Cover Time in Edge-Uniform Stochastically-Evolving Graphs

  • Ioannis LamprouEmail author
  • Russell Martin
  • Paul Spirakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10616)


We define a general model of stochastically evolving graphs, namely the Edge-Uniform Stochastically-Evolving Graphs. In this model, each possible edge of an underlying general static graph evolves independently being either alive or dead at each discrete time step of evolution following a (Markovian) stochastic rule. The stochastic rule is identical for each possible edge and may depend on the past \(k \ge 0\) observations of the edge’s state.

We examine two kinds of random walks for a single agent taking place in such a dynamic graph: (i) The Random Walk with a Delay (RWD), where at each step the agent chooses (uniformly at random) an incident possible edge (i.e. an incident edge in the underlying static graph) and then it waits till the edge becomes alive to traverse it. (ii) The more natural Random Walk on what is Available (RWA) where the agent only looks at alive incident edges at each time step and traverses one of them uniformly at random. Our study is on bounding the cover time, i.e. the expected time until each node is visited at least once by the agent.

For RWD, we provide the first upper bounds for the cases \(k = 0, 1\) by correlating RWD with a simple random walk on a static graph. Moreover, we present a modified electrical network theory capturing the \(k = 0\) case and a mixing-time argument toward an upper bound for the case \(k = 1\).

For RWA, we derive the first upper bounds for the cases \(k = 0, 1\), too, by reducing RWA to an RWD-equivalent walk with a modified delay. Finally, for the case \(k = 1\), we prove that when the underlying graph is complete, then the cover time is \(\mathcal {O}(n\log n)\) (i.e. it matches the cover time on the static complete graph) under only a mild condition on the edge-existence probabilities determined by the stochastic rule.


Dynamic graphs Random walk Cover time Stochastically-evolving network Edge-independent 



We would like to acknowledge an anonymous reviewer who identified an important technical error in a previous version of this extended abstract and another anonymous reviewer who suggested the use of Theorem 1 as an alternative to electrical network theory and several other useful modifications.


  1. 1.
    Aleliunas, R., Karp, R., Lipton, R., Lovasz, L., Rackoff, C.: Random walks, universal traversal sequences and the complexity of maze problems. In: 20th IEEE Annual Symposium on Foundations of Computer Science, pp. 218–223 (1979)Google Scholar
  2. 2.
    Avin, C., Koucký, M., Lotker, Z.: How to explore a fast-changing world (cover time of a simple random walk on evolving graphs). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5125, pp. 121–132. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-70575-8_11CrossRefGoogle Scholar
  3. 3.
    Bar-Ilan, J., Zernik, D.: Random leaders and random spanning trees. In: Bermond, J.-C., Raynal, M. (eds.) WDAG 1989. LNCS, vol. 392, pp. 1–12. Springer, Heidelberg (1989). doi: 10.1007/3-540-51687-5_27CrossRefGoogle Scholar
  4. 4.
    Baumann, H., Crescenzi, P., Fraigniaud, P.: Parsimonious flooding in dynamic graphs. In: Proceedings of 28th ACM Symposium on Principles of Distributed Computing (PODC 2009), pp. 260–269. ACM (2009)Google Scholar
  5. 5.
    Bui, M., Bernard, T., Sohier, D., Bui, A.: Random walks in distributed computing: a survey. In: Böhme, T., Larios Rosillo, V.M., Unger, H., Unger, H. (eds.) IICS 2004. LNCS, vol. 3473, pp. 1–14. Springer, Heidelberg (2006). doi: 10.1007/11553762_1CrossRefGoogle Scholar
  6. 6.
    Chandra, A.K., Raghavan, P., Ruzzo, W.L., Smolensky, R.: The electrical resistance of a graph captures its commute and cover times. In: Proceedings of 21t Annual ACM Symposium on Theory of Computing (STOC 1989), pp. 574–586. ACM (1989)Google Scholar
  7. 7.
    Clementi, A.E.F., Macci, C., Monti, A., Pasquale, F., Silvestri, R.: Flooding time in edge-Markovian dynamic graphs. In: PODC 2008, pp. 213–222. ACM (2008)Google Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Clementi, A., Monti, A., Pasquale, F., Silvestri, R.: Communication in dynamic radio networks. In: PODC 2007, pp. 205–214. ACM (2007)Google Scholar
  10. 10.
    Doyle, P.G., Snell, J.L.: Random Walks and Electric Networks (2006)Google Scholar
  11. 11.
    Habib, M., McDiarmid, C., Ramirez-Alfonsin, J., Reed, B.: Probabilistic Methods for Algorithmic Discrete Mathematics. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Hoffmann, T., Porter, M.A., Lambiotte, R.: Random walks on stochastic temporal networks. In: Holme, P., Saramäki, J. (eds.) Temporal Networks. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-36461-7_15CrossRefGoogle Scholar
  13. 13.
    Michail, O.: An introduction to temporal graphs: an algorithmic perspective. Internet Math. 12(4), 239–280 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  15. 15.
    Ramiro, V., Lochin, E., Snac, P., Rakotoarivelo, T.: Temporal random walk as a lightweight communication infrastructure for opportunistic networks. In: Proceeding of IEEE International Symposium on a World of Wireless, Mobile and Multimedia Networks, pp. 1–6 (2014)Google Scholar
  16. 16.
    Starnini, M., Baronchelli, A., Barrat, A., Pastor-Satorras, R.: Random walks on temporal networks. Phys. Rev. E 85, 056115 (2012)CrossRefGoogle Scholar
  17. 17.
    Wald, A.: Sequential Analysis. Wiley, New York (1947)zbMATHGoogle Scholar
  18. 18.
    Yamauchi, Y., Izumi, T., Kamei, S.: Mobile agent rendezvous on a probabilistic edge evolving ring. In: Proceedings of 3rd International Conference on Networking and Computing (ICNC 2012), pp. 103–112 (2012)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ioannis Lamprou
    • 1
    Email author
  • Russell Martin
    • 1
  • Paul Spirakis
    • 1
    • 2
  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  2. 2.Computer Technology Institute and Press “Diophantus” (CTI)PatrasGreece

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