Hybrid balance theory: Heider balance under higher-order interactions

Heider's balance theory in signed networks, which consists of friendship or enmity relationships, is a model that relates the type of relationship between two people to the third person. In this model, there is an assumption of the independence of triadic relations, which means that the balance or imbalance of one triangle does not affect another and the energy only depends on the number of each type of triangle. There is evidence that in real network data, in addition to third-order interactions (Heider balance), higher-order interactions also play a role. One step beyond the Heider balance, the effect of quartic balance has been studied by removing the assumption of triangular independence. The application of quartic balance results in the influence of the balanced or imbalanced state of neighboring triangles on each specific one. Here, a question arises as to how the Heider balance is affected by the existence of quartic balance (fourth order). To answer this question, we presented a model which has both third- and fourth-order interactions and we called it a hybrid balance theory. The phase diagram obtained from the mean-field approximation shows there is a threshold for higher-order interaction strength, below which a third-order interaction dominates and there are no imbalance triangles in the network, and above this threshold, squares effectively determine the balance state in which the imbalance triangles can survive. The solution of the mean-field indicates that we have a first-order phase transition in terms of the random behavior of agents (temperature) which is in accordance with the Monte Carlo simulation results.

Figure 1

Left: Triadic interactions between an agent (blue) and its neighbors. Right: Quartic interactions between an agent and other agents that do not have direct interactions, necessarily. In hybrid balance theory (presented model), both interactions exist.

Figure 2

Left column: Simultaneous solution of Eq. (9) for $g=0.1$ and $T=250,500,1000$. Right column: The flow diagram for visualizing the convergence (divergence) from stable (unstable) fixed points. The red square and blue circle indicate the stable and unstable fixed points, respectively. The number of nodes is $n=100$.

Figure 3

The phase diagram for for the number of simultaneous solutions for Eq. (9) in $(T,g)$ space. This diagram divides into two regions by $g_c$. In the region $g>g_c$ the squares have more energy than triangles and have a dominant role and in region $g<g_c$ the triangles have the dominant role. Inset: The two-solution region becomes very narrow in log-login the log-log plot, but it exists in the linear plot with approximately fixed width. The number of nodes is $n=100$.

Figure 4

Behavior of critical temperature ($T_c$) versus size of fully connected network ($n$). The lines are fitted to critical temperature values in different sizes. In triangle dominant regime ($g<g_c$ densely dashed green) the line slope is approximately one and by increasing the factor $g$ the line slope increases. Upon reaching the dominant square regime ($g>g_c$ loosely dashed magenta), the line slope becomes equal to two and remains constant even if $g$ increases. The two lines of $g=0.9(>g_c)$ and $g=50(>g_c)$ (blue dash dotted) are parallel to each other, indicated by an equal slope.

Figure 5

Comparison of our mean-field approximation and Monte Carlo simulations (blue solid and red dashed lines) with different boundary conditions (squares, triangles, and circles) for $p,q,o$, and $E$. (a) $g=0.1$, $0\le T\le 1100$ and (b) $g=0.005$, $0\le T\le 220$.

Figure 6

Percentage of jammed states for 2000 ensembles with different $g$ values.