In Pursuit of the Emergence Point: Extracting Phase Transitions in Multi-Agent Communication

Modern multi-agent deep reinforcement learning algorithms have demonstrated empirical success in communication games, yet their black box nature precludes the analytical identification of the transition from random babbling to coordinated signalling. This study introduces an explicitly parameterised, interpretable surrogate model of belief evolution in Lewis signalling games. The proposed ordinary differential equation retains the strategic structure of cheap talk while permitting the closed-form computation of the Jacobian spectrum at the uniform babbling equilibrium. It was proven that the onset of communication corresponded to a supercritical pitchfork bifurcation with a critical threshold determined by the dissipation and sensitivity parameters. Consequently, the leading eigenvalue of the dynamics serves as a detector of the emergence point. The analytical predictions were validated through iterative simulations of Lewis signalling games, illustrating how the critical threshold dictates the consistent and stable transition from stochastic babbling to separating equilibrium. Moreover, a phenomenological experiment demonstrates a possible path toward extending spectral diagnostics to policies parameterised by neural networks in a low-dimensional setting, serving as a bridge towards potential method adaptation for general deep reinforcement learning policies, without fully validating the theoretical framework.