Sublinear time estimation of degree distribution moments: The degeneracy connection

Talya Eden, Dana Ron, C. Seshadhri

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 (non-isolated) vertices, and define (for s > 0) μs = 1/n ∑ν∈2V dνs. Our aim is to estimate μs 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(n1-1/(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 (n1-1/ss1/s1/s + min{α, μ s1/s })) = Õ(n1-1/sα/μ s1/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 μs that has low variance when G does not have large dense subgraphs.

Original languageEnglish
Title of host publication44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
EditorsAnca Muscholl, Piotr Indyk, Fabian Kuhn, Ioannis Chatzigiannakis
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770415
DOIs
StatePublished - 1 Jul 2017
Event44th International Colloquium on Automata, Languages, and Programming, ICALP 2017 - Warsaw, Poland
Duration: 10 Jul 201714 Jul 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume80
ISSN (Print)1868-8969

Conference

Conference44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
Country/TerritoryPoland
CityWarsaw
Period10/07/1714/07/17

Keywords

  • Degree distribution
  • Graph moments
  • Sublinear algorithms

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