Title: Assortative Mixing Equilibria in Social Network Games,
Abstract: It is known that individuals in social networks tend to exhibit homophily
(a.k.a. assortative mixing) in their social ties, which implies that they
prefer bonding with others of their own kind. But what are the reasons for this
phenomenon? Is it that such relations are more convenient and easier to
maintain? Or are there also some more tangible benefits to be gained from this
collective behaviour?
The current work takes a game-theoretic perspective on this phenomenon, and
studies the conditions under which different assortative mixing strategies lead
to equilibrium in an evolving social network. We focus on a biased preferential
attachment model where the strategy of each group (e.g., political or social
minority) determines the level of bias of its members toward other group
members and non-members. Our first result is that if the utility function that
the group attempts to maximize is the degree centrality of the group,
interpreted as the sum of degrees of the group members in the network, then the
only strategy achieving Nash equilibrium is a perfect homophily, which implies
that cooperation with other groups is harmful to this utility function. A
second, and perhaps more surprising, result is that if a reward for inter-group
cooperation is added to the utility function (e.g., externally enforced by an
authority as a regulation), then there are only two possible equilibria,
namely, perfect homophily or perfect heterophily, and it is possible to
characterize their feasibility spaces. Interestingly, these results hold
regardless of the minority-majority ratio in the population.
We believe that these results, as well as the game-theoretic perspective
presented herein, may contribute to a better understanding of the forces that
shape the groups and communities of our society.