Counting Motifs with Graph Sampling

TL;DR

This paper develops statistical methods for estimating motif counts in large networks from sampled data, comparing subgraph and neighborhood sampling schemes, and establishes optimal sampling ratios and bounds.

Contribution

It introduces new estimators and theoretical bounds for motif count estimation under different sampling models, improving understanding of sampling efficiency.

Findings

Neighborhood sampling outperforms subgraph sampling in information content.

Optimal sampling ratios are derived for different motif sizes and sampling schemes.

The proposed estimators are shown to be minimax optimal for certain classes of motifs.

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 $\epsilon$, (a) For subgraph sampling, the optimal sampling ratio $p$ is $\Theta_{k}(\max\{ (s\epsilon^2)^{-\frac{1}{k}}, \; \frac{d^{k-1}}{s\epsilon^{2}} \})$, 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 $O_{k}(\min\{ (\frac{d}{s\epsilon^2})^{\frac{1}{k-1}}, \; \sqrt{\frac{d^{k-2}}{s\epsilon^2}}\})$. 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.

Tables

Table 1. Table 1: Probability mass function of 𝒦 A ​ 𝒦 A ′ subscript 𝒦 𝐴 subscript 𝒦 superscript 𝐴 ′ {\mathcal{K}}_{A}{\mathcal{K}}_{A^{\prime}} for two distinct intersecting edges (excluding zero values).

Graph
Probability	$p{q}^2$	$2p^{2}q$	$p^{2}q$	$p^3$
Value	${\alpha }^2$	$\alpha \beta$	${\alpha }^2$	${\beta }^2$

Table 2. Table 2: The graph H 𝐻 H with ℓ = 5 ℓ 5 \ell=5 and 𝖽 ​ ( H ) = ℓ + 1 = 6 𝖽 𝐻 ℓ 1 6 {\mathsf{d}}(H)=\ell+1=6

Copies	Components
1
2
2
$\frac{\ell +1}{3}$
$2\left(\ell +1\right)$

Equations

$$\mathsf{n}(H,G)=\sum _{H^{\prime }}\mathsf{n}(H,H^{\prime })\mathsf{s}(H^{\prime },G),$$

$$\mathbb{P}[\tilde{G}\simeq H]=\mathsf{s}(H,G)p^{\mathsf{v}(H)}(1-p{)}^{\mathsf{v}(G)-\mathsf{v}(H)}.$$

$$\mathbb{P}[\tilde{G}\cong H]=\mathsf{N}(H,G)p^{{\mathsf{v}}_b(H)}{q}^{\mathsf{v}(G)-{\mathsf{v}}_b(H)}.$$

$${\Theta }_k\left(\max \left\{\frac{1}{{\left(s{ϵ}^2\right)}^{\frac{1}{k}}},\frac{d^{k-1}}{s{ϵ}^2}\right\}\right),$$

$${\Theta }_k\left(\min \left\{{\left(\frac{d}{s{ϵ}^2}\right)}^{\frac{1}{k-1}},\sqrt{\frac{d^{k-2}}{s{ϵ}^2}}\right\}\right)$$

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