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Asymptotic distribution of the numbers of vertices and arcs of the giant strong component in sparse random digraphs
Author(s) -
Pittel Boris,
Poole Daniel
Publication year - 2016
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20622
Subject(s) - giant component , combinatorics , mathematics , digraph , component (thermodynamics) , degree (music) , gaussian , realization (probability) , order (exchange) , exponential function , multivariate random variable , discrete mathematics , random graph , random variable , statistics , mathematical analysis , physics , finance , quantum mechanics , acoustics , economics , thermodynamics , graph
Two models of a random digraph on n vertices, D ( n , Prob ( arc ) = p ) and D ( n , number of arcs = m ) are studied. In 1990, Karp for D ( n, p ) and independently T. Łuczak for D ( n,m = cn ) proved that for c > 1, with probability tending to 1, there is an unique strong component of size of order n . Karp showed, in fact, that the giant component has likely size asymptotic to nθ 2 , where θ = θ ( c ) is the unique positive root of 1 − θ = e − c θ. In this paper we prove that, for both random digraphs, the joint distribution of the number of vertices and number of arcs in the giant strong component is asymptotically Gaussian with the same mean vector | V 1 | , n ( θ 2 , ( c − 1 ) θ 2 ) and two distinct 2 × 2 covariance matrices, n B ( c ) and n [ B ( c ) + c( μ ' ( c ) ) T( μ ' ( c ) ) ) ] . To this end, we introduce and analyze a randomized deletion process which terminates at the directed (1, 1)‐core, the maximal digraph with minimum in‐degree and out‐degree at least 1. This (1, 1)‐core contains all non‐trivial strong components. However, we show that the likely numbers of peripheral vertices and arcs in the (1, 1)‐core, those outside the largest strong component, are of polylog order, thus dwarfed by anticipated fluctuations, on the scale of n 1/2 , of the giant component parameters. By approximating the likely realization of the deletion algorithm with a deterministic trajectory, we obtain our main result via exponential supermartingales and Fourier‐based techniques. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 3–64, 2016