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, which 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.In this paper, we focus on stochastically-evolving temporal graphs. We define a model of evolution, where there exists a single stochastic rule, which is applied independently to each edge. Furthermore, 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. Assume now that a single mobile agent is placed on an arbitrary node of a temporal graph evolving under the aforementioned model. Next, the agent performs a simple random walk; at each time step, after the graph instance is fixed according to the model, the agent chooses uniformly at random a node amongst the neighbors of its current node and visits it. The cover time of such a walk is defined as the expected number of time steps until the agent has visited each node at least once. Herein, we prove some first bounds on the cover time for a simple random walk as defined above, mostly via the use of Markovian theory.Random walks constitute a very important primitive in terms of distributed computing. Examples include their use in information dissemination [1] and random network structure [4]; also, see the short survey in [8]. In this work, we consider a single random walk as a fundame...

In the fast evacuation problem, we study the path planning problem for two robots who want to minimize the worst-case evacuation time on the unit disk. The robots are initially placed at the center of the disk. In order to evacuate, they need to reach an unknown point, the exit, on the boundary of the disk. Once one of the robots finds the exit, it will instantaneously notify the other agent, who will make a beeline to it.The problem has been studied for robots with the same speed [12]. We study a more general case where one robot has speed 1 and the other has speed s ≥ 1. We provide optimal evacuation strategies in the case that s ≥ c 2.75 ≈ 2.75 by showing matching upper and lower bounds on the worst-case evacuation time. For 1 ≤ s < c 2.75 , we show (non-matching) upper and lower bounds on the evacuation time with a ratio less than 1.22. Moreover, we demonstrate that a generalization of the two-robot search strategy from [12] is outperformed by our proposed strategies for any s ≥ c 1.71 ≈ 1.71.

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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.