Statistical Properties of Avalanches in Networks

TL;DR

This paper analyzes how the structure of complex networks influences the size and duration of avalanches, revealing that network topology affects cascade statistics and that critical brain dynamics signatures are robust across different network types.

Contribution

It provides a novel characterization of avalanche distributions in individual networks using spectral properties, extending previous mean-field analyses to account for network structure.

Findings

Avalanche size and duration distributions depend on the largest eigenvalue of the network's adjacency matrix.

Results apply to individual networks, not just ensembles, allowing for node-specific analysis.

Critical brain dynamics signatures are robust to complex network topologies.

Abstract

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.