Finding Optimal Sinks for Random Walkers in a Network

In a model of network communication based on a random walk in an undirected graph, what subset of nodes (subject to constraints on the set size), enables the fastest spread of information? In this paper, we assume the dynamics of spread is described by a network consensus process, but to find the most effective seeds we consider the target set of a random walk--the process dual to network consensus spread. Thus an optimal set $A$ minimizes the sum of the expected first hitting times $F(A)$ of random walks that start at nodes outside the set. We introduce a submodular, non-decreasing rank function $\rho$, that permits some comparison between the solution obtained by the classical greedy algorithm and one obtained by our methods. The supermodularity and non-increasing properties of $F$ are used to show that the rank of our solution is at least $(1-\frac{1}{e})$ times the rank of the optimal set. When our approximation has a higher rank than the greedy solution, this can be improved to $(1-\frac{1}{e})(1+\chi)$ where $\chi >0$ is a constant. A non-zero lower bound for $\chi$ can be obtained when the curvature and increments of $\rho$ are known.