Agent-based models of opinion dynamics examine the spread of opinions between entities and allow one to study phenomena such as consensus, polarization, and fragmentation. One examines them on social networks to investigate the effects of network structure on these phenomena. In social networks, some individuals share their ideas and opinions more frequently than others.These disparities can arise from heterogeneous sociabilities, heterogeneous activity levels, differentprevalences to share opinions when engaging in a social-media platform, or something else. To examine the impact of such heterogeneities on opinion dynamics, we generalize the Deffuant–Weisbuch (DW) bounded-confidence model (BCM) of opinion dynamics by incorporating node weights. The node weights allow us to model agents with different probabilities of interacting. Using numerical simulations, we systematically investigate (using a variety of network structures and node-weight distributions) the effects of node weights, which we assign uniformly at random to the nodes. Wedemonstrate that introducing heterogeneous node weights results in longer convergence times andmore opinion fragmentation than in a baseline DW model. One can use the node weights of our BCM to capture a variety of sociological scenarios in which agents have heterogeneous probabilities of interacting with other agents.

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People's opinions change with time as they interact with each other. In a boundedconfidence model (BCM) of opinion dynamics, individuals (which are represented by the nodes of a network) have continuous-valued opinions and are influenced only by neighboring nodes whose opinions are within their confidence bound. In this paper, we formulate and analyze discrete-time BCMs with heterogeneous and adaptive confidence bounds. We introduce two new models: (1) a BCM with synchronous opinion updates that generalizes the Hegselmann-Krause (HK) and (2) a BCM with asynchronous opinion updates that generalizes the Deffuant-Weisbuch (DW) model. We analytically and numerically explore our adaptive BCMs' limiting behaviors, including the confidencebound dynamics, the formation of clusters of nodes with similar opinions, and the time evolution of an "effective graph", which is a time-dependent subgraph of a network with edges between nodes that can currently influence each other. For a wide range of parameters that control the increase and decrease of confidence bounds, we demonstrate for a variety of networks that our adaptive BCMs result in fewer major opinion clusters and longer convergence times than the baseline (i.e., nonadaptive) BCMs. We also show that our adaptive BCMs can have pairs of adjacent nodes that converge to the same opinion but are not able to influence each other. This qualitative behavior does not occur in the associated baseline BCMs.

People often change their views about things as a result of discussing them with other people. To study how people influence each other through their social interactions, researchers use mathematical models of opinions and social influence. In today’s information age, these models can give insights into how to promote accurate information and reduce unwanted influence. In this article, we discuss a simple mathematical model of opinion influence. We briefly illustrate what opinion models can tell us and how researchers try to make them more realistic.