Forward Reachable Sets: Analytically derived properties of connected components for dynamic networks

TL;DR

This paper analytically investigates the properties of forward reachable sets in dynamic networks, deriving formulas for their growth and thresholds, which are relevant for understanding epidemic spread and network connectivity.

Contribution

It provides the first closed-form expressions for the mean and variance of FRS growth rates in stochastic dynamic networks with node and edge dynamics.

Findings

Derived formulas for FRS growth rate mean and variance.

Identified thresholds for FRS growth in networks with node dynamics.

Explored effects of edge duration and degree distribution on FRS properties.

Abstract

Formal analysis of the emergent structural properties of dynamic networks is largely uncharted territory. We focus here on the properties of forward reachable sets (FRS) as a function of the underlying degree distribution and edge duration. FRS are defined as the set of nodes that can be reached from an initial seed via a path of temporally ordered edges; a natural extension of connected component measures to dynamic networks. Working in a stochastic framework, we derive closed-form expressions for the mean and variance of the exponential growth rate of the FRS for temporal networks with both edge and node dynamics. For networks with node dynamics, we calculate thresholds for the growth of the FRS. The effects of finite population size are explored via simulation and approximation. We examine how these properties vary by edge duration and different cross-sectional degree distributions that characterize a range of scientifically interesting normative outcomes (Poisson and Bernoulli). The size of the forward reachable set gives an upper bound for the epidemic size in disease transmission network models, relating this work to epidemic modeling (Ferguson 2000, Eames 2004).