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The naming game (NG) is a classic model for studying the emergence and evolution of language within a population. In this article, we extend the traditional NG model to encompass multiple committed opinions and investigate the system dynamics on the complete graph with an arbitrarily large population and random networks of finite size. For the fully connected complete graph, the homogeneous mixing condition enables us to use mean-field theory to analyze the opinion evolution of the system. However, when the number of opinions increases, the number of variables describing the system grows exponentially. To mitigate this, we focus on a special scenario where the largest group of committed agents compete with a motley of committed groups, each of which is smaller than the largest one, while initially, most of uncommitted agents hold one unique opinion. This scenario is chosen for its recurrence in diverse societies and its potential for complexity reduction by unifying agents from smaller committed groups into one category. Our investigation reveals that when the size of the largest committed group reaches the critical threshold, most of uncommitted agents change their beliefs to this opinion, triggering a phase transition. Further, we derive the general formula for the multiopinion evolution using a recursive approach, enabling investigation into any scenario. Finally, we employ agent-based simulations to reveal the opinion evolution and dominance transition in random graphs. Our results provide insights into the conditions under which the dominant opinion emerges in a population and the factors that influence these conditions.
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Stability Theory of Stochastic Models in Opinion Dynamics
We consider a certain class of nonlinear maps that preserve the probability simplex, i.e., stochastic maps, that are inspired by the DeGroot-Friedkin model of belief/opinion propagation over influence networks. The corresponding dynamical models describe the evolution of the probability distribution of interacting species. Such models where the probability transition mechanism depends nonlinearly on the current state are often referred to as nonlinear Markov chains. In this paper we develop stability results and study the behavior of representative opinion models. The stability certificates are based on the contractivity of the nonlinear evolution in the l1-metric. We apply the theory to two types of opinion models where the adaptation of the transition probabilities to the current state is exponential and linear, respectively–both of these can display a wide range of behaviors. We discuss continuous-time and other generalizations
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- PAR ID:
- 10106348
- Date Published:
- Journal Name:
- IEEE Transactions on Automatic Control
- ISSN:
- 0018-9286
- Page Range / eLocation ID:
- 1 to 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TAC.2019.2912490
