Eigenvalue distribution of large weighted multipartite random sparse graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929516850446336 |
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| author | Vengerovsky, Valentin |
| author_facet | Vengerovsky, Valentin |
| contents | We investigate the distribution of eigenvalues of weighted adjacency matrices from a specific ensemble of random graphs. We distribute $N$ vertices across a fixed number $κ$ of components, with asymptotically $α_j \dot N$ vertices in each component, where the vector $(α_1,α_2, \ldots, α_κ)$ is fixed. Consider a connected graph $Γ$ with $κ$ vertices. We construct a multipartite graph with $N$ vertices, in which all vertices in the $i$-th component are connected to all vertices in the $j$-th component if if $Γ_{ij}=1$. Conversely, if $Γ_{ij}=0$, no edge connects the $i$-th and $j$-th components. In the resulting graph, we independently retain each edge with a probability of $p/N$, where $p$ is a fixed parameter. To each remaining edge, we assign an independent weight with a fixed distribution, that possesses all finite moments. We establish the weak convergence in probability of a random counting measure to a non-random probability measure. Furthermore, the moments of the limiting measure can be derived from a system of recurrence relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18148 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eigenvalue distribution of large weighted multipartite random sparse graphs Vengerovsky, Valentin Mathematical Physics Probability 60B20 We investigate the distribution of eigenvalues of weighted adjacency matrices from a specific ensemble of random graphs. We distribute $N$ vertices across a fixed number $κ$ of components, with asymptotically $α_j \dot N$ vertices in each component, where the vector $(α_1,α_2, \ldots, α_κ)$ is fixed. Consider a connected graph $Γ$ with $κ$ vertices. We construct a multipartite graph with $N$ vertices, in which all vertices in the $i$-th component are connected to all vertices in the $j$-th component if if $Γ_{ij}=1$. Conversely, if $Γ_{ij}=0$, no edge connects the $i$-th and $j$-th components. In the resulting graph, we independently retain each edge with a probability of $p/N$, where $p$ is a fixed parameter. To each remaining edge, we assign an independent weight with a fixed distribution, that possesses all finite moments. We establish the weak convergence in probability of a random counting measure to a non-random probability measure. Furthermore, the moments of the limiting measure can be derived from a system of recurrence relations. |
| title | Eigenvalue distribution of large weighted multipartite random sparse graphs |
| topic | Mathematical Physics Probability 60B20 |
| url | https://arxiv.org/abs/2409.18148 |