Cover Time in Edge-Uniform Stochastically-Evolving Graphs

We define a general model of stochastically-evolving graphs, namely the \emph{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 \emph{Random Walk with a Delay} (\emph{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 \emph{Random Walk on what is Available} (\emph{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 \emph{cover time}, i.e., the expected time until each node is visited at least once by the agent. For \emph{RWD}, we provide a first upper bound for the cases $k = 0, 1$ by correlating \emph{RWD} with a simple random walk on a static graph. Moreover, we present a modified electrical network theory capturing the $k = 0$ case. For \emph{RWA}, we derive some first bounds for the case $k = 0$, by reducing \emph{RWA} to an \emph{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 \emph{RWA} in Edge-Uniform graphs and compare the experimental findings with our theoretical bounds

Construction of near-optimal vertex clique covering for real-world networks

We propose a method based on combining a constructive and a bounding heuristic to solve the vertex clique covering problem (CCP), where the aim is to partition the vertices of a graph into the smallest number of classes, which induce cliques. Searching for the solution to CCP is highly motivated by analysis of social and other real-world networks, applications in graph mining, as well as by the fact that CCP is one of the classical NP-hard problems. Combining the construction and the bounding heuristic helped us not only to find high-quality clique coverings but also to determine that in the domain of real-world networks, many of the obtained solutions are optimal, while the rest of them are near-optimal. In addition, the method has a polynomial time complexity and shows much promise for its practical use. Experimental results are presented for a fairly representative benchmark of real-world data. Our test graphs include extracts of web-based social networks, including some very large ones, several well-known graphs from network science, as well as coappearance networks of literary works' characters from the DIMACS graph coloring benchmark. We also present results for synthetic pseudorandom graphs structured according to the Erdös-Renyi model and Leighton's model