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Steady-State Bifurcation in Nonlinear Opinion Dynamics: Analysis on Canonical Networks

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Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2025, Tutora: Maria Del Carmen Miguel Lopez

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This paper explores how collective opinion patterns emerge through steady-state bifurcations in a nonlinear dynamical system. We study a model that describes the time evolution of opinions in a multi-agent system interacting over a social network. We show how this bifurcation emerges, provided that the attention parameter – which quantifies each agent’s social susceptibility – exceeds a threshold value. This threshold is solely characterized by other parameters of the system and by the largest eigenvalue of the interaction matrix. Moreover, we see how, near the threshold, the stationary state is approximately proportional to the eigenvector associated with the largest eigenvalue. We apply this framework to canonical networks, including regular, star, Watts-Strogatz, and scale-free graphs. Finally, we investigate the bifurcation unfolding when the agents
in the system hold biased opinions.

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Steady-State Bifurcation in Nonlinear Opinion Dynamics: Analysis on Canonical Networks