Linear stability analysis for large dynamical systems on directed random graphs

TL;DR

This paper develops a linear stability analysis for large dynamical systems on directed random graphs, revealing universal phase transition behavior and stability conditions even with unbounded degree distributions.

Contribution

It provides an exact theory for the leading eigenvalue and eigenvectors of large directed graphs, showing stability can persist with unbounded degrees and characterizing the phase transition.

Findings

Large directed graphs can be stable despite unbounded degree distributions.

The stability transition depends only on mean degree and degree correlation.

Unstable regimes exhibit universal destabilizing eigenmodes.

Abstract

We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems: First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase is universal in the sense that it depends only on a few parameters, such as, the mean degree and a degree correlation coefficient. In addition, in the unstable regime we characterize the nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally tree-like and oriented, as well as, for the right and left eigenvectors associated with the leading eigenvalue. We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory can be extended to graphs with diagonal disorder and to graphs that contain nondirected links. Finally, we discuss the influence of small cycles and how they can destabilize large dynamical systems when they induce strong enough feedback loops.