^{1,a)}and Peter J. Mucha

^{1}

### Abstract

Taking a pragmatic approach to the processes involved in the phenomena of collective opinion formation, we investigate two specific modifications to the coevolving network voter model of opinion formation studied by Holme and Newman [Phys. Rev. E 74, 056108 (2006)]. First, we replace the rewiring probability parameter by a distribution of probability of accepting or rejecting opinions between individuals, accounting for heterogeneity and asymmetric influences in relationships between individuals. Second, we modify the rewiring step by a path-length-based preference for rewiring that reinforces local clustering. We have investigated the influences of these modifications on the outcomes of simulations of this model. We found that varying the shape of the distribution of probability of accepting or rejecting opinions can lead to the emergence of two qualitatively distinct final states, one having several isolated connected components each in internal consensus, allowing for the existence of diverse opinions, and the other having a single dominant connected component with each node within that dominant component having the same opinion. Furthermore, more importantly, we found that the initial clustering in the network can also induce similar transitions. Our investigation also indicates that these transitions are governed by a weak and complex dependence on system size. We found that the networks in the final states of the model have rich structural properties including the small world property for some parameter regimes.

As the study of networks is applied to an ever broadening variety of phenomena, it is important to study the properties of networks, dynamical processes coupled across networks, and the interplay between the two where the coupled dynamics affect the network topology. A minimal mathematical model that has been used to model the social phenomena of collective opinion formation is the coevolving voter model.

^{1,16–26}We introduce two additional attributes to the multi-opinion coevolving voter model, in order to describe processes and networks that are closer to real-world situations within a still relatively simple model. Our model includes a “social environment,” modeling the inherent heterogeneity and asymmetry in relationships within a social group. We also include a path-length-based preference for rewiring that reinforces social clustering. Our inclusion of this second attribute has been influenced by the fact that clustering is a ubiquitous feature of networks and has not been incorporated as a dynamic entity in most coevolving voter models. We explore the consequences of these two additional attributes within the coevolving voter model, comparing and contrasting the behaviors of this only slightly more complicated model with those of the minimal coevolving voter model. Our results highlight the important role of clustering, with possible consequences for future applications of coevolving voter models.

The authors thank Mason Porter and Feng Shi for helpful suggestions and comments. The project described was supported by Award No. R21GM099493 from the National Institute of General Medical Sciences. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of General Medical Sciences or the National Institutes of Health.

I. INTRODUCTION

II. DESCRIPTION OF THE MODEL

A. Basic features of the model

III. PHASE TRANSITIONS

A. Role of social environment in transitions

B. Role of network structure in transitions

IV. CONCLUSIONS

### Key Topics

- Cluster analysis
- 63.0
- Networks
- 33.0
- Network topology
- 11.0
- Collective models
- 10.0
- Social networks
- 6.0

## Figures

Different types of social environment function , where pij is the probability of the jth node accepting the opinion of the ith node: (a) “inflexible,” , so that more links will have lower probabilities of accepting opinions; (b) “flexible,” , so that more links will have higher probabilities of accepting opinions.

Different types of social environment function , where pij is the probability of the jth node accepting the opinion of the ith node: (a) “inflexible,” , so that more links will have lower probabilities of accepting opinions; (b) “flexible,” , so that more links will have higher probabilities of accepting opinions.

A visual representation of the formation of qualitatively distinct consensus states for two different social environments. Both systems start with an initial Watts-Strogatz network (with N = 1000, , and ). (a) Setting and creates an “inflexible” social environment. We observe disintegration of the network into small connected components with each component in internal consensus, i.e., segregated consensus occurs in the network. (b) Setting and creates a “flexible” social environment. We observe a dominant connected component in the final consensus, with size comparable to the initial network, while a large number of the initial opinions go extinct. We refer this kind of final state as a hegemonic consensus.

A visual representation of the formation of qualitatively distinct consensus states for two different social environments. Both systems start with an initial Watts-Strogatz network (with N = 1000, , and ). (a) Setting and creates an “inflexible” social environment. We observe disintegration of the network into small connected components with each component in internal consensus, i.e., segregated consensus occurs in the network. (b) Setting and creates a “flexible” social environment. We observe a dominant connected component in the final consensus, with size comparable to the initial network, while a large number of the initial opinions go extinct. We refer this kind of final state as a hegemonic consensus.

