On Combinatorial Hoeffding Decomposition and Asymptotic Normality of Subgraph Count Statistics

Abstract

Given kand nconsidera graph with k vertices and n “blue” edges. We assume that the set of “blue”edges ${{a}_{n}} = \sum\limits_{{m = 0}}^{{n - 1}} {\underbrace{{{{{\left( q \right)}}_{{n - 1}}}\sum\limits_{{h = 0}}^{\infty } {p\left( {h,m} \right){{q}^{h}}.} }}_{{\mathop{ = }\limits^{{def}} {{H}_{{n,m}}}\left( q \right)}}}$ Is uniformly distributed among n-subsets of $N = \left( {\tfrac{k}{2}} \right)$ Pairs of vertices.Given a graph g, the number N_gof blue copies of g os a U-statistic based on random sample $\mathcal{N}_G as k,n \to \infty .$ normality.