Balance and fragmentation in societies with homophily and social balance

Recent attempts to understand the origin of social fragmentation on the basis of spin models include terms accounting for two social phenomena: homophily—the tendency for people with similar opinions to establish positive relations—and social balance—the tendency for people to establish balanced triadic relations. Spins represent attribute vectors that encode G different opinions of individuals; social interactions between individuals can be positive or negative. Here we present a co-evolutionary Hamiltonian framework that minimizes individuals’ social stress in social networks that have finite connectivity and people with a small number of attributes. We show that such systems always reach stationary, balanced, and fragmented states, if –in addition to homophily– individuals take into account a significant fraction, q, of their triadic relations. Above a critical value, qc, balanced and fragmented states exist for any number of opinions.