The effect of different social environments on a network of N = 1000 nodes with O = 100 opinions initially present. The starting network is an Erdős-Rényi random network (i.e., clustering ). (a) The distribution of component sizes si , the fraction of nodes in the ith (ranked by size) connected component in the final consensus state, is plotted as a function of social environment (with marker sizes proportional si ). Colors indicate , the clustering coefficient of the ith component. The thick bold line in the middle separates the two types of social environment function: on the right we consider flexible social environments, with a single large connected component with size increasing with increasing α; on the left we consider inflexible social environments, observing a decrease of the size of the largest connected component with increasing α, finally leading to its disintegration into several components of comparable sizes. (b) The sizes of the two largest connected components, s 1 and s 2, is plotted versus social environment. Simulations were conducted on 100 realizations of the network and initial opinion distribution, with the plotted component sizes estimated as the means over these realizations. Error bars give the standard deviation of these sizes across realizations.

The effect of different social environments on a network of N = 1000 nodes with O = 100 opinions initially present. The starting network is an Erdős-Rényi random network (i.e., clustering ). (a) The distribution of component sizes si , the fraction of nodes in the ith (ranked by size) connected component in the final consensus state, is plotted as a function of social environment (with marker sizes proportional si ). Colors indicate , the clustering coefficient of the ith component. The thick bold line in the middle separates the two types of social environment function: on the right we consider flexible social environments, with a single large connected component with size increasing with increasing α; on the left we consider inflexible social environments, observing a decrease of the size of the largest connected component with increasing α, finally leading to its disintegration into several components of comparable sizes. (b) The sizes of the two largest connected components, s 1 and s 2, is plotted versus social environment. Simulations were conducted on 100 realizations of the network and initial opinion distribution, with the plotted component sizes estimated as the means over these realizations. Error bars give the standard deviation of these sizes across realizations.

Properties of the largest connected component (hegemonic consensus) for the final state reached for a flexible environment, , with from an initially Erdős-Rényi network. (a) Comparisons of average path length, maximum degree, clustering coefficient and size for different network sizes, with different markers representing different network sizes (see the legend). The initial network is G 0, with initial clustering close to zero, while s 1 here denotes both the largest connected component in the final state and its size (as a fraction of the nodes in the network). We observe that s 1 has a significantly higher clustering coefficient (0.2) whereas it has path length comparable to the initial Erdős-Rényi network G 0, implying that s 1 has small world features. Also, s 1 typically has higher (maximum degree) while its size remains comparable to G 0. (b) The cumulative degree distribution C(k) of the initial network G 0 (dashed lines) is compared with that for s 1 (markers), further showing that s 1 has nodes with higher degrees. In its tail, the cumulative degree distribution of s 1 appears to approximately follow a power law as shown by solid grey line of exponent −8 though the steepness of this line does not preclude other distributions in the tail.

Properties of the largest connected component (hegemonic consensus) for the final state reached for a flexible environment, , with from an initially Erdős-Rényi network. (a) Comparisons of average path length, maximum degree, clustering coefficient and size for different network sizes, with different markers representing different network sizes (see the legend). The initial network is G 0, with initial clustering close to zero, while s 1 here denotes both the largest connected component in the final state and its size (as a fraction of the nodes in the network). We observe that s 1 has a significantly higher clustering coefficient (0.2) whereas it has path length comparable to the initial Erdős-Rényi network G 0, implying that s 1 has small world features. Also, s 1 typically has higher (maximum degree) while its size remains comparable to G 0. (b) The cumulative degree distribution C(k) of the initial network G 0 (dashed lines) is compared with that for s 1 (markers), further showing that s 1 has nodes with higher degrees. In its tail, the cumulative degree distribution of s 1 appears to approximately follow a power law as shown by solid grey line of exponent −8 though the steepness of this line does not preclude other distributions in the tail.

