for each degree class separately, and panel (e) shows the fractions of nodes in each degree class that are S1-but not S2-active. We show the numerical results (averaged over 100 realizations) using symbols and the analytical results given by Eqs. (2)- (3) the case in which R 1 = R 2 = 0.35. This is effectively a single-stage scenario because the S 2 -active nodes are also S 1 active. (See the caption of Fig. 5 for the values of all parameters.) Similar to the previous example, all of the low-degree nodes become active, but now they are also all S 2 active. None of the high-degree nodes become active [see Fig. 5(c) ]. In order to obtain a (true) multi-stage scenario, we reduce the R 1 threshold to 0.22 and show the resulting aggregate dynamics in Fig. 5(b) . The dynamics are initially qualitatively similar to the single-stage case of Fig. 5(a) . However, after some time passes, there is a surge of high-degree nodes that become S 1 -active due to the lower S 1 threshold. They then drive a cascade of S 2 -active nodes [see Fig. 5(d) ]. In panel (e) of Fig. 5 , we show the temporal evolution of the fraction of the S 1 -active nodes that are not S 2 -active. This quantity exhibits a peak for the highdegree nodes but remains at zero for low-degree nodes, indicating that high-degree nodes become S 2 active some time after becoming S 1 active, whereas low-degree nodes become S 2 and S 1 active at the same time. (b) show the aggregate fractions of S1-and S2-active nodes, panels (c) and (d) show these fractions for each degree class separately, and panel (e) shows the fractions of nodes in each degree class that are S1-but not S2-active. We show the numerical results (averaged over 100 realizations) using symbols and the analytical results given by Eqs. (2)- (3) (where we choose the seed uniformly at random). We now present an analytical approximation for the temporal evolution of the fraction of active nodes in our multi-stage model. The method that we employ is based on pairwise interactions between nodes [44, 48] and entails two requirements: (i) for any fixed k, the response functions F i must be non-decreasing functions of both m 1 and m 2 , and (ii) F i (m 1 , m 2 , k) ≥ F i+1 (m 1 , m 2 , k). Condition (i) reflects the effect of positive externalities -i.e., the more active neighbors a node has, the more likely that it becomes S i -active. Condition (ii) follows from the fact that the number of hyper-active nodes should not exceed the number of all active nodes in the system. The situations that we illustrate in this paper satisfy these conditions, which are sensible general assumptions in studying social influence. One computes the density ρ  where Q (i) is a vector of auxiliary variables. We present the functionals H (i) k and G (i) (each of which depends on F i and the network topology) and the derivation of Eqs. (2)- (3) in Appendix A. The fraction φ (i) of nodes that are initially S i -active specifies the initial conditions ρ The aggregate fraction of nodes that are S i -active at time t is , where P k is the degree distribution of the network and k max is the maximum node degree in the network. In Fig. 6 , we compare the analytical predictions of Eqs. (2)- (3) with numerical simulations of both the final values and the temporal evolution of active fractions of nodes. This figure uses two example network ensembles: (i) Erdős-Rényi random graphs (i.e., graphs in which each pair of nodes is connected by an edge with equal, independent probability) with mean degree z = 5; and (ii) uncorrelated random graphs (i.e., random graphs whose joint degree-degree distribution can be expressed in terms of the degree distribution as 2 ) consisting of degree-4 and degree-5 nodes in proportion 1:2. We dub the latter graphs "(4,5)-uncorrelated random networks". We do not plot the final fraction of S 1 -active nodes, as it equals 1 in each of the examples illustrated in Fig. 6 . As one can see in Fig. 6 , Eqs. (2)- (3) correctly predict the final fractions of active nodes in all cases and are in good agreement with the numerically computed temporal evolution. In Fig. 6(a) , the blue dash-dotted curve corresponding to the (4,5)-uncorrelated random network has steps at β = 1/4 and β = 1/3. The reason for these steps is as follows: When β < 1/4, degree-4 nodes can never receive enough influence from their neighbors to overcome the threshold R 2 = 5 to become S 2 -active (see Eq. (1) with k ≡ 1). Therefore, the final fraction of S 2 -active nodes ρ (2) (∞) is given by the fraction of degree-5 nodes in the network (which is equal to 2/3 in this example). When β ∈ [1/4, 1/3), degree-4 nodes become S 2 -active only when all of their neighbors are S 2 -active. Consequently, a finite fraction (about 0.26) of degree-4 nodes becomes S 2 -active, yielding the aggregate value of ρ (2) (∞) ≈ 0.75. We show the temporal evolution of the active fraction of nodes for each degree class for this case in Fig. 6(b,c) . Finally, for β ≥ 1/3, degree-4 nodes become S 2 -active if all of their neighbors are S 1 -active and any three of them are S 2 -active. In this situation, all degree-4 nodes become S 2 -active at the end of the cascade. (2)-(3) (curves) for (a) the final fractions of S2-active nodes as functions of the bonus influence β for ensembles of (4,5)-uncorrelated