Spectra of random graphs with discrete scale invariance

Random graphs defined by an occurrence probability invariant under node aggregation have been identified recently in the context of network renormalization. The invariance property requires to draw edges with a specific probability that, in the annealed case, depends on a necessarily infinite-mean node fitness. The diverging mean determines many properties that are uncommon in models with independent edges, but at the same time widespread in real-world networks. Here, we focus on the leading eigenvalues and eigenvectors of the adjacency matrix of the model, where the n nodes are assigned a Pareto( α )-distributed fitness with 0 < α < 1. We find that the leading eigenvalues are of order √n n , alternate in sign and are located at the intersection between the real axis and a logarithmic spiral in the complex plane, which we characterize analytically in terms of the Gamma function. We also calculate the associated eigenvectors, finding complex-valued scaling exponents and log-periodicity, which are signatures of discrete scale-invariance. In contrast with typical finite-rank behaviour of random graphs, we find that a growing number of the leading eigenvalues emerges from the bulk, whose edge extends up to order √n n and therefore reaches the same scale as that of the structural eigenvalues.