The common randomness capacity of a finite network of channels

Consider a finite number of agents interconnected by an arbitrary network of independent, point-to-point, discrete memoryless channels. The agents wish to generate common randomness by interactive communication over the network. Our main result is an exact characterization of the common randomness capacity of such a network, i.e. the maximum number of bits of randomness that all the agents can agree on, per step of communication. As a by-product, we also obtain a description by linear inequalities of the blocking-type polyhedron whose extreme points are precisely the incidence vectors of all arborescences in a digraph, with a prescribed root of out-degree 1.