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August 2020 Estimating the number of connected components in a graph via subgraph sampling
Jason M. Klusowski, Yihong Wu
Bernoulli 26(3): 1635-1664 (August 2020). DOI: 10.3150/19-BEJ1147


Learning properties of large graphs from samples has been an important problem in statistical network analysis since the early work of Goodman (Ann. Math. Stat. 20 (1949) 572–579) and Frank (Scand. J. Stat. 5 (1978) 177–188). We revisit a problem formulated by Frank (Scand. J. Stat. 5 (1978) 177–188) of estimating the number of connected components in a large graph based on the subgraph sampling model, in which we randomly sample a subset of the vertices and observe the induced subgraph. The key question is whether accurate estimation is achievable in the sublinear regime where only a vanishing fraction of the vertices are sampled. We show that it is impossible if the parent graph is allowed to contain high-degree vertices or long induced cycles. For the class of chordal graphs, where induced cycles of length four or above are forbidden, we characterize the optimal sample complexity within constant factors and construct linear-time estimators that provably achieve these bounds. This significantly expands the scope of previous results which have focused on unbiased estimators and special classes of graphs such as forests or cliques.

Both the construction and the analysis of the proposed methodology rely on combinatorial properties of chordal graphs and identities of induced subgraph counts. They, in turn, also play a key role in proving minimax lower bounds based on construction of random instances of graphs with matching structures of small subgraphs.


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Jason M. Klusowski. Yihong Wu. "Estimating the number of connected components in a graph via subgraph sampling." Bernoulli 26 (3) 1635 - 1664, August 2020.


Received: 1 August 2018; Revised: 1 June 2019; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193938
MathSciNet: MR4091087
Digital Object Identifier: 10.3150/19-BEJ1147

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability


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Vol.26 • No. 3 • August 2020
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