Multilayer directed random networks: Scaling of spectral properties
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Tapia-Labra, G. Hernández-Sánchez, M. Méndez-Bermúdez, J. A. |
| author_facet | Tapia-Labra, G. Hernández-Sánchez, M. Méndez-Bermúdez, J. A. |
| contents | Motivated by the wide presence of multilayer networks in both natural and human-made systems, within a random matrix theory (RMT) approach, in this study we compute eigenfunction and spectral properties of multilayer directed random networks (MDRNs) in two setups composed by $M$ layers of size $N$: A line and a complete graph (node-aligned multiplex network). First, we numerically demonstrate that the normalized localization length $β$ of the eigenfunctions of MDRNs follows a simple scaling law given by $β=x^*/(1+x^*)$, with $x^*\propto (b_{\rm eff}^2/L)^δ$, $δ\sim 1$ and $b_{\rm eff}$ being the effective bandwidth of the adjacency matrix of the network of size $L=M\times N$. Here, $b_{\rm eff}$ incorporates both intra- and inter-layer edges. Then, we show that other eigenfunction and spectral RMT measures (the inverse participation ratio of eigenfunctions, the ratio between nearest- and next-to-nearest-neighbor eigenvalue distances, and the ratio between consecutive singular-value spacings) of MDRNs also scale with $x^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multilayer directed random networks: Scaling of spectral properties Tapia-Labra, G. Hernández-Sánchez, M. Méndez-Bermúdez, J. A. Disordered Systems and Neural Networks Data Analysis, Statistics and Probability Motivated by the wide presence of multilayer networks in both natural and human-made systems, within a random matrix theory (RMT) approach, in this study we compute eigenfunction and spectral properties of multilayer directed random networks (MDRNs) in two setups composed by $M$ layers of size $N$: A line and a complete graph (node-aligned multiplex network). First, we numerically demonstrate that the normalized localization length $β$ of the eigenfunctions of MDRNs follows a simple scaling law given by $β=x^*/(1+x^*)$, with $x^*\propto (b_{\rm eff}^2/L)^δ$, $δ\sim 1$ and $b_{\rm eff}$ being the effective bandwidth of the adjacency matrix of the network of size $L=M\times N$. Here, $b_{\rm eff}$ incorporates both intra- and inter-layer edges. Then, we show that other eigenfunction and spectral RMT measures (the inverse participation ratio of eigenfunctions, the ratio between nearest- and next-to-nearest-neighbor eigenvalue distances, and the ratio between consecutive singular-value spacings) of MDRNs also scale with $x^*$. |
| title | Multilayer directed random networks: Scaling of spectral properties |
| topic | Disordered Systems and Neural Networks Data Analysis, Statistics and Probability |
| url | https://arxiv.org/abs/2410.08221 |