On the variational problem for upper tails in sparse random graphs
What is the probability that the number of triangles in Gn,p, the Erdős‐Rényi random graph with edge density p, is at least twice its mean? Writing it as exp[−r(n,p)], already the order of the rate function r(n, p) was a longstanding open problem when p = o(1), finally settled in 2012 by Chatterjee...
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| Published in: | Random structures & algorithms Vol. 50; no. 3; pp. 420 - 436 |
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| Format: | Journal Article |
| Language: | English |
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| ISSN: | 1042-9832, 1098-2418 |
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| Abstract | What is the probability that the number of triangles in Gn,p, the Erdős‐Rényi random graph with edge density p, is at least twice its mean? Writing it as exp[−r(n,p)], already the order of the rate function r(n, p) was a longstanding open problem when p = o(1), finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that r(n,p)≍n2p2log(1/p) for p≳lognn; the exact asymptotics of r(n, p) remained unknown. The following variational problem can be related to this large deviation question at p≳lognn: for δ > 0 fixed, what is the minimum asymptotic p‐relative entropy of a weighted graph on n vertices with triangle density at least (1 + δ)p3? A beautiful large deviation framework of Chatterjee and Varadhan (2011) reduces upper tails for triangles to a limiting version of this problem for fixed p. A very recent breakthrough of Chatterjee and Dembo extended its validity to n−α≪p≪1 for an explicit α > 0, and plausibly it holds in all of the above sparse regime.
In this note we show that the solution to the variational problem is min{12δ2/3 , 13δ} when n−1/2 ≪ p≪1 vs. 12δ2/3 when n−1≪p≪n−1/2 (the transition between these regimes is expressed in the count of triangles minus an edge in the minimizer). From the results of Chatterjee and Dembo, this shows for instance that the probability that Gn,p for n−α≤p≪1 has twice as many triangles as its expectation is exp[−r(n,p)] where r(n,p)∼13n2p2log(1/p). Our results further extend to k‐cliques for any fixed k, as well as give the order of the upper tail rate function for an arbitrary fixed subgraph when p≥n−α. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 420–436, 2017 |
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| AbstractList | What is the probability that the number of triangles in [Formulaomitted], the Erds-Renyi random graph with edge density p, is at least twice its mean? Writing it as [Formulaomitted], already the order of the rate function r(n, p) was a longstanding open problem when p=o(1), finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that [Formulaomitted] for [Formulaomitted]; the exact asymptotics of r(n, p) remained unknown. The following variational problem can be related to this large deviation question at [Formulaomitted]: for delta >0 fixed, what is the minimum asymptotic p-relative entropy of a weighted graph on n vertices with triangle density at least (1+ delta )p super(3)? A beautiful large deviation framework of Chatterjee and Varadhan (2011) reduces upper tails for triangles to a limiting version of this problem for fixed p. A very recent breakthrough of Chatterjee and Dembo extended its validity to [Formulaomitted] for an explicit alpha >0, and plausibly it holds in all of the above sparse regime. In this note we show that the solution to the variational problem is [Formulaomitted] when [Formulaomitted] vs. [Formulaomitted] when [Formulaomitted] (the transition between these regimes is expressed in the count of triangles minus an edge in the minimizer). From the results of Chatterjee and Dembo, this shows for instance that the probability that [Formulaomitted] for [Formulaomitted] has twice as many triangles as its expectation is [Formulaomitted] where [Formulaomitted]. Our results further extend to k-cliques for any fixed k, as well as give the order of the upper tail rate function for an arbitrary fixed subgraph when [Formulaomitted]. Random Struct. Alg., 50, 420-436, 2017 What is the probability that the number of triangles in G n ,p, the Erds-Rényi random graph with edge density p, is at least twice its mean? Writing it as exp [-r (n ,p )], already the order of the rate function r(n, p) was a longstanding open problem when p=o(1), finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that r (n ,p )n 2 p 2 log (1 /p ) for p log n n; the exact asymptotics of r(n, p) remained unknown. The following variational problem can be related to this large deviation question at p log n n: for [delta]>0 fixed, what is the minimum asymptotic p-relative entropy of a weighted graph on n vertices with triangle density at least (1+[delta])p3? A beautiful large deviation framework of Chatterjee and Varadhan (2011) reduces upper tails for triangles to a limiting version of this problem for fixed p. A very recent breakthrough of Chatterjee and Dembo extended its validity to n -[alpha] p 1 for an explicit [alpha]>0, and plausibly it holds in all of the above sparse regime. In this note we show that the solution to the variational problem is min {1 2 [delta]2 /3 ,1 3 [delta]} when n -1 /2 p 1 vs. 1 2 [delta]2 /3 when n -1 p n -1 /2 (the transition between these regimes is expressed in the count of triangles minus an edge in the minimizer). From the results of Chatterjee and Dembo, this shows for instance that the probability that G n ,p for n -[alpha] ≤p 1 has twice as many triangles as its expectation is exp [-r (n ,p )] where r (n ,p )1 3 n 2 p 2 log (1 /p ). Our results further extend to k-cliques for any fixed k, as well as give the order of the upper tail rate function for an arbitrary fixed subgraph when p ≥n -[alpha]. