  Recent attempts to understand the origin of social fragmentation are based on
spin models which 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
multiple (binary) opinions of individuals and social interactions between
individuals can be positive or negative. Recent work suggests that large
systems of N >> 1 individuals never reach a balanced state (where unbalanced
triads with one or three hostile links remain), provided the number of
attributes for each agent is less than O(N^2) [Phys. Rev. Lett. 125, 078302].
Here we show that this statement is overly restrictive. Within a Hamiltonian
framework that minimizes individuals' social stress, we demonstrate that
stationary, balanced, but fragmented states can be reached for any number of
attributes, if, in addition to homophily, individuals take into account a
significant fraction, q, of their triadic relations. Above a critical value
q_c, balanced states result. This result also holds for sparse realistic social
networks. Finally, in the limit of small q, our result agrees with that of
[Phys. Rev. Lett. 125, 078302].
