TY - JOUR

T1 - Upper tails and independence polynomials in random graphs

AU - Bhattacharya, Bhaswar B.

AU - Ganguly, Shirshendu

AU - Lubetzky, Eyal

AU - Zhao, Yufei

N1 - Funding Information:
This work was initiated when the first author was an intern at the Theory Group of Microsoft Research, Redmond. E.L. was supported in part by NSF grant DMS-1513403 and Y.Z. was supported by a Microsoft Research Ph.D. Fellowship. We thank the anonymous referee for helpful comments that greatly improved the exposition of the paper.
Publisher Copyright:
© 2017 Elsevier Inc.

PY - 2017/10/15

Y1 - 2017/10/15

N2 - The upper tail problem in the Erdős–Rényi random graph G∼Gn,p asks to estimate the probability that the number of copies of a graph H in G exceeds its expectation by a factor 1+δ. Chatterjee and Dembo showed that in the sparse regime of p→0 as n→∞ with p≥n−α for an explicit α=αH>0, this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where H is a clique. Here we extend the latter work to any fixed graph H and determine a function cH(δ) such that, for p as above and any fixed δ>0, the upper tail probability is exp[−(cH(δ)+o(1))n2pΔlog(1/p)], where Δ is the maximum degree of H. As it turns out, the leading order constant in the large deviation rate function, cH(δ), is governed by the independence polynomial of H, defined as PH(x)=∑iH(k)xk where iH(k) is the number of independent sets of size k in H. For instance, if H is a regular graph on m vertices, then cH(δ) is the minimum between [formula omitted] and the unique positive solution of PH(x)=1+δ.

AB - The upper tail problem in the Erdős–Rényi random graph G∼Gn,p asks to estimate the probability that the number of copies of a graph H in G exceeds its expectation by a factor 1+δ. Chatterjee and Dembo showed that in the sparse regime of p→0 as n→∞ with p≥n−α for an explicit α=αH>0, this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where H is a clique. Here we extend the latter work to any fixed graph H and determine a function cH(δ) such that, for p as above and any fixed δ>0, the upper tail probability is exp[−(cH(δ)+o(1))n2pΔlog(1/p)], where Δ is the maximum degree of H. As it turns out, the leading order constant in the large deviation rate function, cH(δ), is governed by the independence polynomial of H, defined as PH(x)=∑iH(k)xk where iH(k) is the number of independent sets of size k in H. For instance, if H is a regular graph on m vertices, then cH(δ) is the minimum between [formula omitted] and the unique positive solution of PH(x)=1+δ.

KW - Large deviations

KW - Sparse random graphs

KW - Subgraph counts

KW - Upper tails

KW - Variational problems

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U2 - 10.1016/j.aim.2017.08.003

DO - 10.1016/j.aim.2017.08.003

M3 - Article

AN - SCOPUS:85028024910

SN - 0001-8708

VL - 319

SP - 313

EP - 347

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -