The spectrum of dense kernel-based random graphs

Abstract

Kernel-based random graphs (KBRGs) are a broad class of random graph models that account for inhomogeneity among vertices. We consider KBRGs on a discrete $d-$dimensional torus $\mathbf{V}_N$ of size $N^d$. Conditionally on an i.i.d.~sequence of {Pareto} weights $(W_i)_{i\in \mathbf{V}_N}$ with tail exponent $\tau-1>0$, we connect any two points $i$ and $j$ on the torus with probability $$p_{ij}= \frac{\kappa_{\sigma}(W_i,W_j)}{\|i-j\|^{\alpha}} \wedge 1$$ for some parameter $\alpha>0$ and $\kappa_{\sigma}(u,v)= (u\vee v)(u \wedge v)^{\sigma}$ for some $\sigma\in(0,\tau-1)$. We focus on the adjacency operator of this random graph and study its empirical spectral distribution. For $\alpha<d$ and $\tau>2$, we show that a non-trivial limiting distribution exists as $N\to\infty$ and that the corresponding measure $\mu_{\sigma,\tau}$ is absolutely continuous with respect to the Lebesgue measure. $\mu_{\sigma,\tau}$ is given by an operator-valued semicircle law, whose Stieltjes transform is characterised by a fixed point equation in an appropriate Banach space. We analyse the moments of $\mu_{\sigma,\tau}$ and prove that the second moment is finite even when the weights have infinite variance. In the case $\sigma=1$, corresponding to the so-called scale-free percolation random graph, we can explicitly describe the limiting measure and study its tail.