We revisit the classic problem of estimating the degree distribution moments of an undirected graph. Consider an undirected graph G = (V, E) with n (non-isolated) vertices, and define (for Our aim is to estimate µ s within a multiplicative error of (1 + ε) (for a given approximation parameter ε > 0) in sublinear time. We consider the sparse graph model that allows access to: uniform random vertices, queries to the degree of any vertex, and queries to a random neighbor of any vertex. Feige (SICOMP 06) and Goldreich-Ron (RSA 08) proved O( √ n) bounds for the s = 1 case (the average degree). Gonen-Ron-Shavitt (SIDMA 11) extended this result to all integral s > 0, by designing an O(n 1−1/(s+1) ) algorithm. (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-Shavitt, and has a much simpler proof. More importantly, the running time of this algorithm is connected to the arboricity of G: the minimum number of forests into which E can be partitioned. For the family of graphs with arboricity at most α, it has a query complexity of Thus, for the class of bounded arboricity graphs (which includes planar graphs, all minor closed families, and preferential attachment graphs), we can estimate the average degree in O(1) queries, and can estimate the variance of the degree distribution in O( √ n) queries. This is a major improvement over the previous worst-case bounds. Our key insight is that a certain estimator for µ s has significantly smaller variance when G does not have large dense subgraphs.
Eden et al. proposed an algorithm to estimate the average degree usingÕ(1) queries REF .
Sublinear Time Estimation of Degree Distribution Moments: The Arboricity Connection
SIAM J. Discrete Math.
SIAM J. Discrete Math.
Mathematics Computer Science