The evolution of system variables with decreasing number of discordant edges. Each variable is plotted at the last time step when that number of discordant edges, , was present in the system. The black line and panel (b) correspond to simulations starting at the highest possible clustering coefficient whereas the red dotted line and panel (a) correspond to simulations starting at the negligible clustering coefficient obtained with a random network of independent edges. In (a) and (b), each color corresponds to one of the opinions, with width indicating the number of nodes holding that opinion. The wide width of cyan at the end in (a) represents the formation of a hegemonic consensus (one large connected component of size comparable to the initial network). We do not observe a similar transition in (b) even though the only difference in this simulation is the large initial clustering coefficient. In (c), we plot s 1, the size of the largest connected component. Observe the abrupt drop of the black line in s 1, indicating the disintegration of the network into smaller components (i.e., segregated consensus). In contrast, we do not observe any such transition for the red dotted line, corresponding to the formation of hegemonic consensus. In (d), we show , the average number of iterations of the system between last-observed times for each , with a substantial increase for the black curve near the end. In (e), is the corresponding evolution of the clustering coefficient.

The evolution of system variables with decreasing number of discordant edges. Each variable is plotted at the last time step when that number of discordant edges, , was present in the system. The black line and panel (b) correspond to simulations starting at the highest possible clustering coefficient whereas the red dotted line and panel (a) correspond to simulations starting at the negligible clustering coefficient obtained with a random network of independent edges. In (a) and (b), each color corresponds to one of the opinions, with width indicating the number of nodes holding that opinion. The wide width of cyan at the end in (a) represents the formation of a hegemonic consensus (one large connected component of size comparable to the initial network). We do not observe a similar transition in (b) even though the only difference in this simulation is the large initial clustering coefficient. In (c), we plot s 1, the size of the largest connected component. Observe the abrupt drop of the black line in s 1, indicating the disintegration of the network into smaller components (i.e., segregated consensus). In contrast, we do not observe any such transition for the red dotted line, corresponding to the formation of hegemonic consensus. In (d), we show , the average number of iterations of the system between last-observed times for each , with a substantial increase for the black curve near the end. In (e), is the corresponding evolution of the clustering coefficient.

A phase diagram for s 1 (size of the largest connected component) varying α and in both the inflexible and the flexible social environments. Colors represent the values of s 1 (see the color bar). The left panel belongs to the inflexible social environment regime whereas the right panel belongs to the flexible social environment regime. Observe the disintegration of the largest connected components in the flexible social environment regime (the right panel) for higher values of initial clustering (lower values of s 1 in shades of red). For lower values of we do not observe any such disintegration (higher values of s 1 in shades of blue). In the inflexible regime (the left panel) we observe that values of α dominate the final outcome of the simulation. A network of N = 1000 nodes and with initial opinions was employed for each α and . For visualization, data were interpolated onto a regular grid by a combination of natural neighbor and spline interpolation.

A phase diagram for s 1 (size of the largest connected component) varying α and in both the inflexible and the flexible social environments. Colors represent the values of s 1 (see the color bar). The left panel belongs to the inflexible social environment regime whereas the right panel belongs to the flexible social environment regime. Observe the disintegration of the largest connected components in the flexible social environment regime (the right panel) for higher values of initial clustering (lower values of s 1 in shades of red). For lower values of we do not observe any such disintegration (higher values of s 1 in shades of blue). In the inflexible regime (the left panel) we observe that values of α dominate the final outcome of the simulation. A network of N = 1000 nodes and with initial opinions was employed for each α and . For visualization, data were interpolated onto a regular grid by a combination of natural neighbor and spline interpolation.

The effect of different initial clustering on a network of N = 1000 nodes with O = 100 initial opinions for flexible social environment with . Large initial clustering leads to final states with segregated consensus, contrary to the expected hegemonic consensus for initially unclustered networks in the same flexible social environment. (a) The distribution of component sizes si , the fraction of nodes in the ith (ranked by size) connected component in the final consensus state, is plotted as a function of the initial clustering coefficient, (with marker sizes proportional si ). Colors indicate , the clustering coefficient of the ith component in the final state. (b) The sizes of the two largest connected components, s 1 and s 2, are plotted versus . Simulations were conducted on 100 realizations of the network and initial opinion distribution, with the plotted component sizes estimated as the means over these realizations. Error bars give the standard deviation of these sizes across realizations.

