Figure 4
Computation of the mean field "magnetization" ⟨ s σ ⟩ (in the gauge transformed basis) and energy h mf for a transcriptional ( E.coli-transcr , left panel), a signaling ( EGFR-signal , middle), and a metabolic ( Yeast-metab , right) network as a function of β (interaction strength). The values for the three true networks are depicted in red. In blue and green the same ⟨sσ⟩ and h mf for two alternative networks built on the same graph: the exactly monotone network (i.e., with all J ij > 0), and a network with random sign assignments. The gray shaded areas in the upper plot delimit the region ⟨sσ⟩ ≤ 0.8 i.e., the region in which the response of the system to a generic perturbation results in a low-medium ⟨sσ⟩. ⟨sσ⟩ ≥ 0.8 means that in the gauge transformed basis the state s σ "concentrates" sufficiently well at the ground state, and, correspondingly, the energy is in average sufficiently close to the minimum (lower plots). For the E.coli-transc network the threshold ⟨sσ⟩ ≥ 0.8 is achieved in correspondence of β = 0.71, higher than the β = 0.46, 0.45 of EGFR-signal and Yeast-metab. Similar differences are observed in the other networks, see Table S4 and Fig. S7, and are also confirmed in Metropolis-Montecarlo simulations, see Fig. S9