Universality for critical heavy-tailed network models: Metric structure of maximal components

Abstract

We study limits of the largest connected components (viewed as metric spaces) obtained by critical percolation on uniformly chosen graphs and configuration models with heavy-tailed degrees. For rank-one inhomogeneous random graphs, such results were derived by Bhamidi, van der Hofstad, Sen (2018) [15]. We develop general principles under which the identical scaling limits as [15] can be obtained. Of independent interest, we derive refined asymptotics for various susceptibility functions and the maximal diameter in the barely subcritical regime.

Geometry of the vacant set left by random walk on random graphs, Wright's constants, and critical random graphs with prescribed degrees

“…Then the maximal components in the critical scaling window still belong to the Erdős-Rényi universality class as in (b) above with distances scaling like n −1∕3 . This contrasts drastically with critical percolation on these random graphs where maximal components with distances scaled by n − −3 −1 converge to limiting random fractals [15,16].…”

“…1.2]) is the continuum scaling limits of the maximal components in the critical regime of the so‐called Norros‐Reittu model where the driving weight sequence is assumed to have heavy tails with exponent τ ∈ (3,4). These were extended to critical percolation for the configuration model with heavy tailed degree sequence in . For a full description of this random graph model as well as the corresponding limits we refer the interested reader to .…”

Degree correlations in scale-free random graph models

“…The erased configuration model and the rank-1 inhomogeneous random graph are closely related. They are known to behave similarly for example under critical percolation [4,5], in terms of distances [40] when τ > 3, and in terms of clustering when τ ∈ (2, 3) [39]. The hyperbolic random graph typically shows different behavior, for example in terms of clustering [14,21], or connectivity [7,8].…”

A probabilistic approach to the leader problem in random graphs

“…The problem of establishing universality of the scaling limit of the MST is currently open. One way to approach the problem of universality that has proven to be useful [6,8,10,11] would be to first obtain the scaling limit of the MST constructed on an inhomogeneous random graph that is closely related to the multiplicative coalescent, and then extending it to more general random graph models (including those with heavy tailed degrees) by suitable coupling methods. Carrying out this program requires the extension of the leader result to the setting of the multiplicative coalescent, which is accomplished in this paper.…”

“…↓ almost surely, and further, 8) with respect to the topology on l 2 ↓ . For any component  of G(x (n) , t ), let l n () (resp.…”