  As shown recently (arXiv:0801.3056), several types of neuronal complex
networks involving non-linear integration-and-fire dynamics exhibit an abrupt
activation along their transient regime. Interestingly, such an avalanche of
activation has also been found to depend strongly on the topology of the
networks: while the Erd\H{o}s-R\'eny, Barab\'asi-Albert, path-regular and
path-transformed BA models exhibit well-defined avalanches; Watts-Strogatz and
geographical structures present instead a gradual dispersion of activation
amongst their nodes. The current work investigates such phenomena by
considering a mean-field equivalent model of a network which is strongly
founded on the concepts of concentric neighborhoods and degrees. It is shown
that the hierarchical number of nodes and hierarchical degrees define the
intensity and timing of the avalanches. This approach also allowed the
identification of the beginning activation times during the transient dynamics,
which is particularly important for community identification (arXiv:0801.4269,
arXiv:0801.4684). The main concepts and results in this work are illustrated
with respect to theoretical and real-world (\emph{C. elegans}) networks.
Several results are reported, including the identification of secondary
avalanches, the validation of the equivalent model, the identification of the
possible universality of the avalanches for most networks (depending only on
the network size), as well as the identification of the fact that different
avalanches can be obtained by locating the activation source at different
neurons of the \emph{C. elegans} network.
