Subgraph Counts in Random Graphs Using Incomplete U-statistics methods

Nowicki, Krzysztof; Wierman, John C.

doi:10.1016/s0167-5060(08)70797-0

Cited by 20 publications

(26 citation statements)

References 7 publications

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“…The number of subgraphs isomorphic to G and containing a fixed edge, is given by n − 2 l − 2 2k a (l − 2)! , see [NW88], p.307. Therefore we can estimate…”

Section: Now We Can State Our Resultmentioning

confidence: 99%

“…because lim n→∞ VZ = 1, see [NW88]. We need Lemma 4.1 to bound d(n): d(n) Now we consider an upper bound for the upper tail P (W − EW ≥ ε EW ) = P Z ≥ ε EW cn,p .…”

Section: Now We Can State Our Resultmentioning

confidence: 99%

“…The subgraph count W is a sum of dependent random variables, for which the exact calculation of the variance is tedious. In [NW88], the authors approximated W by a projection of W , which is a sum of independent random variables. For this sum the variance calculation is elementary, proving the denominator (1.1) in the definition of Z.…”

mentioning

confidence: 99%

“…Poisson convergent if and only ifnp d(G) n→∞ −→ 0 or n β(G) p n→∞ −→ 0 .Here d(G) denotes the density of the graph G andβ(G) := max v G − v H e G − e H : H ⊂ G . <···<κ k ≤( n 2 ) 1 {(eκ 1 ,...,eκ k )∼G} k i=1 X κ i − p k has asymptotic standard normal distribution, if np k−1 n→∞ −→ ∞ and n 2 (1 − p) n→∞ −→ ∞, seeNowicki, Wierman,[NW88]. For G be an arbitrary graph with at least one edge, Rucińskiproved in [Ruc88] that W −E(W ) √ V(W ) converges in distribution to a standard normal distribution if and only if np m(G) n→∞ −→ ∞ and n 2 (1 − p)…”

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confidence: 99%

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Moderate Deviations in a Random Graph and for the Spectrum of Bernoulli Random Matrices

Döring

Eichelsbacher

2009

Electron. J. Probab.

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We prove a moderate deviation principle for subgraph count statistics of Erdős-Rényi random graphs. This is equivalent in showing a moderate deviation principle for the trace of a power of a Bernoulli random matrix. It is done via an estimation of the log-Laplace transform and the Gärtner-Ellis theorem. We obtain upper bounds on the upper tail probabilities of the number of occurrences of small subgraphs. The method of proof is used to show supplemental moderate deviation principles for a class of symmetric statistics, including non-degenerate U-statistics with independent or Markovian entries.Here (e κ 1 , . . . , e κ k ) denotes the graph with edges e κ 1 , . . . , e κ k present and A ∼ G denotes the fact that the subgraph A of the complete graph is isomorphic to G. We assume G to be a

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“…The number of subgraphs isomorphic to G and containing a fixed edge, is given by n − 2 l − 2 2k a (l − 2)! , see [NW88], p.307. Therefore we can estimate…”

Section: Now We Can State Our Resultmentioning

confidence: 99%

“…because lim n→∞ VZ = 1, see [NW88]. We need Lemma 4.1 to bound d(n): d(n) Now we consider an upper bound for the upper tail P (W − EW ≥ ε EW ) = P Z ≥ ε EW cn,p .…”

Section: Now We Can State Our Resultmentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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Moderate Deviations in a Random Graph and for the Spectrum of Bernoulli Random Matrices

Döring

Eichelsbacher

2009

Electron. J. Probab.

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“…Using Stein's method [16], Barbour, Karoński and Ruciński [2] obtained strong quantitative bounds on the error in the central limit theorem for subgraph counts. A technique from the asymptotic theory of statistics, known as U-statistics, was applied by Nowicki and Wierman [13] to obtain a central limit theorem for subgraph counts, although, in a slightly less general setting than the theorem of Ruciński. Janson [8] used a similar method with several applications, including central limit theorems for the joint distribution of various graph statistics.…”

Section: Central Limit Theoremsmentioning

confidence: 99%

A local central limit theorem for triangles in a random graph

Gilmer

Kopparty

2016

Random Struct. Alg.

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A limit theorem for small cliques in inhomogeneous random graphs

Hladký

Pelekis

Šileikis

2021

Journal of Graph Theory

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The theory of graphons comes with a natural sampling procedure, which results in an inhomogeneous variant of the Erdős-Rényi random graph, called W -random graphs. We prove, via the method of moments, a limit theorem for the number of r-cliques in such random graphs. We show that, whereas in the case of dense Erdős-Rényi random graphs the fluctuations are normal of order n r−1 , the fluctuations in the setting of W -random graphs may be of order n 0, r−1 , or n r−0.5 .Furthermore, when the fluctuations are of order n r−0.5 they are normal, while when the fluctuations are of order n r−1 they exhibit either normal or a particular type of chi-square behavior whose parameters relate to spectral properties of W . These results can also be deduced from a general setting, based on the projection method. In addition to providing alternative proofs, our approach makes direct links to the theory of graphons.

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Subgraph Counts in Random Graphs Using Incomplete U-statistics methods

Cited by 20 publications

References 7 publications

Moderate Deviations in a Random Graph and for the Spectrum of Bernoulli Random Matrices

Moderate Deviations in a Random Graph and for the Spectrum of Bernoulli Random Matrices

A local central limit theorem for triangles in a random graph

A limit theorem for small cliques in inhomogeneous random graphs

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