  Models of simple excitable dynamics on graphs are an efficient framework for
studying the interplay between network topology and dynamics. This subject is a
topic of practical relevance to diverse fields, ranging from neuroscience to
engineering. Here we analyze how a single excitation propagates through a
random network as a function of the excitation threshold, that is, the relative
amount of activity in the neighborhood required for an excitation of a node.
Using numerical simulations and analytical considerations, we can understand
the onset of sustained activity as an interplay between topological cycle
statistics and path statistics. Our findings are interpreted in the context of
the theory of network reverberations in neural systems, which is a question of
long-standing interest in computational neuroscience.
