Year: 2026
Type: article
Abstract: 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 $$\sqrt{n}$$ 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 $$\sqrt{n}$$ n and therefore reaches the same scale as that of the structural eigenvalues.
Source: npj Complexity
Institution: Leiden University
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,690
OpenAlex ID: W4416258788
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