The effect of different initial clustering on a network of N = 1000 nodes with O = 100 initial opinions for flexible social environment with . Large initial clustering leads to final states with segregated consensus, contrary to the expected hegemonic consensus for initially unclustered networks in the same flexible social environment. (a) The distribution of component sizes si , the fraction of nodes in the ith (ranked by size) connected component in the final consensus state, is plotted as a function of the initial clustering coefficient, (with marker sizes proportional si ). Colors indicate , the clustering coefficient of the ith component in the final state. (b) The sizes of the two largest connected components, s 1 and s 2, are plotted versus . Simulations were conducted on 100 realizations of the network and initial opinion distribution, with the plotted component sizes estimated as the means over these realizations. Error bars give the standard deviation of these sizes across realizations.

Variation of s 1 (size of largest connected component) and s 2 (size of second largest connected component) with α for inflexible social environments, . Different shapes and colors of the markers represent networks of different sizes [see legend in (g)]. In (a) we observe multiple transitions in s 1 collapse onto a similar curve (inset) for rescaling α by . A second transition is observed near [dashed grey vertical lines in (b) and (c)], where a best fit to the data changes from a polynomial to power law, as indicated by the values of ϵ, the errors between the fitted function and the s 1 data points. This second transition appears to be collocated with a transition in s 2 appearing in (d), with an abrupt decreasing of s 2 after (dashed grey vertical line). Similar to (b) and (c), the best fit to the s 2 data changes from a polynomial to power law [see (e) and (f)]. In (g), we plot the Shanon entropy H of the 10 largest connected components versus α, observing that H tends to saturate near (dashed grey vertical line) and decreases for higher α.

Variation of s 1 (size of largest connected component) and s 2 (size of second largest connected component) with α for inflexible social environments, . Different shapes and colors of the markers represent networks of different sizes [see legend in (g)]. In (a) we observe multiple transitions in s 1 collapse onto a similar curve (inset) for rescaling α by . A second transition is observed near [dashed grey vertical lines in (b) and (c)], where a best fit to the data changes from a polynomial to power law, as indicated by the values of ϵ, the errors between the fitted function and the s 1 data points. This second transition appears to be collocated with a transition in s 2 appearing in (d), with an abrupt decreasing of s 2 after (dashed grey vertical line). Similar to (b) and (c), the best fit to the s 2 data changes from a polynomial to power law [see (e) and (f)]. In (g), we plot the Shanon entropy H of the 10 largest connected components versus α, observing that H tends to saturate near (dashed grey vertical line) and decreases for higher α.

Variation in the size of the largest connected component s 1 with initial clustering coefficient for flexible social environment, , with . When is multiplied to the data for different system sizes appears to collapse onto a single curve. The inset curve shows the fits to data without scaling, with vertical lines indicating the scales of the transition points.

Variation in the size of the largest connected component s 1 with initial clustering coefficient for flexible social environment, , with . When is multiplied to the data for different system sizes appears to collapse onto a single curve. The inset curve shows the fits to data without scaling, with vertical lines indicating the scales of the transition points.

The sizes of different connected components in the consensus state for networks of N = 1500 nodes. (a) Sizes of connected components v. their ordered (by decreasing size) indices. As initial clustering of the network (color bar) is increased, there is emergence of smaller components of comparable sizes. (b) The values of the exponents, , of the slopes fitted to the sizes of components in the final consensus state v. indices at each value of [the thick red line in (a) is an example for ]. In (b), observe the decrease in the slope and error bars for higher initial clusterings, indicating the formation of several components of comparable sizes.

The sizes of different connected components in the consensus state for networks of N = 1500 nodes. (a) Sizes of connected components v. their ordered (by decreasing size) indices. As initial clustering of the network (color bar) is increased, there is emergence of smaller components of comparable sizes. (b) The values of the exponents, , of the slopes fitted to the sizes of components in the final consensus state v. indices at each value of [the thick red line in (a) is an example for ]. In (b), observe the decrease in the slope and error bars for higher initial clusterings, indicating the formation of several components of comparable sizes.

## Tables

A voter model on a coevolving network with clustering and heterogeneous levels of influence.

A voter model on a coevolving network with clustering and heterogeneous levels of influence.

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