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Forward reachable sets: Analytically derived properties of connected components for dynamic networks

Published online by Cambridge University Press:  29 June 2017

BENJAMIN ARMBRUSTER ,
LI WANG  and
MARTINA MORRIS
BENJAMIN ARMBRUSTER
Affiliation:
Northwestern University, Industrial Engineering and Management Sciences, Evanston, IL, USA (e-mail: armbrusterb@gmail.com)
LI WANG
Affiliation:
Statistics, University of Washington, Seattle, WA, USA (e-mails: lxwang@uw.edu, morrism@u.washington.edu)
MARTINA MORRIS
Affiliation:
Statistics, University of Washington, Seattle, WA, USA (e-mails: lxwang@uw.edu, morrism@u.washington.edu)
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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).

Keywords

networksdynamic networknetwork transmissionreachable setreachabilityepidemic size
Type
Research Article
Information
Network Science , Volume 5 , Special Issue 3: Networks and Health , September 2017 , pp. 328 - 354
Copyright
Copyright © Cambridge University Press 2017 

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