  It is not uncommon for certain social networks to divide into two opposing
camps in response to stress. This happens, for example, in networks of
political parties during winner-takes-all elections, in networks of companies
competing to establish technical standards, and in networks of nations faced
with mounting threats of war. A simple model for these two-sided separations is
the dynamical system dX/dt = X^2 where X is a matrix of the friendliness or
unfriendliness between pairs of nodes in the network. Previous simulations
suggested that only two types of behavior were possible for this system: either
all relationships become friendly, or two hostile factions emerge. Here we
prove that for generic initial conditions, these are indeed the only possible
outcomes. Our analysis yields a closed-form expression for faction membership
as a function of the initial conditions, and implies that the initial amount of
friendliness in large social networks (started from random initial conditions)
determines whether they will end up in intractable conflict or global harmony.
