The stripping process can be slow: Part I

Gao, Pu; Molloy, Michael

doi:10.1002/rsa.20760

Cited by 7 publications

(35 citation statements)

References 42 publications

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“…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%

“…So we want the rate at which new light vertices arise to be less than one. It is well‐known that when c is close to

c_{r}^{*}

Section: Proof Of Theorem 6(c)mentioning

confidence: 99%

“…It is well‐known that when c is close to

c_{r}^{*}

, that rate is close to one; much of the work in was to bound it away from one. The following bound comes from the analysis in but is not stated explicitly there:Lemma There is a constant K > 0 such that a.a.s. at every step of SLOW‐STRIP, the degrees of the remaining vertices are such that the expected number of heavy vertices that become light in that step is at most

1 - K n^{- δ / 2}

.Proof In each such step, we remove the only remaining copy of the light vertex at the front of

scriptQ

and r – 1 uniformly chosen copies.…”

Section: Proof Of Theorem 6(c)mentioning

confidence: 99%

“…at every step of SLOW‐STRIP, the degrees of the remaining vertices are such that the expected number of heavy vertices that become light in that step is at most

1 - K n^{- δ / 2}

.Proof In each such step, we remove the only remaining copy of the light vertex at the front of

scriptQ

and r – 1 uniformly chosen copies. The proof of Lemma 16 in establishes that each time we remove a uniformly chosen copy, the probability that a heavy vertex becomes light (ie, that we remove a copy of a degree two vertex) is at most

1 / (r - 1) - Θ (n^{- δ / 2})

. (See Definition 30, line (34) and the discussion preceding line (29) of and let k = 2.)…”

Section: Proof Of Theorem 6(c)mentioning

confidence: 99%

“…The proof of Lemma 16 in establishes that each time we remove a uniformly chosen copy, the probability that a heavy vertex becomes light (ie, that we remove a copy of a degree two vertex) is at most

1 / (r - 1) - Θ (n^{- δ / 2})

. (See Definition 30, line (34) and the discussion preceding line (29) of and let k = 2.) This yields the lemma.…”

Section: Proof Of Theorem 6(c)mentioning

confidence: 99%

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Inside the clustering window for random linear equations

Gao

Molloy

2017

Random Struct Algorithms

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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.

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“…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%

“…So we want the rate at which new light vertices arise to be less than one. It is well‐known that when c is close to

c_{r}^{*}

Section: Proof Of Theorem 6(c)mentioning

confidence: 99%

“…It is well‐known that when c is close to

c_{r}^{*}

1 - K n^{- δ / 2}

.Proof In each such step, we remove the only remaining copy of the light vertex at the front of

scriptQ

and r – 1 uniformly chosen copies.…”

Section: Proof Of Theorem 6(c)mentioning

confidence: 99%

“…at every step of SLOW‐STRIP, the degrees of the remaining vertices are such that the expected number of heavy vertices that become light in that step is at most

1 - K n^{- δ / 2}

.Proof In each such step, we remove the only remaining copy of the light vertex at the front of

scriptQ

1 / (r - 1) - Θ (n^{- δ / 2})

. (See Definition 30, line (34) and the discussion preceding line (29) of and let k = 2.)…”

Section: Proof Of Theorem 6(c)mentioning

confidence: 99%

1 / (r - 1) - Θ (n^{- δ / 2})

. (See Definition 30, line (34) and the discussion preceding line (29) of and let k = 2.) This yields the lemma.…”

Section: Proof Of Theorem 6(c)mentioning

confidence: 99%

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Inside the clustering window for random linear equations

Gao

Molloy

2017

Random Struct Algorithms

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The Satisfiability Threshold For Random Linear Equations

et al. 2020

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Let A be a random m × n matrix over the finite field F q with precisely k non-zero entries per row and let y ∈ F m q be a random vector chosen independently of A. We identify the threshold m/n up to which the linear system Ax = y has a solution with high probability and analyse the geometry of the set of solutions. In the special case q = 2, known as the random k-XORSAT problem, the threshold was determined by [Dubois and Mandler 2002 for k = 3, Pittel and Sorkin 2016 for k > 3], and the proof technique was subsequently extended to the cases q = 3,4 [Falke and Goerdt 2012]. But the argument depends on technically demanding second moment calculations that do not generalise to q > 3. Here we approach the problem from the viewpoint of a decoding task, which leads to a transparent combinatorial proof.

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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

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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

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The stripping process can be slow: Part I

Cited by 7 publications

References 42 publications

Inside the clustering window for random linear equations

Inside the clustering window for random linear equations

The Satisfiability Threshold For Random Linear Equations

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

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