Dynamical influence processes on networks: General theory and applications to social contagion

We study binary state dynamics on a network where each node acts in response to the average state of its neighborhood. Allowing varying amounts of stochasticity in both the network and node responses, we find different outcomes in random and deterministic versions of the model. In the limit of a large, dense network, however, we show that these dynamics coincide. We construct a general mean field theory for random networks and show this predicts that the dynamics on the network are a smoothed version of the average response function dynamics. Thus, the behavior of the system can range from steady state to chaotic depending on the response functions, network connectivity, and update synchronicity. As a specific example, we model the competing tendencies of imitation and non-conformity by incorporating an off-threshold into standard threshold models of social contagion. In this way we attempt to capture important aspects of fashions and societal trends. We compare our theory to extensive simulations of this “limited imitation contagion” model on Poisson random graphs, finding agreement between the mean-field theory and stochastic simulations.

💡 Research Summary

The paper investigates a broad class of binary-state dynamics on networks, where each node updates its state (0 or 1) based on the average activity of its neighbors. Formally, a node i computes the fraction φ_i(t) of active neighbors using the adjacency matrix and then applies a response function f_i(·) to decide its next state. The authors first present the deterministic, fixed‑network version of the model and then explore four stochastic variants obtained by combining (i) fixed or rewired networks (F or R) and (ii) deterministic or probabilistic response functions (D or P). Rewiring corresponds to drawing a fresh Erdős‑Rényi graph G(N, k̄/N) at each time step, while probabilistic responses are generated from a joint distribution P(φ_on, φ_off) over two thresholds, yielding an expected response function \bar f(φ)=∫∫P f(φ; φ_on, φ_off) dφ_on dφ_off.

Update synchronicity is controlled by a parameter α∈