[Paper Review] Critical Percolation on Random Networks with Prescribed Degrees

Souvik Dhara|arXiv (Cornell University)|Sep 10, 2018

Complex Network Analysis Techniques115 references3 citations

TL;DR

This paper establishes the scaling limit of critical percolation clusters in random networks with prescribed degrees, showing convergence in the Gromov-Hausdorff-Prokhorov (GHP) topology by proving a global lower mass-bound property. Using a coupling with an augmented multiplicative coalescent and leveraging near-Feller properties, the authors demonstrate that the limiting metric spaces are compact almost surely and that global functionals like diameter converge.

ABSTRACT

Random graphs have played an instrumental role in modelling real-world networks arising from the internet topology, social networks, or even protein-interaction networks within cells. Percolation, on the other hand, has been the fundamental model for understanding robustness and spread of epidemics on these networks. From a mathematical perspective, percolation is one of the simplest models that exhibits phase transition, and fascinating features are observed around the critical point. In this thesis, we prove limit theorems about structural properties of the connected components obtained from percolation on random graphs at criticality. The results are obtained for random graphs with general degree sequence, and we identify different universality classes for the critical behavior based on moment assumptions on the degree distribution.

Motivation & Objective

• To establish the scaling limit of critical percolation clusters in random networks with given degree sequences.
• To bridge Gromov-weak convergence and GHP convergence by proving a global lower mass-bound property.
• To show that the limiting metric spaces of critical percolation clusters are compact almost surely.
• To extend convergence results to global functionals such as diameter by proving tightness of rescaled component sizes.
• To generalize prior assumptions on degree sequences, allowing slowly varying functions and broader power-law exponents.

Proposed method

• Introduces Algorithm 3.4, a dynamic edge-creation process that retains open half-edges, enabling repeated pairing and generating a graph with extra 'bad edges' compared to the original process.
• Establishes a natural coupling between the original and augmented graphs such that the original is almost surely a subgraph of the augmented one.
• Defines a scaled process $ \bar{\mathbf{Z}}_{n}^{o,{\scriptscriptstyle\mathrm{scl}}} $ that evolves as a standard augmented multiplicative coalescent.
• Applies the near-Feller property of the augmented multiplicative coalescent to ensure joint convergence of component sizes and surplus edges.
• Uses Theorem 26.2 to prove weak convergence of the augmented coalescent process under initial condition convergence.
• Applies the global lower mass-bound property to ensure compactness of the limiting metric space and convergence of global functionals.

Experimental results

Research questions

• RQ1 Does the scaling limit of critical percolation clusters in random networks with prescribed degrees converge in the GHP topology?
• RQ2 Can the global lower mass-bound property be established for critical percolation clusters under general degree sequences?
• RQ3 How does the presence of 'bad edges' in the augmented graph affect the convergence of component sizes and surplus edges?
• RQ4 What conditions on the degree sequence ensure the tightness of rescaled component sizes and convergence of the diameter?
• RQ5 Can the limiting metric space of critical percolation clusters be shown to be compact almost surely?

Key findings

• The sequence $ (\mathscr{C}_{\scriptscriptstyle(i)}, n^{-\eta}, \boldsymbol{w}) $ satisfies the global lower mass-bound property for each fixed $ i \geq 1 $, ensuring tightness of rescaled component sizes.
• Under Assumption 5.1 and condition (5.6), the largest components of $ \mathrm{UM}_n(\boldsymbol{d}) $ also satisfy the global lower mass-bound.
• Critical percolation clusters of $ \mathrm{CM}_n(\boldsymbol{d}) $ and $ \mathrm{UM}_n(\boldsymbol{d}) $ satisfy the global lower mass-bound under condition (5.10), which defines the critical window.
• Gromov-weak convergence combined with the global lower mass-bound implies GHP convergence, ensuring the limiting metric space is compact almost surely.
• The diameter of the rescaled critical clusters converges almost surely, confirming convergence of global functionals.
• The assumptions on the degree sequence are generalized beyond power laws, allowing $ \theta_i \in [L_1(i)i^{-a_1}, L_2(i)i^{-a_2}] $ with $ a_1, a_2 \in (1/3, 1/2) $, which is less restrictive than prior work.