Bounded-Confidence Models of Opinion Dynamics with Neighborhood Effects

TL;DR

This work extends bounded-confidence models by incorporating neighborhood-based, transitive influence into both opinion dynamics and network adaptation. It formalizes neighborhood DW (NDW) and neighborhood HK (NHK) models with a mixing parameter $\sigma$ that balances direct dyadic influence and transitive neighbor influence, and introduces a transitive-homophily rewiring rule with threshold $\zeta$. Through extensive simulations on diverse networks, the study shows that neighborhood effects reshape opinion clustering and coevolving network structure, typically reducing the spectral gap and degree assortativity and producing nonmonotonic dynamics in discordant ties. The framework provides a more nuanced, realistic depiction of how opinions diffuse through social neighborhoods and adapt the underlying network, with implications for polarization, consensus formation, and information spread. Limitations include homogeneous confidence bounds and agent homogeneity; future work could explore heterogeneity, multidimensional opinions, and integration with epidemiological dynamics. $\epsilon$, $\rho$, $\sigma$, and $\zeta$ govern the interplay of local and transitive influences and network rewiring, enabling richer phenomenology in real-world social systems.

Abstract

We generalize bounded-confidence models (BCMs) of opinion dynamics by incorporating neighborhood effects. In a BCM, interacting agents influence each other through dyadic influence if their opinions are sufficiently similar to each other. In our "neighborhood BCMs" (NBCMs), interacting agents are influenced both by each other's opinions and by the opinions of the agents in each other's neighborhoods. Our NBCMs thus include both the usual dyadic influence between agents and a "transitive influence", which encodes the influence of an agent's neighbors, when determining whether or not an interaction changes the opinions of agents. In this transitive influence, an individual's opinion is influenced by a neighbor when, on average, the opinions of the neighbor's neighbors are sufficiently similar to its own opinion. We formulate both neighborhood Deffuant--Weisbuch (NDW) and neighborhood Hegselmann--Krause (NHK) BCMs. We build further on our NBCMs by introducing a neighborhood-based network adaptation in which a network coevolves with agent opinions by changing its structure through "transitive homophily". In this network evolution, an agent breaks a tie to one of its neighbors and then rewires that tie to a new agent, with a preference for agents with a mean neighbor opinion that is closer to its own opinion. Using numerical simulations on a variety of types of networks, we explore how the qualitative opinion dynamics and network properties of our adaptive NDW model change as we adjust the relative proportions of dyadic and transitive influence. In our numerical experiments, we find that incorporating neighborhood effects into the opinion dynamics and the network-adaptation rewiring strategy tends to reduce the spectral gap and degree assortativity of networks. (This is a shortened version of the paper's abstract.)

Figures (9)

