Configuration barrier towards parity-time symmetry in randomly connected mesoscopic sets on a graph

We address the issue of dissipative vs. non-dissipative behavior in a mesoscopic set of coupled elements such as oscillators, with one half having gain and the other half having losses. We introduce a graph with coupling as the graph edges in given fixed number and gain/loss elements as its nodes. This relates to parity-time symmetry, notably in optics, e.g. set of coupled fibers, and more generally to the issue of taming divergence related to imaginary parts of eigenvectors in various network descriptions, for instance biochemical, neuronal, ecological. We thus look for the minimization of the imaginary part of all eigenvalues altogether, with a collective figure of merit. As more edges than gain/loss pairs are introduced, the unbroken cases , i.e., stable cases with real eigenvalues in spite of gain and loss, become statistically very scarce. A minimization from a random starting point by moving one edge at a time is studied, amounting to investigate how the hugely growing configuration number impedes the attainment of the desired minimally-dissipative target. The minimization path and its apparent stalling point are analyzed in terms of network connectivity metrics. We expand in the end on the relevance in biochemical signaling networks and the so-called “stability-optimized circuits” relevant to neural organization. Graphical abstract