[Paper Review] On the robustness of random k-cores

Cristiane M. Sato|arXiv (Cornell University)|Mar 9, 2012

Limits and Structures in Graph Theory12 references3 citations

TL;DR

This paper analyzes the robustness of random k-cores by studying the impact of removing a single random edge on the k-core structure. It proves that when the average degree c is near k, the k-core typically collapses entirely after one edge deletion, while for c above a critical threshold c′, the k-core remains largely intact, with high probability retaining almost all vertices. The results are derived using coupling techniques, random walks, and the differential equation method for degree sequences in random multigraphs and simple graphs.

ABSTRACT

The k-core of a graph is its maximal subgraph with minimum degree at least k. In this paper, we address robustness questions about k-cores. Given a k-core, remove one edge uniformly at random and find its new k-core. We are interested in how many vertices are deleted from the original k-core to find the new one. This can be seem as a measure of robustness of the original k-core. We prove that, if the initial k-core is chosen uniformly at random from the k-cores with n vertices and m edges, its robustness depends essentially on its average degree c. We prove that, if c converges to k, then the new k-core is empty with probability 1+o(1). We define a constant c(k)' such that when k+epsilon < c < c(k)'- epsilon, the new k-core is empty with probability bounded away from zero and, if c > c(k)'+ psi with psi = omega(n^{-1/4}), psi(n) > 0 and c is bounded, then the probability that the new k-core has less than n-h(n) vertices goes to zero, for every h(n) = omega(1/psi).

Motivation & Objective

• To understand how robust the k-core structure is in random graphs after the deletion of a single random edge.
• To determine the threshold behavior of k-core resilience based on average degree c.
• To characterize the probability of complete k-core collapse or retention after edge deletion, depending on c.
• To extend results from random multigraphs to simple graphs via coupling techniques.
• To establish a critical threshold c′ that separates fragile from robust k-core behavior.

Proposed method

• Uses a random deletion algorithm to iteratively remove vertices of degree < k to compute the k-core.
• Employs coupling between the deletion process and a random walk to analyze the number of vertices removed.
• Applies the differential equation method to model the evolution of vertex degrees during deletion.
• Uses degree sequence models for multigraphs and proves equivalence to simple graphs via probabilistic coupling.
• Defines c′ as a function of μk,ck and the cumulative distribution function fk(μ), linking it to the phase transition in k-core stability.
• Leverages results from [12] on random graph models to bound probabilities of edge inclusion and graph realizability.

Experimental results

Research questions

• RQ1 What is the probability that removing a single random edge causes the entire k-core to vanish?
• RQ2 How does the average degree c of a random k-core influence its resilience to edge deletion?
• RQ3 Is there a critical threshold c′ such that for c > c′, the k-core remains large after edge deletion, while for c < c′, it collapses with positive probability?
• RQ4 How does the behavior change as c approaches k from above?
• RQ5 Can the robustness of simple k-cores be inferred from multigraph models via coupling?

Key findings

• If c → k, then the new k-core is empty with probability 1 + o(1), indicating extreme fragility.
• For k + ε < c < c′ − ε with ε > 0, the probability that the new k-core is empty is bounded away from zero.
• If c > c′ + ψ(n) with ψ(n) = ω(n−1/4) and ψ(n) > 0, then with high probability the new k-core retains at least n − h(n) vertices for any h(n) = ω(ψ(n)−1).
• When c → k, the deletion process removes Θ(n) vertices with probability approaching 1, and the k-core collapses entirely a.a.s.
• For c ≤ c′ − ε, the deletion algorithm removes either Θ(n) vertices or at most h(n) vertices with high probability, and the probability of removing all vertices is bounded away from zero.
• The coupling between multigraph and simple graph deletion processes ensures that results on multigraphs extend to simple graphs for t(n) = o(1/ξ(n)) steps, with ξ(n) = o(1).