Individual bias and fluctuations in collective decision making from algorithms to Hamiltonians.*

We explore a spin model for binary collective decision-making in higher organisms proposed by [Hartnett et al, Phys. Rev. Lett.116 038701 (2016)]. We investigate possible Hamiltonians of interactions and look for the equilibrium state via explicit calculation of their partition function. We show that, depending on the assumptions about the nature of social interactions, two different Hamiltonians can be formulated, which can be solved using different methods. Our approach yields exact solutions for the thermodynamics of the model on complete graphs, with our general analytical predictions validated through individual-based simulations. These simulations also enable us to investigate how system size and initial conditions impact collective decision-making in finite-sized systems, particularly in terms of convergence to metastable states.

Furthermore, we extend our analysis to the spin model on Erdos-Renyi random graphs. We discuss various measures of susceptibility, which help identify (quasi-)critical points in simulations, and explore the network's response to a periodic external field.