Asymptotic normality of the k-core in random graphs

Abstract

We study the k-core of a random (multi)graph on n vertices with a given degree sequence. In our previous paper [Random Structures Algorithms 30 (2007) 50-62] we used properties of empirical distributions of independent random variables to give a simple proof of the fact that the size of the giant k-core obeys a law of large numbers as n → ∞. Here we develop the method further and show that the fluctuations around the deterministic limit converge to a Gaussian law above and near the threshold, and to a non-normal law at the threshold. Further, we determine precisely the location of the phase transition window for the emergence of a giant k-core. Hence, we deduce corresponding results for the k-core in G(n, p) and G(n, m).

On percolation in random graphs with given vertex degrees

“…[11]. In any case, it follows from Theorem 5.6 that the size of the transition window is O(n −1/2 ) for G(n, d) too, and not smaller.…”

“…The k-core of a graph G, denoted by Core k (G), is the largest induced subgraph of G with minimum vertex degree at least k. (Note that the k-core may be empty.) The question whether a non-empty k-core exists in a random graph has attracted a lot of attention for various models of random graphs since the pioneering papers by Bollobás [2], Luczak [14] and Pittel, Spencer and Wormald [17] for G(n, p) and G(n, m); in particular, the case of G(n, d) and G * (n, d) with given degree sequences have been studied by several authors, see Janson and Luczak [10,11] and the references given there.…”

“…The latter result can easily be transformed into the following analogue of [11,Theorem 1.4]; the asymptotic variance σ 2 is given by explicit but rather complicated formulas in [11]. Theorem 5.6.…”

Susceptibility in subcritical random graphs

“…We can interpret the asymptotic covariances and the polynomials P k,l in Section 6 by introducing the size-biased Borel distribution B λ defined by 10) and thus, by (6.1),…”

Cores of random graphs are born Hamiltonian

“…That is, in a typical random graph process, the k-core remains empty until the addition of a single edge creates a chain reaction culminating in a linear-sized core. This fascinating phenomenon triggered a long line of works (see, e.g., [9][10][11][12]16,17,[26][27][28]30,31]). Most notably, the tour-de-force by Pittel, Spencer and Wormald [28] determined the asymptotic threshold for its emergence to be p = c k /n for c k = inf λ>0 λ/P(Poisson(λ) ≥ k − 1) = k + √ k log k + O( √ k), as well as characterized various properties of the core upon its creation, e.g., its size, edge density, etc.…”