Concentration Inequalities for Incomplete U-statistics over Arbitrary Sampling Graphs

Let $X_1,X_2,\dots ,X_n$ be independent random vectors. For a directed graph $G=(V,E)$ with vertex set $V=\{1,2,\dots ,n\}$ and a collection of bivariate kernels $\{h_e:e\in E\}$, we consider $$U=\sum _{e=(i,j)\in E}{h}_e(X_i,X_j)\mathrm{.}$$ This framework generalizes incomplete U-statistics by allowing the random vectors to be non-identically distributed, the kernels to be asymmetric and edge-dependent, and the sampling structure to be specified by an arbitrary graph. We derive several concentration inequalities for $U-\mathbb{E}U$. The main proof strategy exploits edge-coloring results from graph theory and relates the tail behavior of $U$ to the chromatic index of $G$. This approach is elementary, transparent, and readily adaptable to broader settings, including U-statistics of order $m>2$ and statistics involving doubly indexed random vectors.