• Figure 1: A schematic illustration of opinion updates of two adjacent nodes, $i$ and $j$, in the pure (i.e., baseline) DW model, pure NDW model, and the NDW model with neighborhood-tuning parameter $\sigma = 0.5$ (i.e., a "mixed" NDW model). All edges have unit weight and convergence parameter $\rho = 1$.
• Figure 2: Several examples of opinion dynamics in our neighborhood DW (NDW) model with homophilic rewiring. In each panel, we show one simulation on a network. The neighborhood-tuning parameter $\sigma$, the confidence bound $\epsilon$, and the discordance threshold $\zeta$ have the values (a) $\sigma = 0$, $\epsilon = 0.1$, and $\zeta = 0.4$; (b) $\sigma = 0.1$, $\epsilon = 0.2$, and $\zeta = 0.2$; (c) $\sigma = 0$, $\epsilon = 0.2$, and $\zeta = 0.4$; and (d) $\sigma = 0.1$, $\epsilon = 0.1$, and $\zeta = 0.3$. At each discrete time, we consider $f = 0.2 N$ dyads, where $N = 100$ is the size (i.e., the number of nodes) of the network. The convergence parameter is $\rho = 0.3$. We choose edges uniformly at random; if an edge is discordant, we rewire it using the homophilic rewiring strategy \ref{['eq:opn_rewiring']}. It is possible for the same node to rewire multiple times. In each simulation, the initial network is a $G(N,p)$ Erdős--Rényi (ER) graph with an independent, homogeneous probability $p = 0.3$ of an edge between each pair of nodes. We initialize each node opinion to a uniformly random value in the interval $[0,1]$. All depicted simulations use the same initial network and the same set of initial opinions. We terminate a simulation either when it reaches our stopping criterion or when $t_{\text{max}} = 2000$ time steps have elapsed (whichever occurs first). We color the opinion trajectories of each node according to its opinion value at the end of a simulation. Any two nodes whose opinions differ by at least the confidence bound $\epsilon$ are in different colors.
• Figure 3: We examine network adaptation as a function of time in our NDW model by plotting (a) the number of discordant edges, (b) the spectral gap of the associated adjacency matrix, (c) a degree-assortativity coefficient, (d) the fraction of edges in the chosen dyads that rewire (i.e., that are discordant), (e) the fraction of nodes in the chosen dyads that update their opinions, and (f) the mean number of connected components. We compare the network properties for a pure DW opinion-update rule (i.e., $\sigma = 1$), a mixed DW and NDW opinion-update rule with $\sigma = 0.5$, and a pure NDW opinion-update rule (i.e., $\sigma = 0$). In each simulation, the initial network is a $G(N,p)$ ER network with $N = 50$ nodes and an independent, homogeneous probability $p = 0.3$ of an edge between each pair of nodes. For each network, we initialize each node opinion to a uniformly random value in $[0,1]$. We plot means of 20 simulations, with the same 20 initial networks and sets of initial opinions for each panel. The shaded regions indicate the standard error. The confidence bound is $\epsilon = 0.1$, the discordance threshold is $\zeta = 0.2$, the number of edges that we choose at each discrete time for interaction is $f = 0.2 N$, and the convergence parameter is $\rho = 0.3$. We terminate a simulation either when it reaches our stopping criterion or when $t_{\text{max}} = 2000$ time steps have elapsed (whichever occurs first).
• Figure 4: We illustrate the dependence of several final opinion-profile properties in our NDW model with homophilic rewiring on the confidence bound $\epsilon$ for different values of (a, b, c, d) the neighborhood-tuning parameter $\sigma$ and (e, f, g, h) the discordance threshold $\zeta$. We show (a, e) the number of opinion clusters, (b, f) the fraction of nodes in the largest opinion cluster, (c, g) the fraction of nodes in the second-largest opinion cluster, and (d, h) the dispersion index $\Delta$. In each simulation, the initial network is a $G(N,p)$ ER graph with $N = 50$ nodes and an independent, homogeneous probability $p = 0.3$ of an edge between each pair of nodes. In (a, b, c, d), the discordance threshold is $\zeta = 0.2$. In (e, f, g, h), the neighborhood-tuning parameter is $\sigma = 0.5$. For each network, we initialize each node opinion to a uniformly random value in $[0,1]$. The number of edges that we select at each discrete time for agents to interact is $f = 0.2N$, and the convergence parameter is $\rho = 0.3$. We plot means of 5 simulations, with the same 5 initial networks and the same sets of initial opinions for each panel. The shaded regions indicate the standard error.
• Figure 5: We simulate our adaptive NDW model with homophilic rewiring on three types of networks. We consider (top) a Holme--Kim power-law graph with clustering, (middle) a Newman--Watts--Strogatz (NWS) small-world graph, and (bottom) the Zachary Karate Club graph. For each type of graph, we plot (a, d, g) the number of discordant edges in the network, (b, e, h) the spectral gap of the network's adjacency matrix, and (c, f, i) a degree-assortativity coefficient. We compare the network properties for the baseline adaptive DW model (i.e., the neighborhood-tuning parameter is $\sigma = 1$), a mixed adaptive NDW model (with $\sigma = 0.5$), and a pure adaptive NDW (i.e., $\sigma = 0$). For the Holme--Kim graph, we start with 5 isolated nodes, add a new node to attach to 5 existing nodes using linear preferential attachment, and incorporate additional edges for triadic closure with probability $\tilde{p} = 0.3$. For the NWS small-world graph, each node is adjacent to $6$ nearest neighbors and the probability of adding a new edge for each edge is $\tilde{p} = 0.3$. The Holme--Kim and NWS graphs have $N = 50$ nodes, and the Zachary Karate Club graph has $N = 34$ nodes. We initialize each node opinion to a uniformly random value in the interval $[0,1]$. Each curve in each panel is the mean of 5 simulations with different initial opinions. We use the same Holme--Kim graph and the same NWS graph for all simulations. The shaded regions indicate the standard error. The confidence bound is $\epsilon = 0.1$, the discordance threshold is $\zeta = 0.2$, the number of edges that we choose at each discrete time for interaction is $f = 0.2 N$, and the convergence parameter is $\rho = 0.3$. We terminate a simulation either when it reaches our stopping criterion or when $t_{\text{max}} = 2000$ time steps have elapsed (whichever occurs first).