On limit behavior of maximum vertex degree in a conditional configuration graph near critical points

Abstract

Abstract We consider configuration graphs with N vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with parameter τ > 0. There are two critical values of this parameter: τ = 1 and τ = 2. The properties of a graph change significantly when τ = τ(N) passes these points as N → ∞. Let G N,n be the subset of random graphs under the condition that sum of degrees of its vertices is equal to n. The limit theorem for the maximum vertex degree in G N,n as N,n → ∞ and τ → 1 or τ → 2 is proved.