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Network science is a rapidly expanding field, with a large and growing bodyof work on network-based dynamical processes. Most theoretical results in thisarea rely on the so-called \emph{locally tree-like approximation}. This is,however, usually an `uncontrolled' approximation, in the sense that themagnitudes of the error are typically unknown, although numerical results showthat this error is often surprisingly small. In this paper, we place thisapproximation on more rigorous footing by calculating the magnitude ofdeviations away from tree-based theories in the context of discrete-timecritical network cascades with re-excitable nodes. We discuss the conditionsunder which tree-like approximations give good results for calculating networkcriticality, and also explain the reasons for deviation from thisapproximation, in terms of the density of certain kinds of network motifs.Using this understanding, we derive results for network criticality that applyto general networks that explicitly do not satisfy the locally tree-likeapproximation. In particular, we focus on the bi-parallel motif, the smallestmotif relevant to the failure of a tree-based theory in this context, and wederive the corrections due to such motifs on the conditions for criticality. Weverify our claims on computer-generated networks, and we confirm that ourtheory accurately predicts the observed deviations from criticality. Using ourtheory, we explain why numerical simulations often show that deviations from atree-based theory are surprisingly small. More specifically, we show that thesedeviations are negligible for networks whose average degree is even modestlylarge compared to one, justifying why tree-based theories appear to work wellfor most real-world networks.

Critical Network Cascades with Re-excitable nodes: Why tree-like approximations usually work, when they breakdown, and how to correct them