The Stripping Process Can be Slow: Part II

Gao, Pu

doi:10.1137/15m1054948

Cited by 5 publications

(6 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…But where the arguments in made use of facts about the 2‐core of a hypergraph with density a constant

c > c_{r}^{*}

, this paper requires analogous results for the much more difficult range

c = c_{r}^{*} + o (1)

. Those results were derived in ; the results of this paper were the motivation for those two papers.…”

Section: Resultsmentioning

confidence: 88%

“…21, Theorem 43] (for c = c r,k + n −δ ) and[22, Theorem 6] (for c = c r,k − n −δ ), and part (b) is from[21, Theorem 5] (for c = c r,k + n −δ ) and[22, Theorem 4] (for c = c r,k − n −δ ).Theorem 16 Let r, k ≥ 2, (r, k) = (2, 2) be fixed. There is a constant κ > 0 such that: for any constant 0 < δ < 1 2 , if c = c r,k ± n −δ ,then (a) a.a.s.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Inside the clustering window for random linear equations

Gao

Molloy

2017

Random Struct Algorithms

Self Cite

View full text Add to dashboard Cite

We study a random system of cn linear equations over n variables in GF(2), where each equation contains exactly r variables; this is equivalent to r‐XORSAT. Previous work has established a clustering threshold, cr∗ for this model: if c=cr∗−ϵ for any constant ϵ>0 then with high probability all solutions form a well‐connected cluster; whereas if c=cr∗+ϵ, then with high probability the solutions partition into well‐connected, well‐separated clusters (with probability tending to 1 as n→∞). This is part of a general clustering phenomenon which is hypothesized to arise in most of the commonly studied models of random constraint satisfaction problems, via sophisticated but mostly nonrigorous techniques from statistical physics. We extend that study to the range c=cr∗+o(1), and prove that the connectivity parameters of the r‐XORSAT clusters undergo a smooth transition around the clustering threshold.

show abstract

“…But where the arguments in made use of facts about the 2‐core of a hypergraph with density a constant

c > c_{r}^{*}

, this paper requires analogous results for the much more difficult range

c = c_{r}^{*} + o (1)

. Those results were derived in ; the results of this paper were the motivation for those two papers.…”

Section: Resultsmentioning

confidence: 88%

mentioning

confidence: 99%

Inside the clustering window for random linear equations

Gao

Molloy

2017

Random Struct Algorithms

Self Cite

View full text Add to dashboard Cite

show abstract

“…Cain and Wormald [8] determined the asymptotic distribution of vertex degrees within the k-core. Further research has focussed for example on the robustness of the core against edge deletion [35] and how quickly the peeling process arrives at the core [1,18,19,23]. There are many more results in the literature for cores in random graphs, see e.g.…”

Section: Motivationmentioning

confidence: 99%

Loose Cores and Cycles in Random Hypergraphs

Cooley

Kang²,

Zalla³

2022

Electron. J. Combin.

View full text Add to dashboard Cite

Inspired by the study of loose cycles in hypergraphs, we define the loose core in hypergraphs as a structurewhich mirrors the close relationship between cycles and $2$-cores in graphs. We prove that in the $r$-uniform binomial random hypergraph $H^r(n,p)$, the order of the loose core undergoes a phase transition at a certain critical threshold and determine this order, as well as the number of edges, asymptotically in the subcritical and supercritical regimes. Our main tool is an algorithm called CoreConstruct, which enables us to analyse a peeling process for the loose core. By analysing this algorithm we determine the asymptotic degree distribution of vertices in the loose core and in particular how many vertices and edges the loose core contains. As a corollary we obtain an improved upper bound on the length of the longest loose cycle in $H^r(n,p)$.

show abstract

“…Additionally, establishing a bivariate central limit theorem, Janson and Luczak [17] studied the joint limiting distribution of the order and size of the k-core. Further aspects of the problems that have been studied include the 'depth' of the peeling process as well as the width of the critical window [7,13,14].…”

Section: Introductionmentioning

confidence: 99%

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

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

The Stripping Process Can be Slow: Part II

Cited by 5 publications

References 26 publications

Inside the clustering window for random linear equations

Inside the clustering window for random linear equations

Loose Cores and Cycles in Random Hypergraphs

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

Contact Info

Product

Resources

About