Extremal spectral properties of random graphs

In this work we study statistical properties of the extreme eigenstates of the randomly weighted adjacency matrices of random graphs. We focus on two random graph models: Erdős-Rényi graphs and random geometric graphs. Indeed, the adjacency matrices of both graph models are diluted versions of the Gaussian orthogonal ensemble (GOE) of random-matrix theory such that a transition from the Poisson ensemble (PE) to the GOE is observed by increasing the graph average degree 〈k〉. First, we write expressions for the spectral density in terms of 〈k〉 for the regimes below and above the percolation threshold. Then we show that the distributions of the largest λ₁ and second-largest λ₂ eigenvalues approach the Tracy-Widom distribution of type 1 for 〈k〉≫1, while 〈λ₁〉=sqrt[2〈k〉]. Additionally, we demonstrate that the distributions of the normalized distance between λ₁ and λ₂, the distribution of the ratio between higher consecutive eigenvalue spacings, and the distributions of the inverse participation ratios of the extreme eigenstates display a clear PE-to-GOE transition as a function of 〈k〉. We conclude that any of these distributions can be effectively used to probe the delocalization transition of the graph models without the need of the full spectrum.