Consensus of discrete-time opinion dynamics with stubbornness under fixed and switching topologies

Dianxingzi Chen; Linglong Du; Yu Huang; Dianxingzi Chen; Linglong Du; Yu Huang

doi:10.3934/nhm.2026077

Networks and Heterogeneous Media

2026, Volume 21, Issue 6: 2037-2067. doi: 10.3934/nhm.2026077

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Research article

Consensus of discrete-time opinion dynamics with stubbornness under fixed and switching topologies

School of Mathematics and Statistics, Donghua University, Shanghai, P. R. China

Received: 11 June 2026 Revised: 01 October 2026 Accepted: 08 October 2026 Published: 19 October 2026

We consider a discrete-time consensus dynamics of an opinion model with stubbornness under fixed and switching Topologies. Regarding the underlying network structures, both static and dynamically switching interaction topologies are considered. For the fixed topology, we prove that as long as the system parameters and initial states meet certain conditions, the presence of a rooted directed graph suffices for consensus. For the dynamically switching topology, under similar parametric conditions, consensus is guaranteed provided that the union of switched directed graphs over a bounded time interval contains a directed spanning tree. Numerical simulations are established to show the consensus and non-consensus phenomenon.
- discrete opinion model,
- consensus behavior,
- Rayleigh friction,
- static topology,
- dynamically switching topology
Citation: Dianxingzi Chen, Linglong Du, Yu Huang. Consensus of discrete-time opinion dynamics with stubbornness under fixed and switching topologies[J]. Networks and Heterogeneous Media, 2026, 21(6): 2037-2067. doi: 10.3934/nhm.2026077

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Abstract

We consider a discrete-time consensus dynamics of an opinion model with stubbornness under fixed and switching Topologies. Regarding the underlying network structures, both static and dynamically switching interaction topologies are considered. For the fixed topology, we prove that as long as the system parameters and initial states meet certain conditions, the presence of a rooted directed graph suffices for consensus. For the dynamically switching topology, under similar parametric conditions, consensus is guaranteed provided that the union of switched directed graphs over a bounded time interval contains a directed spanning tree. Numerical simulations are established to show the consensus and non-consensus phenomenon.

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© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)