Eigenvalue spectral tails and localisation properties of asymmetric networks

In contrast to the neatly bounded spectra of densely populated large random matrices, sparse random matrices often exhibit unbounded eigenvalue tails on the real and imaginary axis, called Lifshitz tails. In the case of asymmetric matrices, concise mathematical results have proved elusive. In this work, we present an analytical approach to characterising these tails. We exploit the fact that eigenvalues in the tail region have corresponding eigenvectors that are exponentially localised on highly-connected hubs of the network associated to the random matrix. We approximate these eigenvectors using a series expansion in the inverse connectivity of the hub, where successive terms in the series take into account further sets of next-nearest neighbours. By considering the ensemble of such hubs, we are able to characterise the eigenvalue density and the extent of localisation in the tails of the spectrum in a general fashion. As such, we classify a number of different asymptotic behaviours in the Lifshitz tails, as well as the leading eigenvalue and the inverse participation ratio. We demonstrate how an interplay between matrix asymmetry, network structure, and the edge-weight distribution leads to the variety of observed behaviours.