Certified symmetry breaking in Laplacian networks with Hebbian memory: abrupt at high gain, smooth only in a low-gain window

In networks of processes with mutual inhibition and Hebbian memory, the uniform state can lose its symmetry either continuouslyor through a jump. We study this transition in networks of N processes coupled through a graph, with a slow symmetric memory,generalizing a minimal three-process model. The system is a gradient-like flow, and the character of the symmetry breaking reducesto two spectral invariants of the graph and two functions of the parameters. We prove that the breaking is abrupt on every graphwhenever G[(νmax − κ)0.72016 + θ] ≥ 2.62822. The activity threshold sA = 0.72015697954... is certified with interval arithmeticand is the exact value of the bound. Smooth breaking is confined to a narrow low-gain strip: among connected graphs with up tonine nodes, it occurs only up to an activity of 0.6902. For a negative threshold we give a closed-form bound. On the complete graphwith three nodes we certify, at 18 values of the memory parameter, that the axial saddle is the lowest point on the boundary of thebasin of the symmetric state Keywords: symmetry breaking, transcritical bifurcation, graph Laplacian, Hebbian memory, interval arithmetic, equilibriumcensus, basin depth