Percolation and localisation: Sub-leading eigenvalues of the nonbacktracking matrix
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| Format: | Preprint |
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2025
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| _version_ | 1866915128070373376 |
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| author | Martin, James Rogers, Tim Zanetti, Luca |
| author_facet | Martin, James Rogers, Tim Zanetti, Luca |
| contents | The spectrum of the nonbacktracking matrix associated to a network is known to contain fundamental information regarding percolation properties of the network. Indeed, the inverse of its leading eigenvalue is often used as an estimate for the percolation threshold. However, for many networks with nonbacktracking centrality localised on a few nodes, such as networks with a core-periphery structure, this spectral approach badly underestimates the threshold. In this work, we study networks that exhibit this localisation effect by looking beyond the leading eigenvalue and searching deeper into the spectrum of the nonbacktracking matrix. We identify that, when localisation is present, the threshold often more closely aligns with the inverse of one of the sub-leading real eigenvalues: the largest real eigenvalue with a "delocalised" corresponding eigenvector. We investigate a core-periphery network model and determine, both theoretically and experimentally, a regime of parameters for which our approach closely approximates the threshold, while the estimate derived using the leading eigenvalue does not. We further present experimental results on large scale real-world networks that showcase the usefulness of our approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_17774 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Percolation and localisation: Sub-leading eigenvalues of the nonbacktracking matrix Martin, James Rogers, Tim Zanetti, Luca Physics and Society Social and Information Networks 60K35 (Primary) 05C50, 05C82 (Secondary) The spectrum of the nonbacktracking matrix associated to a network is known to contain fundamental information regarding percolation properties of the network. Indeed, the inverse of its leading eigenvalue is often used as an estimate for the percolation threshold. However, for many networks with nonbacktracking centrality localised on a few nodes, such as networks with a core-periphery structure, this spectral approach badly underestimates the threshold. In this work, we study networks that exhibit this localisation effect by looking beyond the leading eigenvalue and searching deeper into the spectrum of the nonbacktracking matrix. We identify that, when localisation is present, the threshold often more closely aligns with the inverse of one of the sub-leading real eigenvalues: the largest real eigenvalue with a "delocalised" corresponding eigenvector. We investigate a core-periphery network model and determine, both theoretically and experimentally, a regime of parameters for which our approach closely approximates the threshold, while the estimate derived using the leading eigenvalue does not. We further present experimental results on large scale real-world networks that showcase the usefulness of our approach. |
| title | Percolation and localisation: Sub-leading eigenvalues of the nonbacktracking matrix |
| topic | Physics and Society Social and Information Networks 60K35 (Primary) 05C50, 05C82 (Secondary) |
| url | https://arxiv.org/abs/2501.17774 |