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Exponential Random Graphs as Models of Overlay Networks

Part of: Special processes Operations research and management science Discrete mathematics in relation to computer science Graph theory

Published online by Cambridge University Press:  14 July 2016

M. Draief ,
A. Ganesh  and
L. Massoulié
M. Draief*
Affiliation:
Imperial College London
A. Ganesh*
Affiliation:
University of Bristol
L. Massoulié*
Affiliation:
Thomson Research
*
∗Postal address: Imperial College London, South Kensington Campus, London, SW7 2AZ, UK. Email address: m.draief@imperial.ac.uk
∗∗Postal address: University of Bristol, University Walk, Bristol, BS8 1TW, UK. Email address: a.ganesh@bristol.ac.uk
∗∗∗Postal address: Thomson Research, Boulogne, 92648, France. Email address: laurent.massoulie@thomson.net
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Abstract

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In this paper we give an analytic solution for graphs with n nodes and E = cn log n edges for which the probability of obtaining a given graph G is µn (G) = exp (- β ∑i=1ndi2), where di is the degree of node i. We describe how this model appears in the context of load balancing in communication networks, namely peer-to-peer overlays. We then analyse the degree distribution of such graphs and show that the degrees are concentrated around their mean value. Finally, we derive asymptotic results for the number of edges crossing a graph cut and use these results (i) to compute the graph expansion and conductance, and (ii) to analyse the graph resilience to random failures.

Keywords

Exponential random graphpeer-to-peer networkoverlay optimisationload balancingdegree distributiongraph cutexpansionconductancefailure resilience

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 68R10: Graph theory (including graph drawing) 90B18: Communication networks 05C07: Vertex degrees 05C80: Random graphs 05C85: Graph algorithms 05C90: Applications
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 199 - 220
Copyright
Copyright © Applied Probability Trust 2009 

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