  We consider a toy model of interacting extrovert and introvert agents
introduced earlier by Liu et al [Europhys. Lett. {\bf 100} (2012) 66007]. The
number of extroverts, and introverts is $N$ each. At each time step, we select
an agent at random, and allow her to modify her state. If an extrovert is
selected, she adds a link at random to an unconnected introvert. If an
introvert is selected, she removes one of her links. The set of $N^2$ links
evolves in time, and may be considered as a set of Ising spins on an $N \times
N$ square-grid with single-spin-flip dynamics. This dynamics satisfies detailed
balance condition, and the probability of different spin configurations in the
steady state can be determined exactly. The effective hamiltonian has
long-range multi-spin couplings that depend on the row and column sums of
spins. If the relative bias of choosing an extrovert over an introvert is
varied, this system undergoes a phase transition from a state with very few
links to one in which most links are occupied. We show that the behavior of the
system can be determined exactly in the limit of large $N$. The behavior of
large fluctuations in the total numer of links near the phase transition is
determined. We also discuss two variations, called egalitarian and elitist
agents, when the agents preferentially add or delete links to their least/
most-connected neighbor. These shows interesting cooperative behavior.
