Sublinear time estimation of degree distribution moments: The degeneracy connection

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 s > 0) μ_s = 1/n ∑_ν∈2V d_ν^s. 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 for the degree of any vertex, and queries for a neighbor of any vertex. For the case of s = 1 (the average degree), Õ(√n) queries suffice for any constant ϵ (Feige, SICOMP 06 and Goldreich-Ron, RSA 08). Gonen-Ron-Shavitt (SIDMA 11) extended this result to all integral s > 0, by designing an algorithms that performs Õ(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-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 α, it has a query complexity of Õ (n^1-1/s/μ _s^1/s (α^1/s + min{α, μ _s^1/s })) = Õ(n^1-1/sα/μ _s^1/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 Õ(1) queries, and can estimate the variance of the degree distribution in Õ(√n) queries. This is a major improvement over the previous worst-case bounds. Our key insight is in designing an estimator for μ_s that has low variance when G does not have large dense subgraphs.