Fig. 2: Stability of random interaction networks.
a For random interaction networks (red markers,`rand'), the maximal real part of the eigenvalues of the interaction matrix \({\lambda }_{M,{r}_{\max }}\) scales with \(\sqrt{N}\). Surprisingly, for random directed acyclic networks (blue markers,`DAG'), \({\lambda }_{M,{r}_{\max }}\) also scales approximately with \(\sqrt{N}\). In both of these cases, increasing the interaction strength from \({{{\Omega }}}_{\max }=1.5\) (circles) to \({{{\Omega }}}_{\max }=2\) (triangles) increases \({\lambda }_{M,{r}_{\max }}\). These results suggest that the system will become unstable (i.e., \({\mathrm{log}\,}_{10}({\lambda }_{M,{r}_{\max }})\) exceeds 0, indicated by the black dashed line) when N or \({{{\Omega }}}_{\max }\) becomes too large. Each data point is obtained from an average of 10 randomly drawn networks, with error bars indicating the interquartile range. Each random network is constructed by randomly selecting ρN2 interactions from N(N − 1) possibilities, with half of the interactions chosen to be upregulating and the remaining half to be downregulating. The construction of DAGs is described in (c). For each regulatory interaction, fold change is chosen uniformly between 1 and \({{{\Omega }}}_{\max }\). [Other parameters: ρ = 0.01, h = 1]. b When systems go out of stability, dynamics of protein concentrations c exhibit oscillatory (left, \({{{\Omega }}}_{\max }=20\)) followed by chaotic behavior (right, \({{{\Omega }}}_{\max }=200\)) as interaction strengths are increased. [Other parameters: N = 200, ρ = 0.2, h = 1, fully random network, time t is in units of 1/kp.] c Random directed acyclic networks are constructed by randomly drawing connections between proteins (red circles represent TFs, blue circles represent non-TFs). If a drawn connection creates a loop (e.g., the gray arrow with a cross on it), it is rejected.
