Clustering Does Not Always Imply Latent Geometry
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909561110134784 |
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| author | Aliakbarisani, Roya Boguñá, Marián Serrano, M. Ángeles |
| author_facet | Aliakbarisani, Roya Boguñá, Marián Serrano, M. Ángeles |
| contents | The latent space approach to complex networks has revealed fundamental principles and symmetries, enabling geometric methods. However, the conditions under which network topology implies geometricity remain unclear. We provide a mathematical proof and empirical evidence showing that the multiscale self-similarity of complex networks is a crucial factor in implying latent geometry. Using degree-thresholding renormalization, we prove that any random scale-free graph in a $d$-dimensional homogeneous and isotropic manifold is self-similar when interactions are pairwise. Hence, both clustering and self-similarity are required to imply geometricity. Our findings highlight that correlated links can lead to finite clustering without self-similarity, and therefore without inherent latent geometry. The implications are significant for network mapping and ensemble equivalence between graphs and continuous spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05369 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Clustering Does Not Always Imply Latent Geometry Aliakbarisani, Roya Boguñá, Marián Serrano, M. Ángeles Physics and Society Disordered Systems and Neural Networks Statistical Mechanics The latent space approach to complex networks has revealed fundamental principles and symmetries, enabling geometric methods. However, the conditions under which network topology implies geometricity remain unclear. We provide a mathematical proof and empirical evidence showing that the multiscale self-similarity of complex networks is a crucial factor in implying latent geometry. Using degree-thresholding renormalization, we prove that any random scale-free graph in a $d$-dimensional homogeneous and isotropic manifold is self-similar when interactions are pairwise. Hence, both clustering and self-similarity are required to imply geometricity. Our findings highlight that correlated links can lead to finite clustering without self-similarity, and therefore without inherent latent geometry. The implications are significant for network mapping and ensemble equivalence between graphs and continuous spaces. |
| title | Clustering Does Not Always Imply Latent Geometry |
| topic | Physics and Society Disordered Systems and Neural Networks Statistical Mechanics |
| url | https://arxiv.org/abs/2503.05369 |