Sublinear Time Estimation of Degree Distribution Moments: The Degeneracy Connection.

revisit the classic problem of estimating the degree distribution moments of an undirected graph. Consider an undirected graph $G=(V,E)$ with $n$ vertices, and define (for $s u003e 0$) $mu_s = frac{1}{n}cdotsum_{v V} d^s_v$. Our aim is to estimate $mu_s$ within a multiplicative error of $(1+epsilon)$ (for a given approximation parameter $epsilonu003e0$) in sublinear time. consider the sparse graph model that allows access to: uniform random vertices, queries for the degree of any vertex, and queries for a neighbor of any vertex. For the case of $s=1$ (the average degree), $widetilde{O}(sqrt{n})$ queries suffice for any constant $epsilon$ (Feige, SICOMP 06 and Goldreich-Ron, RSA 08). Gonen-Ron-Shavitt (SIDMA 11) extended this result to all integral $s u003e 0$, by designing an algorithms that performs $widetilde{O}(n^{1-1/(s+1)})$ queries. We design a new, significantly simpler algorithm for this problem. In the worst-case, it exactly matches the bounds of Gonen-Ron-Shavitt, and has a much simpler proof. More importantly, the running time of this algorithm is connected to the degeneracy of $G$. This is (essentially) the maximum density of an induced subgraph. For the family of graphs with degeneracy at most $alpha$, it has a query complexity of $widetilde{O}left(frac{n^{1-1/s}}{mu^{1/s}_s} Big(alpha^{1/s} + min{alpha,mu^{1/s}_s}Big)right) = widetilde{O}(n^{1-1/s}alpha/mu^{1/s}_s)$. Thus, for the class of bounded degeneracy graphs (which includes all minor closed families and preferential attachment graphs), we can estimate the average degree in $widetilde{O}(1)$ queries, and can estimate the variance of the degree distribution in $widetilde{O}(sqrt{n})$ queries. This is a major improvement over the previous worst-case bounds. Our key insight is in designing an estimator for $mu_s$ that has low variance when $G$ does not have large dense subgraphs.