Homophily based on few attributes can impede structural balance

Two complementary mechanisms are thought to shape social groups: homophily between agents and structural balance in connected triads. Here we consider $N$ fully connected agents, where each agent has $G$ underlying attributes, and the similarity between agents in attribute space (i.e., homophily) is used to determine the link weight between them. To incorporate structural balance we use a triad-updating rule where only one attribute of one agent is changed intentionally in each update, but this also leads to accidental changes in link weights and even link polarities. The link weight dynamics in the limit of large $G$ is described by a Fokker-Planck equation from which the conditions for a phase transition to a fully balanced state with all links positive can be obtained. This "paradise state" of global cooperation is, however, difficult to achieve requiring $G > O(N^2)$ and $p>0.5$, where the parameter $p$ captures a willingness to consensus. Allowing edge weights to be a consequence of attributes naturally captures homophily and reveals that many real-world social systems would have a subcritical number of attributes necessary to achieve structural balance.