  We characterize the distributions of size and duration of avalanches
propagating in complex networks. By an avalanche we mean the sequence of events
initiated by the externally stimulated `excitation' of a network node, which
may, with some probability, then stimulate subsequent firings of the nodes to
which it is connected, resulting in a cascade of firings. This type of process
is relevant to a wide variety of situations, including neuroscience, cascading
failures on electrical power grids, and epidemology. We find that the
statistics of avalanches can be characterized in terms of the largest
eigenvalue and corresponding eigenvector of an appropriate adjacency matrix
which encodes the structure of the network. By using mean-field analyses,
previous studies of avalanches in networks have not considered the effect of
network structure on the distribution of size and duration of avalanches. Our
results apply to individual networks (rather than network ensembles) and
provide expressions for the distributions of size and duration of avalanches
starting at particular nodes in the network. These findings might find
application in the analysis of branching processes in networks, such as
cascading power grid failures and critical brain dynamics. In particular, our
results show that some experimental signatures of critical brain dynamics
(i.e., power-law distributions of size and duration of neuronal avalanches),
are robust to complex underlying network topologies.
