Title: Critical behaviors in contagion dynamics,
Abstract: We study the critical behavior of a general contagion model where nodes are
either active (e.g. with opinion A, or functioning) or inactive (e.g. with
opinion B, or damaged). The transitions between these two states are determined
by (i) spontaneous transitions independent of the neighborhood, (ii)
transitions induced by neighboring nodes and (iii) spontaneous reverse
transitions. The resulting dynamics is extremely rich including limit cycles
and random phase switching. We derive a unifying mean-field theory.
Specifically, we analytically show that the critical behavior of systems whose
dynamics is governed by processes (i-iii) can only exhibit three distinct
regimes: (a) uncorrelated spontaneous transition dynamics (b) contact process
dynamics and (c) cusp catastrophes. This ends a long-standing debate on the
universality classes of complex contagion dynamics in mean-field and
substantially deepens its mathematical understanding.