The spectrum of dense kernel-based random graphs

TL;DR

This work analyzes the adjacency spectrum of dense kernel-based random graphs (KBRGs) on a discrete torus with Pareto weights. For $0<\alpha<d$, $\tau>2$, and $\sigma\in(0,\tau-1)$, the authors prove the empirical spectral distribution of the properly scaled adjacency matrix converges in probability to a deterministic measure $\mu_{\sigma,\tau}$, which is symmetric and absolutely continuous, and is characterized via an operator-valued semicircle framework and a fixed-point Stieltjes transform equation. In the special case $\sigma=1$ (scale-free percolation), they explicitly identify $\mu_{1,\tau}$ as the free multiplicative convolution $\mu_{sc}\boxtimes \mu_W$ and establish power-law tails with exponent $2(\tau-1)$. The paper also proves the finiteness and non-degeneracy of the second moment of the limit, and develops a contraction-based analytic description of the Stieltjes transform, offering concrete insight into the limit’s structure and tail behavior. The methods combine truncation, Gaussianisation, moment methods with free probability tools, and fixed-point analyses in Banach spaces, providing a robust approach to dense, inhomogeneous random graphs with heavy-tailed weights.

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 $τ-1>0$, we connect any two points $i$ and $j$ on the torus with probability $$p_{ij}= \frac{κ_σ(W_i,W_j)}{\|i-j\|^α} \wedge 1$$ for some parameter $α>0$ and $κ_σ(u,v)= (u\vee v)(u \wedge v)^σ$ for some $σ\in(0,τ-1)$. We focus on the adjacency operator of this random graph and study its empirical spectral distribution. For $α<d$ and $τ>2$, we show that a non-trivial limiting distribution exists as $N\to\infty$ and that the corresponding measure $μ_{σ,τ}$ is absolutely continuous with respect to the Lebesgue measure. $μ_{σ,τ}$ 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 $μ_{σ,τ}$ and prove that the second moment is finite even when the weights have infinite variance. In the case $σ=1$, corresponding to the so-called scale-free percolation random graph, we can explicitly describe the limiting measure and study its tail.

Figures (6)

• Figure 1: Eigenvalue distribution for two KBRG realizations
• Figure 2: KBRG eigenvalue distribution and $P_N G_N P_N$ distribution.
• Figure 3: Negative of the log-empirical survival function and tails of Theorem \ref{['theorem:tail']} for $x\geq 1.5$.
• Figure 4: ESD for $\tilde{\mathbf{A}}_{N,m,g}$.
• Figure 5: Left: closed walk on $[4]$. Right: graph associated to $\gamma\pi=\{\{1,3\},\{2\},\{4\}\}$. The root is in red.

Theorems & Definitions (53)

• Theorem 2.1: Limiting spectral distribution
• Theorem 2.2: Limiting ESD for $\sigma=1$
• Theorem 2.3: Non-degeneracy of the limiting measure
• Theorem 2.4: Absolute continuity
• Theorem 2.5: Stieltjes transform
• Remark 2.6: Higher dimensions
• Remark 2.7: Sparse case
• Proposition 3.1: Hoffman-Wielandt inequality Bai-silverstein
• Lemma 3.2
• Lemma 3.3