Core forging and local limit theorems for the k-core of random graphs

Abstract

We establish a multivariate local limit theorem for the order and size as well as several other parameters of the k-core of the Erdős-Rényi random graph. The proof is based on a novel approach to the k-core problem that replaces the meticulous analysis of the 'peeling process' by a generative model of graphs with a core of a given order and size. The generative model, which is inspired by the Warning Propagation message passing algorithm, facilitates the direct study of properties of the core and its connections with the mantle and should therefore be of interest in its own right.The formula (1.5) determines the asymptotic probability that the order and size X , Y of the k-core attain specific values within O( n) of their expectations. Hence, Theorem 1.1 provides a bivariate local limit theorem for the order and size of the k-core. This result is significantly stronger than a mere central limit theorem stating that X , Y converge jointly to a bivariate Gaussian because (1.5) actually yields the asymptotic point probabilities. Still it is worthwhile pointing out that Theorem 1.1 immediately implies a central limit theorem. Corollary 1.2. Suppose that k ≥ 3 and d > d k , let Q be the matrix from (1.4) and let X , Y be the order and size of the k-core of G. Then n −1/2 ((X − np(1− q)), 2(Y − mp 2 )/d) converges in distribution to a bivariate Gaussian with mean 0 and covariance matrix Q −1 .A statement similar to Corollary 1.2 was previously established by Janson and Luczak [17] via a careful analysis of the peeling process. However, they did not obtain an explicit formula for the covariance matrix. Indeed, although the formula for Q is a bit on the lengthy side, the only non-algebraic quantity is p = p(d, k), the solution to the fixed point equation. By contrast, the formula of Janson and Luczak implicitly characterises the covariance matrix in terms of another stochastic process, and they do not provide a local limit theorem.The number d k from (1.2) does, of course, coincide with the k-core threshold first derived in [26]. The formula given in that paper looks a bit different but we pointed out the equivalence in [4]. In fact, it is very easy to show 2

The size of the giant high‐order component in random hypergraphs

“…The ‐core is the (unique) maximal subgraph of minimum degree at least . The threshold for the existence of a giant ‐core for was determined by Pittel, Spencer, and Wormald , while the interactions between core and non‐core vertices were described in .…”

Warning Propagation: stability and subcriticality

“…For an upper bound, we will slightly simplify the process of changes made by WP to obtain WP * G t 0 = WP * G 0 from G t 0 . 5 We will reveal the information in G t 0 a little at a time as needed.…”

“…One classical tool is the differential equations method [27], which was used in the seminal k-core paper of Pittel, Spencer and Wormald [23] as well as in the analysis of Unit Clause Propagation [1]. Other approaches include branching processes [25], enumerative methods [5], or birth-death processes [16,17].…”

Phase transitions in graphs on orientable surfaces

“…More generally, given k ≥ 2, the k-core of a graph G is the largest subgraph of G of minimum degree at least k. Like the core, the k-core can be constructed by a peeling process that recursively removes vertices of degree less than k. The order and size of the k-core of G(n, m) has been determined in a seminal paper by Pittel, Spencer, and Wormald [58]. Following Pittel, Spencer, and Wormald, the k-core has been extensively studied [23,24,41,46,49,60]. The most striking results in this area are the astonishing theorem by Luczak [49] that the k-core for k ≥ 3 jumps to linear order at the very moment it becomes non-empty, the central limit theorem by Janson and Luczak [41], and the local limit theorem by Coja-Oghlan, Cooley, Kang, and Skubch [23] that described-in addition to the order and size-several other parameters of the k-core of G(n, m).…”

“…In [24], the same authors used a 5-type branching process in order to determine the local structure of the k-core. In terms of global structure, [23] provides a randomised algorithm that constructs a random graph with given order and size of the k-core.…”