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Learning properties of large graphs from samples has been an importantproblem in statistical network analysis since the early work of Goodman\cite{Goodman1949} and Frank \cite{Frank1978}. We revisit a problem formulatedby Frank \cite{Frank1978} of estimating the number of connected components in alarge graph based on the subgraph sampling model, in which we randomly sample asubset of the vertices and observe the induced subgraph. The key question iswhether accurate estimation is achievable in the \emph{sublinear} regime whereonly a vanishing fraction of the vertices are sampled. We show that it isimpossible if the parent graph is allowed to contain high-degree vertices orlong induced cycles. For the class of chordal graphs, where induced cycles oflength four or above are forbidden, we characterize the optimal samplecomplexity within constant factors and construct linear-time estimators thatprovably achieve these bounds. This significantly expands the scope of previousresults which have focused on unbiased estimators and special classes of graphssuch as forests or cliques.  Both the construction and the analysis of the proposed methodology rely oncombinatorial properties of chordal graphs and identities of induced subgraphcounts. They, in turn, also play a key role in proving minimax lower boundsbased on construction of random instances of graphs with matching structures ofsmall subgraphs.

Estimating the Number of Connected Components in a Graph via Subgraph Sampling