In this paper, we generalize our previous results on generating common rand
omness at two terminals to a situation where any finite number of agents, i
nterconnected by an arbitrary network of independent, point-to-point, discr
ete memoryless channels, wish to generate common randomness by interactive
communication over the network. Our main result is an exact characterizatio
n of the common randomness capacity of such a network, i.e., the maximum nu
mber of bits of randomness that all the agents can agree on per step of com
munication. As a by-product, we also obtain a purely combinatorial result,
viz,, a characterization of (the incidence vectors of) the spanning arbores
cences rooted at a specified vertex in a digraph, and having exactly one ed
ge exiting the root, as precisely the extreme points of a certain unbounded
convex polyhedron, described by a system of linear inequalities.