Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T15:58:23.164Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential Random Graphs as Models of Overlay Networks

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

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

[1] Ball, F. and Barbour, A. (1990). Poisson approximation for some epidemic models. J. Appl. Prob. 27, 479–490.Google Scholar
[2] Besag, J. E. (1974). Spatial interaction and the statistical analysis of lattice systems. J. R. Statist. Soc. Ser. B 36, 192–236.Google Scholar
[3] Bollobàs, B. (2001). Random Graphs (Camb. Stud. Adv. Math. 73). Cambridge Univeristy Press.CrossRefGoogle Scholar
[4] Brémaud, P. (1999). Markov chains, Gibbs fields, Monte Carlo Simulation, and Queues. Springer, New York.CrossRefGoogle Scholar
[5] Chung, F. (1996). Laplacians of graphs and Cheeger's inequalities. In Combinatorics, Paul Erdös is Eighty, Vol. 2 (Keszthely, 1993), János Bolyai Mathematical Society, Budapest, pp. 157–172.Google Scholar
[6] Draief, M., Ganesh, A. and Massoulié, L. (2008). Thresholds for virus spread on networks. Ann. Appl. Prob. 18, 359–378.CrossRefGoogle Scholar
[7] Durrett, R. (2007). Random Graph Dynamics. Cambridge University Press.Google Scholar
[8] Erdös, P. and Rényi, A. (1960). On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5, 17–61.Google Scholar
[9] Frank, O. and Strauss, D. (1986). Markov graphs. J. Amer. Statist. Assoc. 81, 832–842.CrossRefGoogle Scholar
[10] Ganesh, A. J., Kermarrec, A.-M. and Massoulié, L. (2003). Network awareness and failure resilience in self-organizing overlay networks. In Proc. IEEE Symp. Reliab. Distributed Systems, IEEE, pp. 47–55.Google Scholar
[11] Ganesh, A. J. and Massoulié, L. (2003). Failure resilience in balanced overlay networks. In Proc. 41st Allerton Conf. Commun., Control Comput., IEEE.Google Scholar
[12] Ganesh, A. J., Kermarrec, A.-M. and Massoulié, L. (2003). Probabilistic reliable dissemination in large-scale systems. IEEE Trans. Parallel Distributed Systems 14, 248–258.Google Scholar
[13] Ganesh, A. J., Massoulié, L. and Towsley, D. (2005). The effect of network topology on the spread of epidemics. In Proc. IEEE INFOCOM 2005, IEEE, pp. 1455–1466 Google Scholar
[14] Gkantsidis, C., Mihail, M. and Saberi, A. (2003). Conductance and congestion in power law graphs. In Proc. ACM SIGMETRICS 2003, ACM, pp. 148–159.Google Scholar
[15] Holland, P. W. and Leinhardt, S. (1981). An exponential family of probability densities for directed graphs. J. Amer. Statist. Assoc. 76, 33–65.Google Scholar
[16] Janson, S. (2009). The probability that a random multigraph is simple. To appear in Combinatorics Prob. Comput. Google Scholar
[17] Lovász, L. (1996). Random walks on graphs: a survey. In Combinatorics, Paul Erdös is Eighty, Vol. 2, (Keszthely, 1993), János Bolyai Mathematical Society, Budapest, pp. 353–397.Google Scholar
[18] Lua, E. K. et al. (2005). A survey and comparison of peer-to-peer overlay network schemes. IEEE Commun. Surveys Tutorials 7, 72–93.Google Scholar
[19] McDonald, D. (1979). A local limit theorem for large deviations of sums of independent, nonidentically distributed random variables. Ann. Prob. 7, 526–531.CrossRefGoogle Scholar
[20] McKay, B. D. and Wormald, N. C. (1990). Asymptotic enumeration by degree sequence of graphs of high degree. Europ. J. Combinatorics 11, 565–580.Google Scholar
[21] Metropolis, N. et al. (1953). Equations of state calculations by fast computing machines. J. Chemical Physics 21, 1087–1092.Google Scholar
[22] Mohar, B. (1997). Some applications of Laplace eigenvalues of graphs. In Graph Symmetry, eds Hahn, G. and Sabidussi, G., Kluwer, Dordrecht, pp. 225–275.CrossRefGoogle Scholar
[23] Park, J. and Newman, M. E. J. (2004). Statistical mechanics of networks. Phys. Rev. E 70, 066117.Google Scholar
[24] Snijders, T. A. B. (2002). Markov chain Monte Carlo estimation of exponential random graph models. J. Social Structure 3, 1–40.Google Scholar
[25] Solomonoff, R. and Rapoport, A. (1951). Connectivity of random nets. Bull. Math. Biophysics 13, 107–117.Google Scholar
[26] Steinmetz, R. and Wehrle, K. (2005). What is this peer-to-peer about? In Peer-to-Peer Systems and Applications, Springer, Berlin, pp. 9–16.Google Scholar
[27] Van der Hofstad, R. (2009). Random Graphs and Complex Networks. Available at http://www.win.tue.nl/rhofstad/NotesRGCN2009.pdf.Google Scholar