Using First Hitting Times to Find Sets that Maximize the Convergence Rate to Consensus

In a model of communication in a social network described by a simple consensus model, we pose the problem of finding a subset of nodes with given cardinality and fixed consensus values that enable the fastest convergence rate to equilibrium of the values of the remaining nodes. Given a network topology and a subset, called the stubborn nodes, the equilibrium exists and is a convex sum of the initial values of the stubborn nodes. The value at a non-stubborn node converges to its consensus value exponentially with a rate constant determined by the expected first hitting time of a random walker starting at the node and ending at the first stubborn node it visits. In this paper, we will use the sum of the expected first hitting times to the stubborn nodes as an objective function for a minimization problem. Its solution is a set with the fastest convergence rate. We present a polynomial time method for obtaining approximate solutions of the optimization problem for fixed cardinality less than that of a reference vertex cover. Under the assumption that the transition matrix for the random walk is irreducible and reversible, we also obtain an upper bound for the expected first hitting time and therefore an upper bound on the rate of convergence to consensus, using results from the mixing theory of Markov chains