[Paper Review] Counting Motifs with Graph Sampling

Jason M. Klusowski, Yihong Wu|arXiv (Cornell University)|Feb 21, 2018

Complex Network Analysis Techniques29 references3 citations

TL;DR

This paper studies statistical estimation of motif counts (induced subgraphs) in large networks using subgraph and neighborhood sampling. It establishes optimal sampling ratios for estimating motif counts with multiplicative error, showing that neighborhood sampling significantly outperforms subgraph sampling, with results depending only on motif size and graph degree, not topology.

ABSTRACT

Applied researchers often construct a network from a random sample of nodes in order to infer properties of the parent network. Two of the most widely used sampling schemes are subgraph sampling, where we sample each vertex independently with probability $p$ and observe the subgraph induced by the sampled vertices, and neighborhood sampling, where we additionally observe the edges between the sampled vertices and their neighbors. In this paper, we study the problem of estimating the number of motifs as induced subgraphs under both models from a statistical perspective. We show that: for any connected $h$ on $k$ vertices, to estimate $s=\mathsf{s}(h,G)$, the number of copies of $h$ in the parent graph $G$ of maximum degree $d$, with a multiplicative error of $ε$, (a) For subgraph sampling, the optimal sampling ratio $p$ is $Θ_{k}(\max\{ (sε^2)^{-\frac{1}{k}}, \; \frac{d^{k-1}}{sε^{2}} \})$, achieved by Horvitz-Thompson type of estimators. (b) For neighborhood sampling, we propose a family of estimators, encompassing and outperforming the Horvitz-Thompson estimator and achieving the sampling ratio $O_{k}(\min\{ (\frac{d}{sε^2})^{\frac{1}{k-1}}, \; \sqrt{\frac{d^{k-2}}{sε^2}}\})$. This is shown to be optimal for all motifs with at most $4$ vertices and cliques of all sizes. The matching minimax lower bounds are established using certain algebraic properties of subgraph counts. These results quantify how much more informative neighborhood sampling is than subgraph sampling, as empirically verified by experiments on both synthetic and real-world data. We also address the issue of adaptation to the unknown maximum degree, and study specific problems for parent graphs with additional structures, e.g., trees or planar graphs.

Motivation & Objective

• To address the statistical challenge of estimating motif counts in large networks when only a sampled subgraph is available.
• To compare the efficiency of subgraph sampling versus neighborhood sampling for motif counting under realistic sampling constraints.
• To derive optimal sampling ratios that minimize estimation error for any connected motif of size k.
• To develop universally optimal estimators—particularly Horvitz-Thompson-type estimators—for motif counting under both sampling models.
• To establish minimax lower bounds and demonstrate the optimality of proposed estimators for small motifs and cliques.

Proposed method

• Proposes a Horvitz-Thompson-type estimator for subgraph sampling, which is shown to be universally optimal across all connected motifs.
• Introduces a family of estimators for neighborhood sampling that generalize and outperform the Horvitz-Thompson estimator.
• Derives optimal sampling ratios using algebraic properties of subgraph counts and minimax risk analysis.
• Establishes matching minimax lower bounds via combinatorial and algebraic techniques to prove optimality of the proposed sampling strategies.
• Analyzes adaptation to unknown maximum degree d using data-driven sampling ratio selection.
• Considers special graph structures (e.g., trees, planar graphs) to refine sampling efficiency under structural constraints.

Experimental results

Research questions

• RQ1 What is the optimal sampling ratio p for subgraph sampling to estimate motif counts with multiplicative error ε?
• RQ2 How does neighborhood sampling improve estimation efficiency compared to subgraph sampling for motif counting?
• RQ3 Can universally optimal estimators be constructed for all connected motifs under both sampling models?
• RQ4 What are the minimax lower bounds for motif counting under subgraph and neighborhood sampling?
• RQ5 How can sampling ratios be adapted when the maximum degree d of the parent graph is unknown?

Key findings

• For subgraph sampling, the optimal sampling ratio is Θk(max{(sε²)^(-1/k), d^{k-1}/(sε²)}), independent of motif topology.
• The Horvitz-Thompson estimator is universally optimal for all connected motifs under subgraph sampling.
• For neighborhood sampling, the proposed estimator achieves a sampling ratio of Ok(min{(d/(sε²))^{1/(k-1)}, √(d^{k-2}/(sε²))}), again independent of motif structure.
• The neighborhood sampling estimator is optimal for all motifs with at most 4 vertices and for cliques of any size.
• Minimax lower bounds match the upper bounds, proving the optimality of the proposed sampling strategies.
• Experiments on synthetic and real-world data confirm that neighborhood sampling provides significantly better estimation accuracy than subgraph sampling.