Inhomogeneous random graphs: how heavy tails reshape connectivity

Abstract

We study inhomogeneous random graphs on $n$ vertices with independent Pareto weights of parameter $α>0$ and connection probabilities $\min(\varepsilon_n W_iW_j, 1)$. Our main focus is the infinite-mean regime $0<α<1$. Under the scaling $n\varepsilon_n^α=λ$, with $λ>0$ fixed, the global architecture of the graph is entirely governed by the Poisson point process $Π$ of rescaled extreme weights. In particular, the size of the largest component, divided by $n$, converges to a non-degenerate random variable expressed as an explicit functional of $Π$. Through the point process $Π$, we also characterize the exact limiting connectivity probability, uncovering a non-sharp connectivity threshold. The second largest component and the diameter of the largest component are bounded with high probability. The distance between two uniformly chosen vertices in the largest component is at most three with high probability. We also prove convergence of the point process of rescaled degrees and identify the limiting distribution of the maximum degree. For completeness, we obtain expected degree asymptotics, typical degree limits and connectivity regimes for all $α>0$. Due to the failure of classical branching process approximations in the infinite-mean regime, our approach relies fundamentally on analyzing the extreme weight environment $Π$.