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- W2991293040 abstract "The article considers an inhomogeneous Erdős-Renyi 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 $ile j$, 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-Renyi random graph under the assumption that $Nvarepsilon_Nto infty$ sufficiently fast. Although the bulk of the spectrum of the adjacency matrix, scaled by $sqrt{Nvarepsilon_N}$, 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)." @default.
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- W2991293040 date "2019-11-19" @default.
- W2991293040 modified "2023-09-27" @default.
- W2991293040 title "Eigenvalues outside the bulk of inhomogeneous ErdH{o}s-Renyi random graphs" @default.
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