Multistable binary decision making on networks

We propose a simple model for a binary decision making process on a graph, motivated by modeling social decision making with cooperative individuals. The model is similar to a random field Ising model or fiber bundle model, but with key differences on heterogeneous networks. For many types of disorder and interactions between the nodes, we predict discontinuous phase transitions with mean field theory which are largely independent of network structure. We show how these phase transitions can also be understood by studying microscopic avalanches, and describe how network structure enhances fluctuations in the distribution of avalanches. We suggest theoretically the existence of a “glassy” spectrum of equilibria associated with a typical phase, even on infinite graphs, so long as the first moment of the degree distribution is finite. This behavior implies that the model is robust against noise below a certain scale, and also that phase transitions can switch from discontinuous to continuous on networks with too few edges. Numerical simulations suggest that our theory is accurate.

💡 Research Summary

The paper introduces a minimalist binary decision‑making model defined on an undirected graph (G=(V,E)). Each node (v) carries a binary variable (x_v\in{0,1}) that is determined by comparing an internal “field” (s_v) with a global control parameter (p): (x_v=\Theta(s_v-p)). The internal field is taken to be a monotonic function of a node‑specific random strength (P_v) and the fraction (q_v) of neighbors that are in state 1, i.e. (s_v=h(P_v,q_v)) with (h) increasing. The authors focus on the cooperative case where a node’s switch to 1 does not decrease the probability that its neighbors are also 1.

A mean‑field (MF) approximation assumes that all nodes experience the same average neighbor activity (q). Under this assumption the self‑consistency equation becomes
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