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Algorithms 2018, 11(10), 149; https://doi.org/10.3390/a11100149

Cover Time in Edge-Uniform Stochastically-Evolving Graphs

1
Department of Computer Science, University of Liverpool, Liverpool L69 3BX, UK
2
Department of Computer Engineering and Informatics, University of Patras, 26504 Patras, Greece
An extended abstract of this article appeared in SSS: Stabilization, Safety, and Security of Distributed Systems, Boston, MA, USA, 5–8 November 2017, LNCS 10616, pp. 441–455, Springer.
*
Author to whom correspondence should be addressed.
Received: 28 February 2018 / Revised: 18 July 2018 / Accepted: 29 September 2018 / Published: 2 October 2018
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Abstract

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 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 a first upper bound 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. For RWA, we derive some first bounds for the case k = 0 , by reducing RWA to an RWD-equivalent walk with a modified delay. Further, we also provide a framework that is shown to compute the exact value of the cover time for a general family of stochastically-evolving graphs in exponential time. Finally, we conduct experiments on the cover time of RWA in edge-uniform graphs and compare the experimental findings with our theoretical bounds. View Full-Text
Keywords: dynamic graphs; random walk; cover time; stochastically-evolving; edge-independent; temporal graphs dynamic graphs; random walk; cover time; stochastically-evolving; edge-independent; temporal graphs
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Lamprou, I.; Martin, R.; Spirakis, P. Cover Time in Edge-Uniform Stochastically-Evolving Graphs. Algorithms 2018, 11, 149.

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