The Interplay Between Dynamics and Networks: Centrality, Communities, and Cheeger Inequality

TL;DR

This paper introduces a unified framework for analyzing dynamic processes on networks, extending traditional measures like conductance and Cheeger's inequality to better identify communities and influential nodes.

Contribution

It generalizes the Laplacian framework to continuous-time biased random walks and epidemic models, providing new tools for network analysis.

Findings

Extended Cheeger's inequality to dynamic processes

Defined a generalized cluster quality measure

Unified framework for various network dynamics

Abstract

We study the interplay between a dynamic process and the structure of the network on which it is defined. Specifically, we examine the impact of this interaction on the quality-measure of network clusters and node centrality. This enables us to effectively identify network communities and important nodes participating in the dynamics. As the first step towards this objective, we introduce an umbrella framework for defining and characterizing an ensemble of dynamic processes on a network. This framework generalizes the traditional Laplacian framework to continuous-time biased random walks and also allows us to model some epidemic processes over a network. For each dynamic process in our framework, we can define a function that measures the quality of every subset of nodes as a potential cluster (or community) with respect to this process on a given network. This subset-quality function generalizes the traditional conductance measure for graph partitioning. We partially justify our choice of the quality function by showing that the classic Cheeger's inequality, which relates the conductance of the best cluster in a network with a spectral quantity of its Laplacian matrix, can be extended from the Laplacian-conductance setting to this more general setting.