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.
Highlights
In the modern era of the Internet, modifications in a network topology can occur extremely frequently and in a disorderly way
We define a general model of stochastically-evolving graphs, where each possible edge evolves independently, but all of them evolve following the same stochastic rule
As a first start and for mathematical convenience, it is formalized as a synchronous system, where all possible edges evolve concurrently in distinct rounds
Summary
In the modern era of the Internet, modifications in a network topology can occur extremely frequently and in a disorderly way. The graph is evolving over a series of (discrete) time steps under a set of deterministic or stochastic rules of evolution. Such a rule can be edge- or graph-specific and may take as input graph instances observed in previous time steps. Our model is general in the sense that the underlying static graph is allowed to be a general connected graph, i.e., with no further constraints on its topology, and the stochastic rule can include any finite number of past observations. Random walks constitute a very important primitive in terms of distributed computing Examples include their use in information dissemination [2] and random network structure [3]; see the short survey in [4]. We consider a single random walk as a fundamental building block for other more distributed scenarios to follow
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call