Spectra of random graphs with discrete scale invariance
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866914042199670784 |
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| author | Catanzaro, Alessio Hazra, Rajat Subhra Garlaschelli, Diego |
| author_facet | Catanzaro, Alessio Hazra, Rajat Subhra Garlaschelli, Diego |
| contents | Random graphs defined by an occurrence probability that is invariant under node aggregation have been identified recently in the context of network renormalization. The invariance property requires that edges are drawn 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 all of order square root of 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 that they display complexvalued scaling exponents and log-periodicity, which are signatures of discrete scale invariance. In contrast with the typical finite-rank behaviour of random graphs with finite-mean variables, we find that a growing number of the leading eigenvalues emerges from the bulk, whose edge extends up to order square root of n and therefore reaches the same scale as that of the structural eigenvalues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectra of random graphs with discrete scale invariance Catanzaro, Alessio Hazra, Rajat Subhra Garlaschelli, Diego Spectral Theory Disordered Systems and Neural Networks Statistical Mechanics Mathematical Physics Probability Random graphs defined by an occurrence probability that is invariant under node aggregation have been identified recently in the context of network renormalization. The invariance property requires that edges are drawn 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 all of order square root of 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 that they display complexvalued scaling exponents and log-periodicity, which are signatures of discrete scale invariance. In contrast with the typical finite-rank behaviour of random graphs with finite-mean variables, we find that a growing number of the leading eigenvalues emerges from the bulk, whose edge extends up to order square root of n and therefore reaches the same scale as that of the structural eigenvalues. |
| title | Spectra of random graphs with discrete scale invariance |
| topic | Spectral Theory Disordered Systems and Neural Networks Statistical Mechanics Mathematical Physics Probability |
| url | https://arxiv.org/abs/2509.12407 |