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 upon previous results, which usually only take into account the mean degree of the network, by allowing for non-trivial 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 formulae for the corrections (due to non-zero network heterogeneity) to well-known results in random matrix theory. Specifically, we derive modified versions of the Wigner semi-circle 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 an directed Barabasi-Albert network. We are thus able to make general deductions about the effect of network heterogeneity on the stability of complex dynamic systems.