Branching Ratios of Input Trees for Directed Multigraphs
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917890160066560 |
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| author | Boldi, Paolo Stewart, Ian |
| author_facet | Boldi, Paolo Stewart, Ian |
| contents | We define the branching ratio of the input tree of a node in a finite directed multigraph, prove that it exists for every node, and show that it is equal to the largest eigenvalue of the adjacency matrix of the induced subgraph determined by all upstream nodes. This real eigenvalue exists by the Perron-Frobenius Theorem for non-negative matrices. We motivate our analysis with simple examples, obtain information about the asymptotics for the limit growth of the input tree, and establish other basic properties of the branching ratio. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06812 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Branching Ratios of Input Trees for Directed Multigraphs Boldi, Paolo Stewart, Ian Combinatorics 05C12, 05C20, 05C38 We define the branching ratio of the input tree of a node in a finite directed multigraph, prove that it exists for every node, and show that it is equal to the largest eigenvalue of the adjacency matrix of the induced subgraph determined by all upstream nodes. This real eigenvalue exists by the Perron-Frobenius Theorem for non-negative matrices. We motivate our analysis with simple examples, obtain information about the asymptotics for the limit growth of the input tree, and establish other basic properties of the branching ratio. |
| title | Branching Ratios of Input Trees for Directed Multigraphs |
| topic | Combinatorics 05C12, 05C20, 05C38 |
| url | https://arxiv.org/abs/2501.06812 |