Figure 4

Sketch illustrating the origin of the noise-induced Jensen’s force. Each node in a sparse network receives an input \(\Lambda \) which is a random variable extracted from some bell-shaped probability distribution function \(P(\Lambda )\) (sketched below the x-axis) with averaged value \(\langle \Lambda \rangle =\gamma (1-2\alpha )s\) and standard deviation \({\sigma }_{s}=(\gamma \sqrt{s(1-s)})/\sqrt{k}\) (see SI-1). The possible outputs \(f(\Lambda )\) are also distributed according to some probability (sketched to the left of the y-axis). Given that around \(\Lambda \approx 0\) the function \(f(\Lambda )\) is locally convex then, as a consequence of Jensen’s inequality for convex functions, \(\langle f(\Lambda )\rangle \ge f(\langle \Lambda \rangle )\) (i.e. the dotted red line is above the blue one). Indeed, while for positive inputs, the transformation is linear, negative ones are mapped into 0 thus creating a net positive Jensen’s force for small values of \(\Lambda \) (or s). The inset shows the Jensen’s force \(F(\tilde{\gamma },s)\equiv \langle f(\Lambda )\rangle > -\,f(\langle \Lambda \rangle )\) computed right at the critical point γ_c for different connectivity values, as a function of s. Note, the negative values for large values of s which stem from the concavity of the function f(x) around x = 1. Note that F decreases as k grows and vanishes in the mean-field limit.