research-article

Characterizing continuous time random walks on time varying graphs

Authors:

Daniel Figueiredo,

Edmundo de Souza e Silva,

Don TowsleyAuthors Info & Claims

ACM SIGMETRICS Performance Evaluation Review, Volume 40, Issue 1

Pages 307 - 318

https://doi.org/10.1145/2318857.2254794

Published: 11 June 2012 Publication History

Abstract

In this paper we study the behavior of a continuous time random walk (CTRW) on a stationary and ergodic time varying dynamic graph. We establish conditions under which the CTRW is a stationary and ergodic process. In general, the stationary distribution of the walker depends on the walker rate and is difficult to characterize. However, we characterize the stationary distribution in the following cases: i) the walker rate is significantly larger or smaller than the rate in which the graph changes (time-scale separation), ii) the walker rate is proportional to the degree of the node that it resides on (coupled dynamics), and iii) the degrees of node belonging to the same connected component are identical (structural constraints). We provide examples that illustrate our theoretical findings.

References

[1]

Utku Günay Acer, Petros Drineas, and Alhussein A. Abouzeid. Random walks in time-graphs. In MobiOpp, pages 93--100, 2010.

Digital Library

[2]

C. Avin, M. Koucky, and Z. Lotker. How to explore a fast-changing world. (on the cover time of dynamic graphs). In ICALP, pages 121--132, 2008.

Digital Library

[3]

Andrea E. F. Clementi, Francesco Pasquale, Angelo Monti, and Riccardo Silvestri. Communication in dynamic radio networks. In PODC, pages 205--214. ACM, 2007.

Digital Library

[4]

Andrea E.F. Clementi, Claudio Macci, Angelo Monti, Francesco Pasquale, and Riccardo Silvestri. Flooding time in edge-markovian dynamic graphs. In PODC, pages 213--222. ACM, 2008.

Digital Library

[5]

Daniel Figueiredo, Philippe Nain, Bruno Ribeiro, Edmundo de Souza e Silva, and Don Towsley. Characterizing random walks on general time varying graphs. Technical Report UM-CS-2012-011, UMass Amherst, 2012.

[6]

P. Franken, D. Konig, U. Arndt, and V. Schmidt. Queues and Point Processes. J. Wileyl, 1982.

[7]

M. Gjoka, M. Kurant, C. Butts, and A. Markopoulou. A walk in Facebook: Uniform sampling of users in online social networks. In INFOCOM, 2010.

[8]

S.M. Hedetniemi, S.T. Hedetniemi, and A.L.Liestman. A survey of gossiping and broadcasting in communication networks. Networks, 18(4):319--349, 1988.

[9]

R. Horn and C. Johnson. Matrix analysis. Technical report, Cambridge University Press, 1985.

Digital Library

[10]

D. Kempe and J. Kleinberg. Protocols and impossibility results for gossip-based communication mechanisms. In FOCS, pages 471--480, 2002.

Digital Library

[11]

A. N. Kolmogorov and S. V. Fomin. Introductory Real Analysis. Dover, 1975.

[12]

Laurent Massoulié, Erwan Le Merrer, Anne-Marie Kermarrec, and Ayalvadi Ganesh. Peer counting and sampling in overlay networks: random walk methods. In PODC, pages 123--132, 2006.

Digital Library

[13]

P. Moyal. A generalized backwards scheme for solving non monotonic stochastic recursions. Arxiv preprint arXiv:1009.1235, Sept. 2010.

[14]

M.E.J. Newman. A measure of betweenness centrality based on random walks. Social Networks, 27(1):39--54, 2005.

[15]

B. Pittel. On spreading a rumor. SIAM Journal on Applied Mathematics, 47(1):213--223, 1987.

Digital Library

[16]

A. H. Rasti, M. Torkjazi, R. Rejaie, N. Duffield, W. Willinger, and D. Stutzbach. Respondent-driven sampling for characterizing unstructured overlays. In INFOCOM Mini-Conference, 2009.

[17]

B. Ribeiro and D. Towsley. Estimating and sampling graphs with multidimensional random walks. In SIGCOMM IMC, Nov 2010.

Digital Library

[18]

Bruno Ribeiro, Daniel Figueiredo, Edmundo de Souza e Silva, and Don Towsley. Characterizing continuous-time random walks on dynamic networks. SIGMETRICS, 39(1):343--344, June 2011.

Digital Library

[19]

Albert N. Shiryaev. Probability. Springer, 2nd edition, 1995.

[20]

Daniel Stutzbach, Reza Rejaie, Nick Duffield, Subhabrata Sen, and Walter Willinger. On unbiased sampling for unstructured peer-to-peer networks. IEEE/ACM Trans. Netw., 17:377--390, April 2009.

