Hierarchical Balance Theory: Emergence of Instability in Follower Layer Below Critical Temperatures

Hierarchy significantly shapes interactions in social structures by organizing individuals or groups based on status, power, or privilege. This study investigates how hierarchy affects structural balance as temperature variations, which measure an individual's average irrationality in society. To address this question, we develop a two-layer balance model, the \enquote{leader layer}, which maintains structural balance exclusively through intra-layer interactions. Conversely, the \enquote{follower layer} maintains structural equilibrium through both inter- and intra-layer interactions. The Hamiltonian of the leading layer is independent, while the follower layer depends on its parameters as well as those of the leading layer. Analytical results from the mean-field approximation and exact Monte Carlo simulations show that instability arises in the equilibrium states of the follower layer when the temperature is below the critical threshold ($T<T_c$), which is different from the structural Heider equilibrium. Furthermore, our findings indicate that the critical temperature is elevated in the follower layer.