Eigenvalue spectra and stability of directed complex networks

Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that
contribute to the stability of dynamical systems with many interacting components. This work explores the
effect that the interaction network between components has on the eigenvalue spectrum. We build on previous
results, which usually only take into account the mean degree of the network, by allowing for nontrivial network
degree heterogeneity. We derive closed-form expressions for the eigenvalue spectrum of the adjacency matrix
of a general weighted and directed network. Using these results, which are valid for any large well-connected
complex network, we then derive compact formulas for the corrections (due to nonzero network heterogeneity) to
well-known results in random matrix theory. Specifically, we derive modified versions of the Wigner semicircle
law, the Girko circle law, and the elliptic law and any outlier eigenvalues. We also derive a surprisingly neat
analytical expression for the eigenvalue density of a directed Barabási-Albert network. We are thus able to
deduce that network heterogeneity is mostly a destabilizing influence in complex dynamical systems.