Critical properties of Heider balance on multiplex networks

Heider's structural balance theory has proven invaluable in comprehending the dynamics of social groups characterized by both friendly and hostile relationships. Since people's relations are rarely single faceted, we investigate Heider balance dynamics on a multiplex network, consisting of several copies of the same agent displaying correlated relations at different layers building the multiplex. Intralayer interactions in our model adhere to Heider dynamics, while interlayer correlations stem from Ising interactions, with the heat-bath dynamics of link signs. Our investigation reveals a multifaceted system with a diverse equilibrium landscape contingent on the coexistence of distinct phases across layers. We observe that, starting from a paradise state with positive links in all layers, an increase in temperature triggers a discontinuous transition to a disordered state akin to single-layer scenarios. The critical temperature surpasses that of the single-layer case, a fact verified through extended mean-field analysis and agent-based simulations. Furthermore, the scenario shifts when one layer exhibits a two-clique configuration instead of a paradise state. This change introduces additional transitions: synchronization of interlayer relations and a transition to the disorder, appearing at a different, lower temperature compared to matching paradise states. This exploration shows the intricate interplay of Heider balance and multiplex interactions.

Figure 1

Example of interactions between edge signs (blue solid line for $+1$ and red dashed line for $-1$) on a duplex network. Edge sign changes are driven by intralayer interactions according to Heider structural balance (with strength $A^{\left(1\right)}$ or $A^{\left(2\right)}$, respectively) as well as interlayer coupling of Ising nature between the edges, with strength $K$. The signs of relations evolve over time, according to the heat-bath dynamics with energy as expressed by Eq. (1) and given temperature $T$.

Figure 2

Mean-field solution for average link polarization $x^{\left(1\right)}$ for various interlayer coupling strengths. The coupled layers undergo a discontinuous transition at a critical temperature $T_c$ that increases with coupling strength. The results are shown for $d=0.2,0.5,1,3$ with the transition point moving from left to right with increasing $d$.

Figure 3

Graphical solution of the mean-field state equation (20) for $L=10$ layers. The blue solid curve is $y=g\left(\langle L^{+}\rangle \right)$ and the red straight line is $y=\langle L^{+}\rangle$. When $T>T_c$ there is a single stable fixed point. When $T<T_c$ there are three fixed points and the inner one is unstable

Figure 4

Mean link polarizations $x^{\left(1\right)}=\langle x_{ij}^{\left(1\right)}\rangle$ and $x^{\left(2\right)}=\langle x_{ij}^{\left(2\right)}\rangle$ (a) and corresponding energies (b) for a duplex network as a function of temperature $T$ for the complete graphs with $N=50$ nodes and a coupling strength $K=50$ (corresponding to $d=1$). The results show averages over 50 independent simulations. Initial conditions were paradise states at both layers, where the overlapping red pluses and yellow crosses correspond to layers 1 and 2, respectively. In contrast to the intralayer energies (overlapping red pluses and yellow crosses), the interlayer energy (black circles) increases slowly when $T>T_c$, with layers remaining partially synchronized even when intralayer disorder sets in.

Figure 5

Critical temperature $T_c$ increases with coupling strength $K$ and saturates to $T\approx L{T}_{c,L=1}$ for high $K/M$ values. The black solid line and black triangles show the mean-field prediction (11) and numerical results for $L=2$, respectively, while the red dotted line and red circles show mean-field prediction (14) and numerical results for $L=3$, respectively.

Figure 6

Critical temperature $T_c$ increases with the number of layers $L$ when coupling $K$ is positive. Black symbols and dotted lines show the results of Monte Carlo simulations, while orange symbols and solid lines show the mean-field prediction [iterating Eq. (20) as a map] at different interlayer coupling strengths for $N=50$ and $A=1$. The critical temperature for a single layer is shown as a closed red circle at $T_c=26.5$.

Figure 7

If both layers of a duplex are in two-clique states, the overlap between the layers can vary from (a) full overlap (with matching order and overlap $o=1$) through (b) partial overlap (with partial matching order and overlap $o=3/4$ in this example) to (c) minimal overlap displayed by orthogonal two-clique states (with mismatched order and overlap $o=1/2$). The same nodes exist in both layers and interlayer interactions exist only between the same link replicas with coupling strength $K$. Blue solid lines represent positive links, red dotted lines represent negative links, and yellow circles enclose agents belonging to different groups in both layers.

