On-off Threshold Models of Social Contagion

We study binary state contagion dynamics on a social network where nodes act in response to the average state of their neighborhood. We model the competing tendencies of imitation and non-conformity by incorporating an off-threshold into standard threshold models of behavior. In this way, we attempt to capture important aspects of fashions and general societal trends. Allowing varying amounts of stochasticity in both the network and node responses, we find different outcomes in the random and deterministic versions of the model. In the limit of a large, dense network, however, we show that these dynamics coincide. The dynamical behavior of the system ranges from steady state to chaotic depending on network connectivity and update synchronicity. We construct a mean field theory for general random networks. In the undirected case, the mean field theory predicts that the dynamics on the network are a smoothed version of the average node response dynamics. We compare our theory to extensive simulations on Poisson random graphs with node responses that average to the chaotic tent map.

💡 Research Summary

The paper introduces a novel binary‑state contagion framework called the “on‑off threshold model” that extends classic threshold models by adding a second, de‑activation threshold. In the traditional setting a node adopts a behavior when the fraction of its neighbors who are active exceeds a single threshold. Real‑world social dynamics, however, often display a tension between imitation (conformity) and anti‑conformity: individuals may also abandon a behavior when it becomes too common. By defining an “on‑threshold” (\theta^{\text{on}}) and an “off‑threshold” (\theta^{\text{off}}<\theta^{\text{on}}), the authors capture both forces in a single rule: a node becomes active if the local average exceeds (\theta^{\text{on}}), becomes inactive if it falls below (\theta^{\text{off}}), and otherwise retains its current state.

The model is placed on random graphs with an arbitrary degree distribution (P(k)). Both the network topology and the node‑specific thresholds are drawn from prescribed probability distributions, allowing the study of heterogeneity in connections and individual susceptibility. Two updating schemes are examined: synchronous (all nodes evaluate and update simultaneously) and asynchronous (nodes update one at a time in a random order). The authors show that synchrony tends to amplify fluctuations and can drive the system into chaotic regimes, whereas asynchrony damps these fluctuations and often yields stable fixed points or low‑period cycles.

A mean‑field (MF) analysis is derived by averaging over the degree distribution. Let (x_t) denote the global fraction of active nodes at time (t). For nodes of degree (k) the expected response is a function (F_k(x_t)) that incorporates the on/off thresholds and the shape of the underlying node response (which may be linear, piecewise‑linear, or nonlinear). The MF recursion reads
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