How does the core sit inside the mantle?

The phase transition in the size of the giant component in random graphs is one of the most well‐studied phenomena in random graph theory. For hypergraphs, there are many possible generalizations of the notion of a connected component. We consider the following: two j‐sets (sets of j vertices) are j‐connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. A hypergraph is j‐connected if all j‐sets are pairwise j‐connected. In this paper, we determine the asymptotic size of the unique giant j‐connected component in random k‐uniform hypergraphs for any k≥3 and 1≤j≤k−1.

show abstract

“…and note that for 0 ≤ i ≤ τ we have p ≤ p i ≤ p τ = (1 + o(1))p by (9). This guarantees in particular that…”

Section: Lemma 20mentioning

confidence: 76%

“…The

ℓ

‐core is the (unique) maximal subgraph of minimum degree at least

ℓ

. The threshold for the existence of a giant

ℓ

‐core for

ℓ \geq 3

was determined by Pittel, Spencer, and Wormald , while the interactions between core and non‐core vertices were described in .…”

Section: Discussionmentioning

confidence: 99%

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

Cooley

Kang

Koch

2018

Random Struct Algorithms

Self Cite

View full text Add to dashboard Cite

show abstract

“…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).…”

mentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

Phase transitions in graphs on orientable surfaces

Kang

Moßhammer

Sprüssel

2020

Random Struct Algorithms

Self Cite

View full text Add to dashboard Cite

Let S g be the orientable surface of genus g and denote by  g (n, m) the class of all graphs on vertex set [n] = {1, … , n} with m edges embeddable on S g . We prove that the component structure of a graph chosen uniformly at random from  g (n, m) features two phase transitions. The first phase transition mirrors the classical phase transition in the Erdős-Rényi random graph G(n, m) chosen uniformly at random from all graphs with vertex set [n] and m edges. It takes place at m = n 2 + O(n 2∕3 ), when the giant component emerges. The second phase transition occurs at m = n + O(n 3∕5 ), when the giant component covers almost all vertices of the graph. This kind of phenomenon is strikingly different from G(n, m) and has only been observed for graphs on surfaces.

show abstract

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

Coja-Oghlan

Cooley

Kang

et al. 2019

Journal of Combinatorial Theory, Series B

Self Cite

View full text Add to dashboard Cite

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

show abstract

How does the core sit inside the mantle?

Cited by 4 publications

References 27 publications

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

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

Phase transitions in graphs on orientable surfaces

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

Contact Info

Product

Resources

About