Figure 8

In the case of coupled paradise and two-clique states, the system shows two transitions: synchronization of layers starting at $T_s$ and transition to disorder starting at $T_o$ and ending at additional characteristic temperature $T_m$. (a) Mean link polarization of 50 independent simulations over a range of temperatures for $N=50$ nodes and a coupling strength $K=25$, with one layer starting in the paradise state (red pluses) and the second layer starting in the two-clique state (yellow crosses). (b) Corresponding mean intralayer energy (red pluses and yellow crosses) and interlayer energy (black circles). (c) Intralayer and interlayer energies for a duplex network starting in a disordered state, showing that spontaneous order appears at any temperature below $T_d$.

Figure 9

When paradise and two-clique states synchronize, it results most often in a paradise state $x^{\left(1\right)}\approx 1$ but two-clique states $x^{\left(1\right)}<1$ are also a possible outcome. The graph shows the distribution of mean link polarization after paradise and two-clique states synchronize at the temperature $T$ where $T_s<T<T_d$.

Figure 10

The final state of the synchronization between two mismatched layers is partially random due to thermal fluctuations. Example trajectories are shown for the synchronization scenario between paradise (red solid line) and two-clique (blue dashed line) layers for $N=50, K=25$, and $T=20$. The synchronized state can be (a) a two-clique state similar to the initial condition of the second layer, (b) a paradise state, or (c) a two-clique state of different sizes, matching neither of the two initial layer states. Each shown scenario is a result of independent agent-based simulation under the same initial conditions and parameters.

Figure 11

The behavior of the system of two partially matching [(a) and (c)] and mismatched [(b) and (d)] layers in two-clique states for $N=50$ and $K=25$ closely resembles the mismatched order of paradise and two-clique layers seen in Fig. 8. (a) and (b) Variation of mean link polarization for two orthogonal two-clique layers over temperature for 50 independent simulations. (c) and (d) Corresponding variation of mean intralayer (red pluses and yellow crosses) and interlayer (black circles) energies over temperature.

Figure 12

In a triplex network, there are two classes of states: (a) and (c) two matching layers and a third mismatched $\left(PCC\right)$ and (b) and (d) all three mutually mismatched layers $\left(PC{C}^{\prime }\right)$. These two state classes show different synchronization temperatures $T_s$. (a) and (b) Mean polarization (blue squares, red pluses, and yellow crosses) of each of the three layers. (c) and (d) Intralayer energies (blue squares, red pluses, and yellow crosses) as well as interlayer energy (green up triangles, teal down triangles, and black circles) tied to each pair of layers for $N=50$ and $K=10$. The $PCC$ state consists of the paradise state in one layer and matching two-clique states in the other two; the $PC{C}^{\prime }$ state consists of the paradise state in one layer and mismatched two-clique states in the other two.

Figure 13

Phase diagram of the duplex system featuring multistability. The five colored areas are blue (below $T_s$), a region where both mismatched and matching order states are stable; green (between $T_s$ and $T_d$), a region where only the matching order state is stable; yellow (between $T_d$ and $T_c$), a region where both matching order and disordered states are stable; red (above $T_c$), a region where only the disordered state is stable; and white (between $T_d$ and $T_s$), a region where mismatched order, matching order, and disordered states are all stable. The boundaries $T_d$ and $T_c$ are fixed, but $T_s, T_o$, and $T_m$ vary for different states. The results were obtained from agent-based simulations for $N=50$ and $A=B=1$ with 50 realizations.

Figure 14

The Heider balance dynamics on a duplex network is multistable, with different states possible depending on the temperature. Gray boxes show a range of temperatures where a given state is stable for a fixed $K$ value, with gray arrows showing possible transitions between these states. Five characteristic temperatures can be distinguished: synchronization at $T_s$, the appearance of disorder at $T_d$, the lower limit of transitions to disorder $T_o$, the upper limit of transitions to matching order $T_m$, and the disappearance of order at $T_c$. Note that the figure is schematic and does not depict actual values for better clarity. The temperature values $T_s$ depend on the exact states they correspond to, since partially matching states with lower overlap have lower temperatures $T_s$. Characteristic temperatures $T_o$ and $T_m$ depend on the initial state and its overlap between the layers, since higher overlap means higher temperature values, up to $T_c$ for $o=1$.

Figure 15

Equation (A11) has three solutions for the auxiliary variable $z$ that depend only on relative interlayer coupling strength $D$. Out of them, only the solution $z>1$ (blue top line) is physically meaningful and corresponds to a positive critical temperature $T_c$.