[Paper Review] Sublinear Time Estimation of Degree Distribution Moments: The Degeneracy Connection

Talya Eden, Dana Ron+1 more|arXiv (Cornell University)|Apr 13, 2016

Complexity and Algorithms in Graphs3 citations

TL;DR

This paper presents a novel sublinear-time algorithm for estimating degree distribution moments in undirected graphs, leveraging graph degeneracy to achieve significantly improved query complexity. It achieves optimal bounds for bounded-degeneracy graphs—such as minor-closed families and preferential attachment networks—estimating average degree in Õ(1) queries and variance in Õ(√n) queries, outperforming prior worst-case bounds.

ABSTRACT

We 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 $s > 0$) $μ_s = \frac{1}{n}\cdot\sum_{v \in V} d^s_v$. 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), $\widetilde{O}(\sqrt{n})$ 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 $\widetilde{O}(n^{1-1/(s+1)})$ 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 $α$, it has a query complexity of $\widetilde{O}\left(\frac{n^{1-1/s}}{μ^{1/s}_s} \Big(α^{1/s} + \min\{α,μ^{1/s}_s\}\Big) ight) = \widetilde{O}(n^{1-1/s}α/μ^{1/s}_s)$. 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 $μ_s$ that has low variance when $G$ does not have large dense subgraphs.

Motivation & Objective

• To design a more efficient and conceptually simpler algorithm for estimating degree distribution moments in sublinear time.
• To connect the query complexity of moment estimation to the degeneracy of the graph, a structural property measuring local density.
• To improve upon prior worst-case query bounds by exploiting low degeneracy in real-world graph families.
• To demonstrate that for graphs with bounded degeneracy, constant-query algorithms exist for average degree estimation.
• To establish that knowledge of degeneracy enables sublinear algorithms where general-purpose methods require Ω(√n) queries.

Proposed method

• The algorithm uses uniform random vertex queries, degree queries, and neighbor queries within the sparse graph model.
• It introduces a new estimator for μs that maintains low variance in graphs without large dense subgraphs, i.e., low degeneracy graphs.
• The query complexity is bounded by Õ(n^{1-1/s} α / μ_s^{1/s}), where α is the degeneracy and μ_s is the s-th moment of the degree distribution.
• The method adapts to the graph's degeneracy, achieving near-constant query complexity for bounded-degeneracy families.
• A key technical insight is that degeneracy controls the variance of the estimator, enabling tighter concentration bounds.
• The algorithm is simpler than prior work by Gonen, Ron, and Shavitt, with a significantly streamlined analysis.

Experimental results

Research questions

• RQ1 Can the query complexity of degree distribution moment estimation be improved by exploiting structural properties like degeneracy?
• RQ2 Is it possible to achieve Õ(1) query complexity for average degree estimation in graphs with bounded degeneracy?
• RQ3 Does the degeneracy of a graph provide a meaningful parameter to bound the complexity of sublinear moment estimation?
• RQ4 Can a simpler algorithm be designed that matches the query complexity of prior work while offering better structural insight?
• RQ5 What is the inherent query complexity of estimating μs when degeneracy is unknown, even for bounded-degeneracy graph families?

Key findings

• For graphs of degeneracy at most α, the algorithm achieves a query complexity of Õ(n^{1-1/s} α / μ_s^{1/s}), which is optimal up to polylogarithmic factors.
• For bounded-degeneracy graphs—such as minor-closed families and preferential attachment graphs—the average degree can be estimated in Õ(1) queries.
• The variance of the degree distribution can be estimated in Õ(√n) queries for such graphs, matching the best-known bounds for general graphs.
• The algorithm matches the worst-case query complexity of Gonen, Ron, and Shavitt’s algorithm but with a much simpler design and analysis.
• Knowledge of degeneracy is essential: any algorithm that must work on all graphs with degeneracy ≤2 must still make Ω(√n) queries in the worst case.
• A lower bound shows that even for degeneracy-2 graphs, no c-valid algorithm can achieve sub-√n query complexity, proving the tightness of the bound.