Skip to main content Accessibility help
×
Home
Hostname: page-component-6f6fcd54b-vl5pb Total loading time: 0.182 Render date: 2021-05-11T02:34:31.701Z Has data issue: true Feature Flags: {}

Exponential Random Graphs as Models of Overlay Networks

Published online by Cambridge University Press:  14 July 2016

M. Draief
Affiliation:
Imperial College London
A. Ganesh
Affiliation:
University of Bristol
L. Massoulié
Affiliation:
Thomson Research
Rights & Permissions[Opens in a new window]

Abstract

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 (- βi=1 n d i 2), where d i 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.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Ball, F. and Barbour, A. (1990). Poisson approximation for some epidemic models. J. Appl. Prob. 27, 479490.CrossRefGoogle Scholar
[2] Besag, J. E. (1974). Spatial interaction and the statistical analysis of lattice systems. J. R. Statist. Soc. Ser. B 36, 192236.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.Google 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. 157172.Google Scholar
[6] Draief, M., Ganesh, A. and Massoulié, L. (2008). Thresholds for virus spread on networks. Ann. Appl. Prob. 18, 359378.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, 1761.Google Scholar
[9] Frank, O. and Strauss, D. (1986). Markov graphs. J. Amer. Statist. Assoc. 81, 832842.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. 4755.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, 248258.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. 14551466 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. 148159.Google Scholar
[15] Holland, P. W. and Leinhardt, S. (1981). An exponential family of probability densities for directed graphs. J. Amer. Statist. Assoc. 76, 3365.CrossRefGoogle Scholar
[16] Janson, S. (2009). The probability that a random multigraph is simple. To appear in Combinatorics Prob. Comput. CrossRefGoogle 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. 353397.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, 7293.Google Scholar
[19] McDonald, D. (1979). A local limit theorem for large deviations of sums of independent, nonidentically distributed random variables. Ann. Prob. 7, 526531.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, 565580.CrossRefGoogle Scholar
[21] Metropolis, N. et al. (1953). Equations of state calculations by fast computing machines. J. Chemical Physics 21, 10871092.CrossRefGoogle Scholar
[22] Mohar, B. (1997). Some applications of Laplace eigenvalues of graphs. In Graph Symmetry, eds Hahn, G. and Sabidussi, G., Kluwer, Dordrecht, pp. 225275.CrossRefGoogle Scholar
[23] Park, J. and Newman, M. E. J. (2004). Statistical mechanics of networks. Phys. Rev. E 70, 066117.CrossRefGoogle ScholarPubMed
[24] Snijders, T. A. B. (2002). Markov chain Monte Carlo estimation of exponential random graph models. J. Social Structure 3, 140.Google Scholar
[25] Solomonoff, R. and Rapoport, A. (1951). Connectivity of random nets. Bull. Math. Biophysics 13, 107117.CrossRefGoogle 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. 916.CrossRefGoogle Scholar
[27] Van der Hofstad, R. (2009). Random Graphs and Complex Networks. Available at http://www.win.tue.nl/rhofstad/NotesRGCN2009.pdf.Google Scholar
You have Access

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Exponential Random Graphs as Models of Overlay Networks
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Exponential Random Graphs as Models of Overlay Networks
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Exponential Random Graphs as Models of Overlay Networks
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *