Critical network cascades with re-excitable nodes: Why treelike approximations usually work, when they break down, and how to correct them.
Network science is a rapidly expanding field, with a large and growing body of work on network-based dynamical processes. Most theoretical results in this area rely on the so-called locally treelike approximation. This is, however, usually an "uncontrolled" approximation, in the sense that the magnitudes of the error are typically unknown, although numerical results show that this error is often surprisingly small. In this paper we place this approximation on more rigorous footing by calculating the magnitude of deviations away from tree-based theories in the context of discrete-time critical network cascades with re-excitable nodes. We discuss the conditions under which tree-like approximations give good results for calculating network criticality, and also explain the reasons for deviation from this approximation, in terms of the density of certain kinds of network motifs. Using this understanding, we derive results for network criticality that apply to general networks that explicitly do not satisfy the locally treelike approximation. In particular, we focus on the biparallel motif, the smallest motif relevant to the failure of a tree-based theory in this context, and we derive the corrections due to such motifs on the conditions for criticality. We verify our claims on computer-generated networks, and we confirm that our theory accurately predicts the observed deviations from criticality. Using our theory, we explain why numerical simulations often show that deviations from a tree-based theory are surprisingly small. More specifically, we show that these deviations are negligible for networks whose average degree is even modestly large compared to one, justifying why tree-based theories appear to work well for most real-world networks.