Date of Award

Spring 1-1-2012

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Applied Mathematics

First Advisor

Juan G. Restrepo

Second Advisor

Jem Corcoran

Third Advisor

Anne Dougherty

Abstract

We characterize the distribution of sizes and durations of avalanches propagating in complex networks. We find that the statistics of avalanches can be characterized in terms of the Perron-Frobenius eigenvalue and eigenvectors 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 in all cases. Our results are specific to individual networks and allow us to find expressions for the distribution of size and duration of avalanches starting at particular nodes. These findings apply more broadly to 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 neuronal avalanches sizes and durations), are robust to complex underlying network topologies. We model avalanches in complex networks by considering a collection of connected nodes where the connection strength between two nodes determines the probability that an excitation is passed from one node to the next. Networks of size $N$ can be identified with a N x N adjacency matrix where the ijth entry represents the connection strength from node i to node j. Networks are separated into three classes: subcritical, critical, and supercritical based on the largest eigenvalue of the adjacency matrix. We are able to determine the distribution for avalanche size and duration for each type of network.

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