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 420-436, 2017 What is the probability that the number of triangles in , the Erdős‐Rényi random graph with edge density p , is at least twice its mean? Writing it as , already the order of the rate function r ( n, p ) was a longstanding open problem when p = o (1), finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that for ; the exact asymptotics of r ( n, p ) remained unknown. The following variational problem can be related to this large deviation question at : for δ > 0 fixed, what is the minimum asymptotic p ‐relative entropy of a weighted graph on n vertices with triangle density at least (1 + δ ) p 3 ? A beautiful large deviation framework of Chatterjee and Varadhan (2011) reduces upper tails for triangles to a limiting version of this problem for fixed p . A very recent breakthrough of Chatterjee and Dembo extended its validity to for an explicit α > 0, and plausibly it holds in all of the above sparse regime. In this note we show that the solution to the variational problem is when vs. when (the transition between these regimes is expressed in the count of triangles minus an edge in the minimizer). From the results of Chatterjee and Dembo, this shows for instance that the probability that for has twice as many triangles as its expectation is where . Our results further extend to k ‐cliques for any fixed k , as well as give the order of the upper tail rate function for an arbitrary fixed subgraph when . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 420–436, 2017 What is the probability that the number of triangles in Gn,p, the Erdős‐Rényi random graph with edge density p, is at least twice its mean? Writing it as exp[−r(n,p)], already the order of the rate function r(n, p) was a longstanding open problem when p = o(1), finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that r(n,p)≍n2p2log(1/p) for p≳lognn; the exact asymptotics of r(n, p) remained unknown. The following variational problem can be related to this large deviation question at p≳lognn: for δ > 0 fixed, what is the minimum asymptotic p‐relative entropy of a weighted graph on n vertices with triangle density at least (1 + δ)p3? A beautiful large deviation framework of Chatterjee and Varadhan (2011) reduces upper tails for triangles to a limiting version of this problem for fixed p. A very recent breakthrough of Chatterjee and Dembo extended its validity to n−α≪p≪1 for an explicit α > 0, and plausibly it holds in all of the above sparse regime. In this note we show that the solution to the variational problem is min{12δ2/3 , 13δ} when n−1/2 ≪ p≪1 vs. 12δ2/3 when n−1≪p≪n−1/2 (the transition between these regimes is expressed in the count of triangles minus an edge in the minimizer). From the results of Chatterjee and Dembo, this shows for instance that the probability that Gn,p for n−α≤p≪1 has twice as many triangles as its expectation is exp[−r(n,p)] where r(n,p)∼13n2p2log(1/p). Our results further extend to k‐cliques for any fixed k, as well as give the order of the upper tail rate function for an arbitrary fixed subgraph when p≥n−α. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 420–436, 2017 |
| Author | Lubetzky, Eyal Zhao, Yufei |
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| Cites_doi | 10.1002/rsa.20381 10.1002/rsa.10031 10.1017/CBO9780511814068 10.1017/S0963548399004095 10.1002/rsa.10113 10.1016/j.ejc.2011.03.014 10.1002/rsa.20382 10.1016/j.jctb.2006.05.002 10.1016/j.aim.2008.07.008 10.1214/aop/1176989534 10.1007/s00039-007-0599-6 10.1002/rsa.20536 10.1002/rsa.20440 10.1007/s004930050052 10.1007/BF02771528 |
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| References | 2007; 17 2004; 142 2015; 47 2006; 96 2001 2012 2000 2002; 20 1999; 19 2004; 24 2008; 219 2011; 32 1992; 20 2012; 41 1978 2012; 40 2001; 10 e_1_2_7_6_1 e_1_2_7_5_1 e_1_2_7_4_1 e_1_2_7_3_1 Lovász L. (e_1_2_7_17_1) 2012 e_1_2_7_9_1 e_1_2_7_8_1 e_1_2_7_7_1 Janson S. (e_1_2_7_12_1) 2000 e_1_2_7_19_1 e_1_2_7_18_1 e_1_2_7_16_1 e_1_2_7_2_1 e_1_2_7_14_1 e_1_2_7_13_1 e_1_2_7_11_1 Janson S. (e_1_2_7_15_1) 2004; 24 e_1_2_7_22_1 e_1_2_7_10_1 e_1_2_7_21_1 e_1_2_7_20_1 |
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| Snippet | What is the probability that the number of triangles in Gn,p, the Erdős‐Rényi random graph with edge density p, is at least twice its mean? Writing it as... What is the probability that the number of triangles in , the Erdős‐Rényi random graph with edge density p , is at least twice its mean? Writing it as ,... What is the probability that the number of triangles in G n ,p, the Erds-Rényi random graph with edge density p, is at least twice its mean? Writing it as exp... What is the probability that the number of triangles in [Formulaomitted], the Erds-Renyi random graph with edge density p, is at least twice its mean? Writing... |
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| SubjectTerms | Asymptotic properties Constraining Density Deviation Graph theory Graphs large deviations sparse random graphs Triangles upper tails of subgraph counts Vibration |
| Title | On the variational problem for upper tails in sparse random graphs |
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