Collective Decision-Making Over Nonlinear Potential Fields

Collective decision-making is a fundamental cognitive process through which groups coordinate actions and resolve competing alternatives. It plays a vital role across domains ranging from animal behavior and human societies to robotics, cybersecurity, and urban systems. Motivated by recent findings in fish schools, where individuals switch between cohesive and exploratory behaviors based on a nonlinear decision landscape, we propose a model of collective decision-making in which each agent evolves according to gradient flow over a bistable potential. Grounded in statistical mechanics, the model employs a double-well energy landscape, a canonical structure that captures transitions between metastable states and naturally encodes the competition between two discrete options, enabling the study of consensus formation or polarization. Unlike classical opinion dynamics models, which describe the continuous evolution of beliefs, our framework assumes that agents possess intrinsic discrete decision states and focuses on transitions between them due to social interactions. Using Lyapunov analysis and spectral graph theory, we derive explicit forward-invariant sublevel sets around each decision equilibrium, yielding closed-form estimates of the basins of attraction and revealing how intrinsic agent dynamics and network topology jointly govern convergence to a shared decision. We demonstrate the effectiveness of our framework through both synthetic simulations and real-world case studies, including collective strike decisions in the Kapferer Tailor Shop network and intermittent navigation behavior in fish schools.