random networks (blue dash-dotted curve) and Erdős-Rényi random graphs with mean degree z = 5 and (b,c) the temporal evolution of active fraction of nodes in each degree class in (4,5)-uncorrelated random networks for β = 0.25. In panel (a), we use the response functions of Eq. (1) with k ≡ 1 and uniform thresholds R1 = 1 and R2 = 5 for (4,5)-uncorrelated random networks (blue dash-dotted curve) and Erdős-Rényi random graphs with mean degree z = 5 (black dashed curve). For the latter, we show using a solid red curve the case in which R2 thresholds are Gaussian-distributed with mean µ = 5 and standard deviation σ = 0.1. The total number of nodes in each network is N = 10 4 . In all cases, we initially S1-activate a fraction φ (1) = 10 −3 of nodes chosen uniformly at random, and we average over 100 numerical realizations of networks and initial conditions. In Fig. 7 , we illustrate the relationship between the final fraction of S 1 -active nodes ρ (1) ∞ , the bonus influence β and the mean-degree z for Erdős-Rényi random graphs. Darker colors indicate larger final activation fractions. The final fraction of S 2 -active nodes is qualitatively similar. Note that when β = 0 (i.e., single-stage dynamics), there is no cascade. A bifurcation analysis gives an analytical estimate of the boundary of the region in which cascades occur. In analogy to the methods developed for single-stage models [44, 48] , we derive a cascade condition from Eqs. (2)-(3). (We present full details in Appendix C.) Briefly, we compute zero eigenvalues of the Jacobian matrix of G evaluated at Q (1) = Q (2) = 0. This yields a closed-7 form expression that approximates the boundary of the cascade region. This approximation, which is crude but given by a closed-form expression (see Appendix C), yields the dashed curve in Fig. 7 . For low z values (towards the left of Fig. 7 ), we find a continuous transition from low to high values of ρ (1) ∞ . In contrast, for higher values of z, we find a discontinuous transition. This jump arises from a saddle-node bifurcation that occurs as z is increased while β is held fixed. This bifurcation can be followed numerically by solving G = 0 and finding zero eigenvalues of the Jacobian evaluated at the corresponding equilibria (see the solid curve in Fig. 7) . We provide full details in Appendix C. ∞ (whose value is indicated by color) calculated from Eqs. (2)- (3) for Erdős-Rényi random graphs. The mean degree z is on the horizontal axis, and the bonus influence β is on the vertical axis. The first threshold is R1 = 0.3, and the second threshold R2 is Gaussian-distributed with mean µ = 0.8 and standard deviation σ = 0.2. The initial seed fractions are φ (1) = 2 × 10 −3 and φ (2) = 0. The dashed curve gives the boundary of the cascade condition, and the solid curve is a numerical continuation of the saddle-node bifurcation. Just as in Watts's single-stage cascade model [7] , numerical simulations using parameter values close to a saddle-node bifurcation are very sensitive to the choice of seed nodes. A recent empirical study [55] has highlighted the fact that real cascades are extremely rare events. This is consistent with the dynamics that occur near the saddle-node bifurcation: a few specific choices of initial seeds produce large-scale cascades but most choices do not. Social movements and other forms of collective action require an ensemble of supporters with different levels of commitment, and social influence can make a crucial difference as to whether or not they succeed. This motivates the development of analytically tractable complex contagion models with multiple stages in which different agents have different levels of influence. In the present paper, we introduced and analyzed such a model, in which we define the level of influence on a node from its neighbors using a general function of the node's degree and the state of its nearest neighbors. We illustrated that this model can exhibit interesting dynamics that is not possible with single-stage cascade models. This includes, in particular, the interplay between the cascades of fans (active nodes) and fanatics (hyper-active nodes), where one cascade can drive the other and vice versa. Our model and our analytical results can be generalized to multi-stage cases with any (finite) number of active states. The model can be developed further to allow one to distinguish the level of a node's commitment and the influence it has on its neighbors. Different levels of commitment and influence have welldocumented importance on social movements, product advertising, and other sociological, political, and economic situations. However, mere observation and data analysis of complex social dynamics make it difficult to discern the relative importance of social influence, homophily, and covariates on the observations [42] . It is therefore imperative to develop new mathematical models to tackle this challenging situation, and we hope that the model we have introduced in this paper will prove beneficial for such efforts. ACKNOWLEDGEMENTS
Melnik et al. REF proposed a model of a multi-stage complex contagion, in which agents at different stages exert different amounts of influence on their neighbours.
Multi-Stage Complex Contagions
Chaos 23, 013124 (2013)
Mathematics Computer Science Physics Medicine