Fragmentation transitions in multistate voter models

Adaptive models of opinion formation among humans can display a fragmentation transition, where a social network breaks into disconnected components. Here we investigate this transition in a class of models with arbitrary number of opinions. In contrast to previous work we do not assume that opinions are equidistant or arranged on a one-dimensional conceptual axis. Our investigation reveals detailed analytical results on fragmentations in a three-opinion model, which are confirmed by agent-based simulations. Furthermore, we show that in certain models the number of opinions can be reduced without affecting the fragmentation points.

Figure 1

State network of a three-state voter model. The nodes in this network are the three opinions $A$, $B$, and $C$. The rewiring rates $p_1,p_2$, or $p_3$, encode the degree of controversy between agents holding the respective pairings of opinions $AB$, $BC$, or $CA$. The ellipse and dashed lines illustrate the example described in the text: partial fragmentation with respect to $A$, i.e., the fragmentation of the corresponding networks of agents in a component where all individuals hold opinion $A$ and a second component where the individuals hold opinions $B$ and $C$. The ongoing dynamics in the latter component may later lead to the disappearance of either opinion $B$ or $C$.

Figure 2

Transitions of $AB$- and $AC$-fans for a degree-regular network with $k=3$ and equiprobable states for the scenario of partial fragmentation with respect to A. Black, white, and gray nodes correspond to agents holding opinion $A$, $B$, and $C$, respectively. The active link densities ${\rho }_1$ and ${\rho }_2$ are assumed to vanish close to the partial fragmentation point, whereas ${\rho }_3$, the density of $BC$-links, can be finite.

Figure 3

Schematic fragmentation diagram for a three-state system in the thermodynamic limit. The region where fragmentation with respect to $A$ occurs depends on the $BC$-link density ${\rho }_3$: If ${\rho }_3=0$, fragmentation with respect to $A$ occurs only in region III. If ${\rho }_3>0$, fragmentation with respect to $A$ can occur additionally in part of the dashed region. For ${\rho }_3={\rho }_{\text{max}}$ the maximal extension of the fragmentation region (II $+$ III) is realized. In region I, no fragmentation with respect to $A$ occurs. If $p_1>p_3$ and $p_2>p_3$, then this fragmentation diagram characterizes the final state of the whole system. In this case, region I is characterized by an active equilibrium where deterministic fragmentation cannot occur (with respect to none of the states) and full fragmentation is reached iff $p_3\ge p_c$, where $p_c$ is the fragmentation threshold for the adaptive two-state voter model.

Figure 4

Numerical phase diagram for the three-state model. Plotted is the density of links, connecting the $A$- and the $BC$-cluster (${\rho }_1+{\rho }_2$) over the rewiring rates $p_1$ and $p_2$. Dark gray regions correspond to fragmentation with respect to $A$. The left panel shows the case ${\rho }_3=0(p_3=0.5)$. This corresponds to an uncoupled system: The critical rewiring rates for $p_1$ and $p_2$ are the same as for the two-state voter model, $p_c=0.46$. We note that this result differs from Ref. [18], where a related model with different update rules was studied. In the right panel, ${\rho }_3$ is maximal ($p_3=0$). Here the active link density in the active cluster leads to an extension of the fragmentation region. Black lines represent analytical results. Link densities were measured at $t=1000$, i.e., much before the active phase decays due to finite-size effects (see Fig. 5, upper left panel). $N$ $=$ 10 000, $\langle k\rangle$= 4, averaged over 20 realizations.

Figure 5

Numerical exploration of fragmentation in the long-term limit. In the upper left panel the time to the first fragmentation or extinction is plotted. The time profile resembles the corresponding fragmentation diagram in Fig. 4, demonstrating that (partial) fragmentation occurs significantly earlier than the decay of the active phase to an absorbing state. In the upper right panel the size of the largest cluster (as fraction of the total number of nodes) in the final state is shown. One can clearly distinguish three domains, corresponding to three types of long-term behavior. Analyzing the size distributions of the largest cluster at the points (a), (b), and (c) (lower panel) reveals that (a) for $p_1<p_c$ and $p_2<p_c$ always consensus is reached, (b) for $p_1>p_c$ or $p_2>p_c$ either consensus or fragmentation is reached, and (c) for parameter settings within the estimated deterministic fragmentation region (compare Fig. 4) fragmentation is always reached. The histograms were obtained from 100 independent network simulations for each parameter setting, in the upper panels we averaged over 20 realizations. In all panels: $p_3=0$, $N$ $=$ 10 000, $\langle k\rangle$= 4.

Figure 6

Fragmentation diagrams for partial fragmentation with respect to state $A$, $C$, or $B$ for $p_1\ge p_2\ge p_3$ in the limit of infinite network size. There are four different cases, according to the conditions given on the left-hand side of the chart. The positions of the points $P=(p_2,p_1),Q=(p_3,p_1),R=(p_2,p_3)$ indicate for each case whether partial fragmentation is reached for the respective state. The final state of the whole system can be deduced from the outcomes for all three states and is given on the right-hand side of the chart. For the first, second, and fourth case the final state can be predicted without ambiguity. In the third case, either partial fragmentation with respect to $A$ or an active equilibrium can be reached, depending on the specific parameter setting (the two possibilities are indicated by two different symbols $\circ$ and $\times$ in the first diagram and a solid and a dashed line in the second diagram of the third row). It can be seen that partial fragmentation is only possible with respect to the state with the largest rewiring rates (here $A$).

Figure 7

Schematic representation of a state network with $G$ states. Ellipses illustrate a fragmentation into two (possibly) active clusters of $s$ and $G-s$ states. Dashed lines correspond to intercluster links with rewiring rates $\left\{p_{i\underline{i}}\right\}$, connecting every state in one cluster with every state in the other cluster. The number of intercluster links, $s(G-s)$, determines the dimension of the (matrix-valued) Jacobian. Solid lines correspond to intracluster links within both clusters with active link densities $\left\{{\rho }_{ij}\right\}$ and $\left\{{\rho }_{\underline{ij}}\right\}$, respectively.

Figure 8

Numerical test of the reduction principle. We run simulations for a five-state system assigning random values to the rewiring rates $r_i$ (inset). Plotted is the density of intercluster links (dashed links) for three different values of $p$. For each $p$ value we run ten independent simulations with randomly chosen $r_i$ and compare the corresponding intercluster link density to the active link density of a two-state model with the same rewiring rate and a ratio 2:3 for the number of nodes in opposite states. It can be seen that the intercluster density does not depend on $r_i$ and that the steady-state value equals the corresponding intercluster density (active link density) in the original two-state voter model. $N=10000,\langle k\rangle =4$.

Figure 9

Schematic representation of the generalized Frobenius inequalities using the example of a five-state system and a specific partial fragmentation into a two-state and a three-state cluster. The inequalities in (a) and (b) represent two different sets of bounding systems obtained from a partition of type (P2), as described in the text. In (a) the bounding systems are three-state systems, obtained by reducing the three-cluster to a single state. The remaining intercluster rewiring rates are the respective maximum (minimum) values of the rewiring rates between each state in the two-cluster and all states in the three-cluster. In (b) the bounding systems are four-state systems, obtained by reducing the two-cluster to a single state. The remaining intercluster rewiring rates are the maximum (minimum) values of the rewiring rates between each state in the three-cluster and all states in the two-cluster. In (c) a special case is shown where the upper and lower three-state bounding systems coincide. In this case the leading eigenvalue of the five-state system equals the leading eigenvalue of the associated three-state system, and the considered fragmentation of the original system is fully captured by the lower-dimensional one.