Skip to main content
SpringerLink
Log in

Information spreading in dynamic graphs

  • Published:
Distributed Computing Aims and scope Submit manuscript

Abstract

We present a general approach to study the flooding time (a measure of how fast information spreads) in dynamic graphs (graphs whose topology changes with time according to a random process). We consider arbitrary ergodic Markovian dynamic graph processes, that is, processes in which the topology of the graph at time \(t\) depends only on its topology at time \(t-1\) and which have a unique stationary distribution. The most well studied models of dynamic graphs are all Markovian and ergodic. Under general conditions, we bound the flooding time in terms of the mixing time of the dynamic graph process. We recover, as special cases of our result, bounds on the flooding time for the random trip model and the random path one; previous analysis techniques provided bounds only in restricted settings for such models. Our result also provides the first bound for the random waypoint model whose analysis had been an important open question. The bound is tight for the most realistic ranges of the network parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. With an abuse of notation, event probabilities such as \(\mathbf {P} \left( e(i,A) =1 \right) \) will be shortly denoted as \(\mathbf {P} \left( e(i,A) \right) \).

  2. We say that an event holds with high probability if it holds with probability at least \(1 - 1/n\).

  3. The level of resolution does not affect the obtained bound on the flooding time, provided the resolution is high enough.

  4. Recall that we are still using the abbreviations like \(I_\tau = I_{\tau M}\).

References

  1. Aldous, D., Fill, J.A.: Reversible Markov Chains and Random Walks on Graphs, Chap. 14 (1999). http://www.stat.berkeley.edu/aldous/RWG/book.html

  2. Avin, C., Koucky, M., Lotker, Z.: How to explore a fast-changing world. In: Proceedings of 35th ICALP’08, LNCS, vol. 5125, pp. 121–132 (2008)

  3. Azar, Y., Broder, A.Z., Karlin, A.R., Upfal, E.: Balanced allocation. SIAM J. Comput. 29(1), 180–200 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baumann, H., Crescenzi, P., Fraigniaud, P.: Parsimonious flooding in dynamic graphs. Distrib. Comput. 24(1), 31–44 (2011)

    Article  MATH  Google Scholar 

  5. Becchetti, L., Clementi, A.E.F., Pasquale, F., Resta, G., Santi, P., Silvestri, R.: Information spreading in opportunistic networks is fast. arXiv:1107.5241v1 (2011)

  6. Bettstetter, C., Resta, G., Santi, P.: The node distribution of the random waypoint mobility model for wireless ad hoc networks. IEEE Trans. Mob. Comput. 2, 257–269 (2003)

    Article  Google Scholar 

  7. Camp, T., Boleng, J., Davies, V.: A survey of mobility models for ad hoc network research. Wirel. Commun. Mob. Comput. 2(5), 483–502 (2002)

    Article  Google Scholar 

  8. Camp, T., Navidi, W., Bauer, N.: Improving the accuracy of random waypoint simulations through steady-state initialization. In: Proceedings of 15th Interantional Conference on Modelling and Simulation, pp. 319–326 (2004)

  9. Chierichetti, F., Lattanzi, S., Panconesi, A.: Rumor spreading in social networks. Theor. Comput. Sci. 412(24), 2602–2610 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Clementi, A., Monti, A., Pasquale, F., Silvestri, R.: Broadcasting in dynamic radio networks. J. Comput. Syst. Sci. 75(4), 213–230 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Clementi, A., Macci, C., Monti, A., Pasquale, F., Silvestri, R.: Flooding time of edge-Markovian evolving graphs. SIAM J. Discrete Math. 24(4), 1694–1712 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Clementi, A., Monti, A., Pasquale, F., Silvestri, R.: Information spreading in stationary markovian evolving graphs. IEEE Trans. Parallel Distrib. Syst. 22(9), 1425–1432 (2011)

    Article  Google Scholar 

  13. Clementi, A., Pasquale, F., Silvestri, R.: Opportunistic manets: high mobility can make up for low transmission power. IEEE/ACM Trans. Netw. 21(2), 610–620 (2013)

    Article  Google Scholar 

  14. Clementi, A., Monti, A., Silvestri, R.: Flooding over Manhattan. Distrib. Comput. 26(1), 25–38 (2013)

    Article  MATH  Google Scholar 

  15. Clementi, A., Monti, A., Silvestri, R.: Modelling mobility: a discrete revolution. Ad Hoc Netw. 9(6), 998–1014 (2011)

    Article  Google Scholar 

  16. Dimitriou, T., Nikoletseas, S., Spirakis, P.: The infection time of graphs. Discrete Appl. Math. 154(8), 2577–2589 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Easley, D., Kleinberg, J.: Networks, Crowds, and Markets. Cambridge University Press, Cambridge (2010)

