Fig. 3: Stability of bipartite networks.
a When constructing a bipartite interaction network, we group the proteins into transcription factors (TFs, red circles) and non-TFs (blue circles), and only allow directed regulatory interactions to go from a TF to a non-TF. b For bipartite networks, there is a critical value for the fraction of inhibitory interactions Pneg (that is slightly > 0.5) below which the maximal real part of the eigenvalues of the interaction matrix \({\lambda }_{M,{r}_{\max }}=0\) and above which \({\lambda }_{M,{r}_{\max }} \, > \, 0\). In the regime where \({\lambda }_{M,{r}_{\max }}=0\) (which can be considered to be deeply stable since it is far from the point \({\lambda }_{M,{r}_{\max }}=1\) where the system becomes unstable), this value of \({\lambda }_{M,{r}_{\max }}\) stays the same even when the number of different proteins N (star markers vs. circles) or interaction strengths \({{{\Omega }}}_{\max }\) (star markers vs. squares) are increased. c When there is an equal fraction of up/downregulatory interactions Pneg = 0.5, \({\lambda }_{M,{r}_{\max }}\) is independent of both N and \({{{\Omega }}}_{\max }\) for bipartite networks (green markers). This is in contrast to fully random networks (‘Random’, red markers) and random directed acyclic graphs (‘DAG’, blue markers) where the system approaches the instability limit (\({\lambda }_{M,{r}_{\max }}=1\)) as N or \({{{\Omega }}}_{\max }\) (circles to triangles) is increased. This implies that a bipartite network structure can maintain and enhance the stability of the system as N or \({{{\Omega }}}_{\max }\) is increased. In both (b) and (c), each data point is obtained from an average of 10 randomly drawn networks, with error bars indicating the interquartile range. [Other parameters: h = 1, ρ = 0.01 for fully random and random DAGs, number of TFs for bipartite networks q = 0.1N].
