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 that cascades --- and hence collective action --- can be driven not only by high-stage influencers but also by low-stage influencers.
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 that cascades -and hence collective action -can be driven not only by high-stage influencers but also by low-stage influencers.
Studying models of cascades allows one to gain insights into a variety of processes ranging from the spread of fads and ideas in social networks to the appearance of cascading failures in infrastructure networks. To date, researchers have mostly considered single-stage cascade models wherein the propagation of a cascade is characterized by a single subpopulation of active agents, [1][2][3][4][5] though some multi-stage models have been examined recently. 6,7 In the usual approach, it is assumed that all active agents exhibit the same amount of influence on their peers. In reality, however, supporters of a cause can vary significantly in their desire and ability to recruit new members. In this paper, we introduce a model of multi-stage cascading dynamics in which agents can exert different amounts of influence on their peers depending on the stage of their adoption (i.e., on the level of their commitment to a certain idea or product). We investigate the dynamics of our multi-stage cascade model on various networks and observe an interplay between cascades -e.g., one cascade driving the other one or vice versa -that cannot be observed in single-stage cascade models. We also provide an analytical method for solving the model that gives a good prediction for the cascade sizes on configuration-model networks.
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. [8][9][10][11] More generally, the effect that other people’s opinions and actions have on the decisions that people make is a crucial sociological consideration, 8,[12][13][14] 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. 2,3,7, A closely related societal concern is that the mechanisms rooted in social interaction can give rise to financial crashes, 37 political revolutions, 38 successful technologies, 39 and cultural market sensations. 40 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 16,41 has led to the use of the term contagion for the spread of social influence. 42 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, social 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. Consequently, social contagions have been called complex contagions 2,17,18 . Key investigations of complex contagions have included examinations of the diffusion of applications on the social networking site Facebook, 43 memes (short textual phrases) on news websites 44 and other social media, 45,46 information on blogs 47 and on the micro-blogging service Twitter, 48 and voting in political elections. 49 Although large data sets have the potential to help provide a better picture of social contagions, analyzing them without accompanying dynamical models offers little hope of distinguishing between underlying causes of individual behavior (social influence versus homophily versus covariates). 50 Statistical methods have been developed to approach the data side of this problem, 51 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
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