  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
