Eigenvalues outside the bulk of inhomogeneous Erdős-Rënyi random graphs
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| Format: | Preprint |
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2019
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| _version_ | 1866913244810051584 |
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| author | Chakrabarty, Arijit Chakraborty, Sukrit Hazra, Rajat Subhra |
| author_facet | Chakrabarty, Arijit Chakraborty, Sukrit Hazra, Rajat Subhra |
| contents | The article considers an inhomogeneous Erdős-Rënyi random graph on $\{1,\ldots, N\}$, where an edge is placed between vertices $i$ and $j$ with probability $\varepsilon_N f(i/N,j/N)$, for $i\le j$, the choice being made independent for each pair. The function $f$ is assumed to be non-negative definite, symmetric, bounded and of finite rank $k$. We study the edge of the spectrum of the adjacency matrix of such an inhomogeneous Erdős-Rényi random graph under the assumption that $N\varepsilon_N\to \infty$ sufficiently fast. Although the bulk of the spectrum of the adjacency matrix, scaled by $\sqrt{N\varepsilon_N}$, is compactly supported, the $k$-th largest eigenvalue goes to infinity. It turns out that the largest eigenvalue after appropriate scaling and centering converge to a Gaussian law, if the largest eigenvalue of $f$ has multiplicity $1$. If $f$ has $k$ distinct non-zero eigenvalues, then the joint distribution of the $k$ largest eigenvalues converge jointly to a multivariate Gaussian law. The first order behaviour of the eigenvectors is derived as a by-product of the above results. The results complement the homogeneous case derived by Erdős et al.(2013). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1911_08244 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Eigenvalues outside the bulk of inhomogeneous Erdős-Rënyi random graphs Chakrabarty, Arijit Chakraborty, Sukrit Hazra, Rajat Subhra Probability The article considers an inhomogeneous Erdős-Rënyi random graph on $\{1,\ldots, N\}$, where an edge is placed between vertices $i$ and $j$ with probability $\varepsilon_N f(i/N,j/N)$, for $i\le j$, the choice being made independent for each pair. The function $f$ is assumed to be non-negative definite, symmetric, bounded and of finite rank $k$. We study the edge of the spectrum of the adjacency matrix of such an inhomogeneous Erdős-Rényi random graph under the assumption that $N\varepsilon_N\to \infty$ sufficiently fast. Although the bulk of the spectrum of the adjacency matrix, scaled by $\sqrt{N\varepsilon_N}$, is compactly supported, the $k$-th largest eigenvalue goes to infinity. It turns out that the largest eigenvalue after appropriate scaling and centering converge to a Gaussian law, if the largest eigenvalue of $f$ has multiplicity $1$. If $f$ has $k$ distinct non-zero eigenvalues, then the joint distribution of the $k$ largest eigenvalues converge jointly to a multivariate Gaussian law. The first order behaviour of the eigenvectors is derived as a by-product of the above results. The results complement the homogeneous case derived by Erdős et al.(2013). |
| title | Eigenvalues outside the bulk of inhomogeneous Erdős-Rënyi random graphs |
| topic | Probability |
| url | https://arxiv.org/abs/1911.08244 |