Eigenvalues outside the bulk of inhomogeneous Erd\H{o}s-R\"enyi random graphs

Understand

The article considers an inhomogeneous Erd\H{o}s-R\"enyi 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 Erd\H{o}s-R\'enyi 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.