[Paper Review] Cluster tails for critical power-law inhomogeneous random graphs

Remco van der Hofstad, S.M. Kliem+1 more|arXiv (Cornell University)|Apr 7, 2014

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes the tail behavior of the largest cluster in critical power-law inhomogeneous random graphs with degree exponent $\tau \in (3,4)$, extending Pittel's Erdős-Rényi tail asymptotics to heavy-tailed networks. Using large deviations and weak convergence techniques, it derives the exact asymptotic decay rate of the probability that the rescaled largest cluster exceeds a large value $u$, showing a stretched exponential tail governed by a thinned Lévy process scaling limit.

ABSTRACT

Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained (see previous work by Bhamidi, van der Hofstad and van Leeuwaarden). It was proved that when the degrees obey a power law with exponent in the interval (3,4), the sequence of clusters ordered in decreasing size and scaled appropriately converges as n goes to infinity to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel for the Erdős-Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

Motivation & Objective

• To understand the tail behavior of the largest cluster in critical inhomogeneous random graphs with power-law degrees having infinite third moment.
• To extend Pittel's exact asymptotic for the Erdős-Rényi graph's largest component to the setting of rank-1 inhomogeneous random graphs with $\tau \in (3,4)$.
• To characterize the probability that the rescaled largest cluster exceeds a large value $u$ in the critical window.
• To establish the scaling limit of cluster sizes via a thinned Lévy process and analyze its tail properties.

Proposed method

• Uses exponential tilting and weak convergence to analyze the scaling limit of cluster sizes in the critical regime.
• Applies large deviations techniques to study the tail behavior of the rescaled largest cluster.
• Employs a thinned Lévy process as the scaling limit of cluster sizes, derived from the exploration process of the graph.
• Analyzes the finite-dimensional distributions and tightness of rescaled processes to prove weak convergence.
• Uses conditional independence and moment bounds to control the variance of the rescaled process.
• Relies on the conditional distribution of vertex weights given large cluster size to derive tail estimates.

Experimental results

Research questions

• RQ1 What is the asymptotic decay rate of the probability that the rescaled largest cluster exceeds a large value $u$ in critical inhomogeneous random graphs with $\tau \in (3,4)$?
• RQ2 How does the tail behavior of the largest cluster differ from the Erdős-Rényi case when degrees have infinite third moment?
• RQ3 What role does the thinned Lévy process play in describing the scaling limit of cluster sizes in this heavy-tailed regime?
• RQ4 Can large deviations techniques be used to derive sharp tail asymptotics in this non-i.i.d. setting?
• RQ5 How does the conditional distribution of vertex weights affect the tail of the largest cluster?

Key findings

• The probability that the rescaled largest cluster exceeds a large value $u$ decays as $\frac{1}{\sqrt{2\pi}u^{3/2}}\mathrm{e}^{-\frac{1}{8}u(u-2\lambda)^2}(1+o(1))$ as $u \to \infty$, extending Pittel's result to the inhomogeneous case.
• The scaling limit of the largest cluster is governed by a thinned Lévy process, which arises as the limit of the exploration process in the critical window.
• The tail asymptotics are derived via exponential tilting and weak convergence arguments, with careful control of conditional moments.
• The variance of the rescaled process is bounded uniformly in $u$, ensuring tightness and convergence to the limiting process.
• The analysis shows that the contribution from vertices with small weights is negligible in the tail, while high-weight vertices dominate the large-deviation behavior.
• The result confirms that the tail behavior is determined by the interplay between the heavy-tailed degree distribution and the critical scaling window, with the decay rate reflecting the geometry of the limiting process.