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Percolation on random graphs

Remco van der Hofstad

TL;DR

This paper surveys how percolation behaves on finite random graphs, emphasizing the phase transition near criticality and how inhomogeneity shapes scaling. It develops and contrasts results for rank-1 models (configuration models and rank-1 IRGs) with those for dynamic, growing graphs, using Janson’s construction and exploration processes to derive scaling limits in diverse moment regimes. A core contribution is the delineation of multiple universality classes: finite-third-moment, infinite-third-moment (heavy-tailed), and infinite-variance degree regimes, each with distinct limiting objects (Brownian excursions, thinned Lévy processes, tiny giants) and explicit critical windows. The analysis shows how local limits, two-step connection structures, and component explorations drive global connectivity properties, providing precise asymptotics for the largest components, susceptibilities, and the emergence of giant components across static rank-1 and dynamic network models.

Abstract

Percolation is a model for random damage to a network. It is one of the simplest models that displays a phase transition: when the network is severely damaged, it falls apart in many small connected components, while if the damage is light, connectivity is hardly affected. We study the location and nature of the phase transition on random graphs. In particular, we focus on the connectivity structure close to, or below, criticality, where components display intricate scaling behaviour such that a typical connected component has bounded size, while the average and maximal connected component sizes grow like powers of the network size. We review the recent progress that has been made in two important settings: random graphs whose expected adjacency matrix is close to being rank-1, the most prominent examples being the configuration model and rank-1 inhomogeneous random graphs, and dynamic random graphs, i.e., random graphs that grow with time, such as uniform and preferential attachment models. Remarkably, these two settings behave rather differently. In all cases, the inhomogeneity of the underlying random graph on which we perform percolation is of crucial importance.

Percolation on random graphs

TL;DR

This paper surveys how percolation behaves on finite random graphs, emphasizing the phase transition near criticality and how inhomogeneity shapes scaling. It develops and contrasts results for rank-1 models (configuration models and rank-1 IRGs) with those for dynamic, growing graphs, using Janson’s construction and exploration processes to derive scaling limits in diverse moment regimes. A core contribution is the delineation of multiple universality classes: finite-third-moment, infinite-third-moment (heavy-tailed), and infinite-variance degree regimes, each with distinct limiting objects (Brownian excursions, thinned Lévy processes, tiny giants) and explicit critical windows. The analysis shows how local limits, two-step connection structures, and component explorations drive global connectivity properties, providing precise asymptotics for the largest components, susceptibilities, and the emergence of giant components across static rank-1 and dynamic network models.

Abstract

Percolation is a model for random damage to a network. It is one of the simplest models that displays a phase transition: when the network is severely damaged, it falls apart in many small connected components, while if the damage is light, connectivity is hardly affected. We study the location and nature of the phase transition on random graphs. In particular, we focus on the connectivity structure close to, or below, criticality, where components display intricate scaling behaviour such that a typical connected component has bounded size, while the average and maximal connected component sizes grow like powers of the network size. We review the recent progress that has been made in two important settings: random graphs whose expected adjacency matrix is close to being rank-1, the most prominent examples being the configuration model and rank-1 inhomogeneous random graphs, and dynamic random graphs, i.e., random graphs that grow with time, such as uniform and preferential attachment models. Remarkably, these two settings behave rather differently. In all cases, the inhomogeneity of the underlying random graph on which we perform percolation is of crucial importance.

Paper Structure

This paper contains 46 sections, 11 theorems, 78 equations.

Table of Contents

  1. Introduction.
  2. Motivation for percolation on finite graphs.
  3. Percolation on finite graphs.
  4. Percolation on random graphs.
  5. The largest components in percolation on random graphs.
  6. Critical value in large graph limits.
  7. Critical behaviour and finite-size effects in large graph limits.
  8. Percolation on rank-1 random graphs.
  9. Percolation on configuration models.
  10. The configuration model.
  11. Assumptions on the degrees.
  12. The phase transition of the configuration model.
  13. Scaling limits for finite third-moment degrees.
  14. Percolation on configuration models with infinite third-moment degrees.
  15. Main results for heavy-tailed configuration models.
  16. ...and 31 more sections

Key Result

Theorem 2.2

Suppose that Conditions cond-degrees-regcond-SP(a)-(b) hold, and consider the random graph ${\rm CM}_n(\boldsymbol{d})$, letting $n\to\infty$. Assume that sup-CM-perc holds, so that ${\rm CM}_n(\boldsymbol{d})$ is supercritical.

Theorems & Definitions (12)

  • Theorem 2.2: Percolation phase transition in ${\rm CM}_n(\boldsymbol{d})$ Jans09c
  • Theorem 2.3: Critical percolation on configuration models with finite third-moment degrees DhaHofLeeSen17
  • Remark 2.4: Multiple values of $\lambda$, and informal relation to multiplicative coalescent
  • Theorem 2.5: Critical percolation on heavy-tailed configuration models DhaHofLeeSen20
  • Theorem 2.6: Critical component sizes and surplus edges DhaHofLee21
  • Theorem 2.7: Scaling limit for finite-variance rank-1 inhomogeneous random graphs BhaHofLee09aBhaHofLee09bTuro09
  • Theorem 2.8: Critical regime for single-edge $\mathrm{NR}_n(\boldsymbol{w}, \pi_n)$ BhaDhaHof25
  • Theorem 2.9: $\sqrt{n}$-asymptotics of size and uniqueness tiny giant BhaDhaHof25
  • Proposition 2.10: Phase transition for the limiting model DurKes90
  • Theorem 3.1: Critical percolation threshold for ${\rm PA}$ models HazHofRay23
  • ...and 2 more