  Motivated by the study of social insects, we introduce a stochastic model
based on interacting particle systems in order to understand the effect of
communication on the division of labor. Members of the colony are located on
the vertex set of a graph representing a communication network. They are
characterized by one of two possible tasks, which they update at a rate equal
to the cost of the task they are performing by either defecting by switching to
the other task or cooperating by anti-imitating a random neighbor in order to
balance the amount of energy spent in each task. We prove that, at least when
the probability of defection is small, the division of labor is poor when there
is no communication, better when the communication network consists of a
complete graph, but optimal on bipartite graphs with bipartite sets of equal
size, even when both tasks have very different costs. This shows a
non-monotonic relationship between the number of connections in the
communication network and how well individuals organize themselves to
accomplish both tasks equally.
