  A gossip process is an iterative process in a multi-agent system where only
two neighboring agents communicate at each iteration and update their states.
The neighboring condition is by convention described by an undirected graph. In
this paper, we consider a general update rule whereby each agent takes an
arbitrary weighted average of its and its neighbor's current states. In
general, the limit of the gossip process (if it converges) depends on the order
of iterations of the gossiping pairs. The main contribution of the paper is to
provide a necessary and sufficient condition for convergence of the gossip
process that is independent of the order of iterations. This result relies on
the introduction of the novel notion of holonomy of local stochastic matrices
for the communication graph. We also provide complete characterizations of the
limit and the space of holonomic stochastic matrices over the graph.
