Can social contagion spread without key players?

Contagion models have been used to study the spread of social behavior among agents of a population, such as information diffusion, social influence, and participation to collective action (e.g., protests). Key players, which are typically high-degree, -k-core or -centrality agents in a networked population, are considered important for spreading social contagions. In this paper, we ask whether contagions can propagate through a population that is void of key players. We use Erdos-Renyi random graphs as a representation of unstructured populations that lack key players, and investigate whether complex contagions - those requiring reinforcement - can spread on them. We demonstrate that two game-theoretic contagion models that utilize common knowledge for collective action can readily spread such contagions, which is a significant difference from classic complex contagion models. We compare contagion dynamics results on unstructured networks to those on more typically-studied, structured social networks to understand the role of network structure. We test a total of 14 networks. The two common knowledge models are also contrasted to understand the effects of different modeling assumptions on dynamics. We show that under a wide range of conditions, these two models produce markedly different results.