  18. Ghosh, B., Leighton, F.T., Maggs, B.M., Muthukrishnan, S., Plaxton, C.G., Rajaraman, R., Richa, A.W., Tarjan, R.E., Zuckerman, D.I.: Tight analyses of two local load balancing algorithms. SIAM J. Comput. 29(1), 29–64 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Grindrod, P., Higham, D.J.: Evolving graphs: dynamical models, inverse problems and propagation. Proc. R. Soc. A 466(2115), 753–770 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jacquet, P., Mans, B., Rodolakis, G.: Information propagation speed in mobile and delay tolerant networks. IEEE Trans. Inf. Theory 56, 5001–5015 (2010)

    Article  MathSciNet  Google Scholar 

  21. Karagiannis, T., Le Boudec, J.-Y., Vojnovic, M.: Power law and exponential decay of inter contact times between mobile devices. In: Proceedings of 13th ACM MOBICOM, pp. 183–194 (2007)

  22. Kesten, H., Sidoravicius, V.: The spread of a rumor or infection in a moving population. Ann. Probab. 33(6), 2402–2462 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Khun, F., Lynch, N., Oshman, R.: Distributed computation in dynamic networks. In: Proceedings of 42nd ACM STOC, pp. 513–522 (2010)

  24. Khun, F., Oshman, R.: Dynamic networks: models and algorithms. ACM SIGACT News 42(1), 82–96 (2011)

    Article  Google Scholar 

  25. Lam, H., Liu, Z., Mitzenmacher, M., Sun, X., Wang, Y.: Information dissemination via random walks in d-dimensional space. In: Proceedings of 23rd ACM-SIAM SODA, pp. 1612–1622 (2012)

  26. Le Boudec, J.-Y., Vojnovic, M.: The random trip model: stability, stationary regime, and perfect simulation. IEEE/ACM Trans. Netw. 14(6), 1153–1166 (2006)

    Article  Google Scholar 

  27. Le Boudec, J.-Y.: Understanding the simulation of mobility models with palm calculus. Perform. Eval. 64(2), 126–147 (2007)

    Article  Google Scholar 

  28. Peres, Y., Sinclair, A., Sousi, P., Stauffer, A.: Mobile geometric graphs: detection, coverage and percolation. In: Proceedings of 22nd ACM-SIAM SODA, pp. 412–428 (2011)

  29. Pettarin, A., Pietracaprina, A., Pucci, G., Upfal, E.: Tight bounds on information dissemination in sparse mobile networks. In: Proceedings of 30th ACM PODC, pp. 355–362 (2011)

  30. Rabani, Y., Sinclair, A., Wanka, R.: Local divergence of Markov chains and the analysis of iterative load-balancing schemes. In: Proceedings of 39th IEEE FOCS, pp. 694–703 (1998)

  31. Spyropoulos, T., Jindal, A., Psounis, K.: An analytical study of fundamental mobility properties for encounter based protocols. Int. J. Auton. Adapt. Commun. Syst. 1(1), 4–40 (2008)

    Article  Google Scholar 

  32. Vojnovic, M., Proutier, A.: Hop limited flooding over dynamic networks. In: Proceedings of 30th IEEE INFOCOM, pp. 685–693 (2011)

  33. Whitbeck, J., Conan, V., de Amorim, M.D.: Performance of opportunistic epidemic routing on edge-Markovian dynamic graphs. IEEE Trans. Commun. 59(5), 1259–1263 (2011)

    Article  Google Scholar 

Download references

Acknowledgments

We thank Francesco Pasquale and Alessandro Pettarin for carefully reading a previous version of our paper and for providing helpful comments. We are also grateful to the anonymous referees that suggested several improvements of the paper: in particular, one of them showed us a simple way to improve our upper bounds by a logarithmic factor.

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria dell’Impresa, Università Tor Vergata di Roma, Roma, Italy

    Andrea Clementi

  2. Dipartimento di Informatica, Sapienza Università di Roma, Roma, Italy

    Riccardo Silvestri

  3. Department of Computer Science, Stanford University, Stanford, CA, USA

    Luca Trevisan

Authors
  1. Andrea Clementi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Riccardo Silvestri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luca Trevisan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrea Clementi.

Additional information

An extended abstract of this work has been presented at 31st Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2012).

Andrea Clementi: Partially supported by Italian MIUR under the PRIN 2010-11 Project ARS TechnoMedia.

Luca Trevisan: Supported by the National Science Foundation under Grant No. CCF-1216642 and by the US-Israel Binational Science Foundation under Grant No. 0378582.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Clementi, A., Silvestri, R. & Trevisan, L. Information spreading in dynamic graphs. Distrib. Comput. 28, 55–73 (2015). https://doi.org/10.1007/s00446-014-0219-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00446-014-0219-2

Keywords

Search

Navigation