Sublinear Time Estimation of Degree Distribution Moments: The Arboricity Connection

Abstract

We revisit the classic problem of estimating the moments of the degree distribution of an undirected simple graph. Consider an undirected simple graph $G=(V,E)$ with $n$ (nonisolated) vertices, and define (for $s > 0$) $M_s= \sum_{v \in V} d^s_v$. Our aim is to estimate $M_s$ within a multiplicative error of $(1+\varepsilon)$ (for a given approximation parameter $\varepsilon>0$) in sublinear time. We 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), $O^*(\sqrt{n})$ queries suffice for any constant $\varepsilon$ [U. Feige, SIAM J. Comput., 35 (2006), pp. 964--984], [O. Goldreich and D. Ron, Random Structures Algorithms, 32 (2008), pp. 473--493]. (We use the $O^*$ notation to suppress dependencies in $\log n$ and $1/\varepsilon$.) Gonen, Ron, and Shavitt [SIAM J. Discrete Math., 25 (2011), pp. 1365--1411] extended this result to all integral $s > 0$ by designing an algorithm that performs $O^*(n^{1-1/(s+1)})$ queries. (Strictly speaking, their algorithm approximates the number of star-subgraphs of a given size, but a slight modification gives an algorithm for moments.) We design a new, significantly simpler algorithm for this problem. In the worst case, it exactly matches the bounds of Gonen, Ron, and Shavitt and has a much simpler proof. More importantly, the running time of this algorithm is connected to the arboricity of $G$. This is (essentially) the maximum density of an induced subgraph. For the family of graphs with arboricity at most $\alpha$, it has a query complexity of $O^*\big(\frac{n \cdot \alpha^{1/s}}{M_s^{1/s}} + \min\big\{\frac{m}{M_s^{1/s}},\frac{m \cdot n^{s-1}}{M_s}\big\}\big)$ which is always upper bounded by $O^*\big(\frac{n\alpha}{M_s^{1/s}}\big)$. Thus, for the class of constant-arboricity graphs (which includes, among others, all minor-closed families and preferential attachment graphs), we can estimate the average degree in $O^*(1)$ queries, and we can estimate the variance of the degree distribution in $O^*(\sqrt{n})$ queries. This is a major improvement over the previous worst-case bounds.