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

TL;DR

This paper investigates the feasibility of estimating the number of connected components in large graphs through subgraph sampling, establishing bounds and efficient estimators specifically for chordal graphs.

Contribution

It characterizes the optimal sample complexity for chordal graphs and introduces linear-time estimators that achieve these bounds, expanding prior work beyond special graph classes.

Findings

Impossible to estimate in sublinear regime for graphs with high-degree vertices or long cycles.

Achieves optimal sample complexity bounds for chordal graphs.

Proposes linear-time estimators that are provably optimal.

Abstract

Learning properties of large graphs from samples has been an important problem in statistical network analysis since the early work of Goodman \cite{Goodman1949} and Frank \cite{Frank1978}. We revisit a problem formulated by Frank \cite{Frank1978} 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 \emph{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.