Size and Duration of Avalanches in Complex Networks
Marshall Carpenter
Applied Mathematics, University of Colorado Boulder
Date and time:
Thursday, November 17, 2011 - 4:00pm
Abstract:
We study the influence of network structure on avalanches of excitation cascading over the nodes of the network. Such excitable networks have been used to model epidemics, social interactions, gene expression, and neuronal networks. Avalanches of activity have been observed in the brains of awake monkeys and in slices of rat cortex, and are thus of particular interest.
The case of a tree-like network can be extended to webs or directed networks. The dynamics of the network are related to the Perron-Frobenius eigenvector and eigenvalue of the adjacency matrix. We are most interested in critical networks which occur when the largest eigenvalue is one because this is most representative of biological systems. For these networks, the distribution of avalanche sizes and durations is a power law and is proportional to the entry of the Perron-Frobenius eigenvector corresponding to the starting node.
When the largest eigenvalue is less than one (subcritical), or greater than one (supercritical), the distribution of avalanche duration follows an exponential decay, and we can numerically find the decay rate. Additionally, the total sizes of avalanches in subcritical networks are studied, and a numerical method can find the percentage of avalanches that achieve sustained activity in supercritical networks.