Sublinear-Time Approximation for Graph Frequency Vectors in Hyperfinite Graphs

In this work, we address the problem of approximating the $k$-disc distribution (``frequency vector") of a bounded-degree graph in sublinear-time under the assumption of hyperfiniteness. We revisit the partition-oracle framework of Hassidim, Kelner, Nguyen, and Onak \cite{hassidim2009local}, and provide a concise, self-contained analysis that explicitly separates the two sources of error: (i) the cut error, controlled by hyperfiniteness parameter $\phi$, which incurs at most $\varepsilon/2$ in $\ell_1$-distance by removing at most $\phi |V|$ edges; and (ii) the sampling error, controlled by the accuracy parameter $\varepsilon$, bounded by $\varepsilon/2$ via $N=\Theta(\varepsilon^{-2})$ random vertex queries and a Chernoff and union bound argument. Combining these yields an overall $\ell_1$-error of $\varepsilon$ with high probability. Algorithmically, we show that by sampling $N=\lceil C\varepsilon^{-2} \rceil$ vertices and querying the local partition oracle, one can in time $poly(d,k,\varepsilon^{-1})$ construct a summary graph $H$ of size $|H|=poly(d^k,1/\varepsilon)$ whose $k$-disc frequency vector approximates that of the original graph within $\varepsilon$ in $\ell_1$-distance. Our approach clarifies the dependence of both runtime and summary-size on the parameter $d$,$k$, and $\varepsilon$.