Digital Library

Cited By

Avena LGüldaş Hvan der Hofstad Rden Hollander FNagy O(2022)Linking the mixing times of random walks on static and dynamic random graphsStochastic Processes and their Applications10.1016/j.spa.2022.07.009153(145-182)Online publication date: Nov-2022
https://doi.org/10.1016/j.spa.2022.07.009
Figueiredo DIacobelli GOliveira RReed BRibeiro R(2021)On a random walk that grows its own treeElectronic Journal of Probability10.1214/20-EJP57426:noneOnline publication date: 1-Jan-2021
https://doi.org/10.1214/20-EJP574
Petit JLambiotte RCarletti T(2019)Classes of random walks on temporal networks with competing timescalesApplied Network Science10.1007/s41109-019-0204-64:1Online publication date: 23-Sep-2019
https://doi.org/10.1007/s41109-019-0204-6
Show More Cited By

Index Terms

Characterizing continuous time random walks on time varying graphs
1. Mathematics of computing
  1. Probability and statistics
    1. Stochastic processes

Recommendations

Characterizing continuous time random walks on time varying graphs
SIGMETRICS '12: Proceedings of the 12th ACM SIGMETRICS/PERFORMANCE joint international conference on Measurement and Modeling of Computer Systems

In this paper we study the behavior of a continuous time random walk (CTRW) on a stationary and ergodic time varying dynamic graph. We establish conditions under which the CTRW is a stationary and ergodic process. In general, the stationary distribution ...
Multiple Random Walks in Random Regular Graphs

We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the analysis for random regular graphs. The cover time of a random walk on a random $r$-regular ...
Random Walks on Randomly Evolving Graphs
Structural Information and Communication Complexity
Abstract
A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution in ...

Comments

Information & Contributors

Information

Published In

cover image ACM SIGMETRICS Performance Evaluation Review

ACM SIGMETRICS Performance Evaluation Review Volume 40, Issue 1

Performance evaluation review

June 2012

433 pages

ISSN:0163-5999

DOI:10.1145/2318857

Issue’s Table of Contents

SIGMETRICS '12: Proceedings of the 12th ACM SIGMETRICS/PERFORMANCE joint international conference on Measurement and Modeling of Computer Systems
June 2012
450 pages
ISBN:9781450310970
DOI:10.1145/2254756
General Chair:
Peter Harrison
Imperial College London, United Kingdom
,
Program Chairs:
Martin Arlitt
HP Labs, USA and University of Calgary, Canada
,
Giuliano Casale
Imperial College London, United Kingdom

Copyright © 2012 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 11 June 2012

Published in SIGMETRICS Volume 40, Issue 1

Check for updates

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

23
Total Citations
View Citations
424
Total Downloads

Downloads (Last 12 months)16
Downloads (Last 6 weeks)2

Reflects downloads up to 08 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Avena LGüldaş Hvan der Hofstad Rden Hollander FNagy O(2022)Linking the mixing times of random walks on static and dynamic random graphsStochastic Processes and their Applications10.1016/j.spa.2022.07.009153(145-182)Online publication date: Nov-2022
https://doi.org/10.1016/j.spa.2022.07.009
Figueiredo DIacobelli GOliveira RReed BRibeiro R(2021)On a random walk that grows its own treeElectronic Journal of Probability10.1214/20-EJP57426:noneOnline publication date: 1-Jan-2021
https://doi.org/10.1214/20-EJP574
Petit JLambiotte RCarletti T(2019)Classes of random walks on temporal networks with competing timescalesApplied Network Science10.1007/s41109-019-0204-64:1Online publication date: 23-Sep-2019
https://doi.org/10.1007/s41109-019-0204-6
Nain PTowsley D(2016)File Dissemination in Dynamic GraphsACM Transactions on Modeling and Performance Evaluation of Computing Systems10.1145/29813442:1(1-23)Online publication date: 18-Nov-2016
https://dl.acm.org/doi/10.1145/2981344
Iacobelli GFigueiredo D(2016)Edge-attractor random walks on dynamic networksJournal of Complex Networks10.1093/comnet/cnw009(cnw009)Online publication date: 21-Jun-2016
https://doi.org/10.1093/comnet/cnw009
Gayraud NPitoura ETsaparas PSharma AAgrawal RGrossglauser M(2015)Diffusion Maximization in Evolving Social NetworksProceedings of the 2015 ACM on Conference on Online Social Networks10.1145/2817946.2817965(125-135)Online publication date: 2-Nov-2015
https://dl.acm.org/doi/10.1145/2817946.2817965
Wehmuth KZiviani AFleury E(2015)A unifying model for representing time-varying graphs2015 IEEE International Conference on Data Science and Advanced Analytics (DSAA)10.1109/DSAA.2015.7344810(1-10)Online publication date: Oct-2015
https://doi.org/10.1109/DSAA.2015.7344810
Lamprou IMartin RSpirakis P(2018)Cover Time in Edge-Uniform Stochastically-Evolving GraphsAlgorithms10.3390/a1110014911:10(149)Online publication date: 2-Oct-2018
https://doi.org/10.3390/a11100149
Silva ASingh ASwami AChampin PGandon FMédini LLalmas MIpeirotis P(2018)Spectral Algorithms for Temporal Graph CutsProceedings of the 2018 World Wide Web Conference10.1145/3178876.3186118(519-528)Online publication date: 10-Apr-2018
https://dl.acm.org/doi/10.1145/3178876.3186118
Shafiq ZLiu A(2017)Cascade size prediction in online social networks2017 IFIP Networking Conference (IFIP Networking) and Workshops10.23919/IFIPNetworking.2017.8264864(1-9)Online publication date: Jun-2017
https://doi.org/10.23919/IFIPNetworking.2017.8264864
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Issue’s Table of Contents