  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 \emph{locally tree-like 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 tree-like
approximation. In particular, we focus on the bi-parallel 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.
