[Paper Review] A Survey and Taxonomy of Graph Sampling

Pili Hu, Wing Cheong Lau|arXiv (Cornell University)|Aug 23, 2013

Complex Network Analysis Techniques54 references126 citations

TL;DR

This survey presents a comprehensive taxonomy of graph sampling techniques, categorizing them by sampling objectives, network types, and approaches such as vertex, edge, and traversal-based sampling. It systematically analyzes which graph properties—like degree distribution, modularity, and conductance—are preserved under different sampling methods, providing theoretical foundations and practical guidelines for efficient graph analysis and algorithm acceleration.

ABSTRACT

Connectivity is one of the important fundamental structural properties of graphs, and a graph drawing D should faithfully represent the connectivity structure of the underlying graph G. This paper investigates connectivity-faithful graph drawing leveraging the famous Nagamochi-Ibaraki (NI) algorithm, which computes a sparsification G_NI, preserving the k-connectivity of a k-connected graph G. Specifically, we first present CFNI, a divide-and-conquer algorithm, which computes a sparsification G_CFNI, which preserves the global k-connectivity of a graph G and the local h-connectivity of the h-connected components of G. We then present CFGD, a connectivity-faithful graph drawing algorithm based on CFNI, which faithfully displays the global and local connectivity structure of G. Extensive experiments demonstrate that CFNI outperforms NI with 66% improvement in the connectivity-related sampling quality metrics and 73% improvement in proxy quality metrics. Consequently, CFGD outperforms a naive application of NI for graph drawing, in particular with 62% improvement in stress metrics. Moreover, CFGD runs 51% faster than drawing the whole graph G, with a similar quality.

Motivation & Objective

• To establish a unified taxonomy of graph sampling methods based on objectives, network types, and sampling approaches.
• To identify which graph properties are preserved under various sampling techniques, enabling reliable estimation on sampled graphs.
• To bridge theoretical analysis with practical implementation by formalizing relations between sampling methods and property preservation.
• To highlight gaps in existing research and advocate for systematic, neutral evaluation frameworks for graph sampling.
• To provide a foundation for accelerating graph algorithms by leveraging preserved properties in sampled graphs.

Proposed method

• Categorizes graph sampling into three dimensions: sampling objectives (e.g., size reduction, exploration), network types (e.g., social, web), and sampling approaches (vertex, edge, traversal-based).
• Introduces a general framework to relate theoretical sampling properties to practical implementations.
• Analyzes 10 traversal-based sampling methods, including BFS, random walk, Metropolis-Hastings, and forest fire, with formal relations between them.
• Classifies graph properties into classical (e.g., degree, density), advanced (e.g., conductance, modularity), and distance metrics for comparison.
• Applies probabilistic and spectral graph theory to derive theoretical results on property preservation, such as unbiased estimation of average degree and network size.
• Proposes extensions to existing theoretical results, particularly for non-uniform edge sampling using effective resistance and edge strong connectivity.

Experimental results

Research questions

• RQ1 Which graph properties are preserved under different sampling techniques, and under what conditions?
• RQ2 How can sampling-based estimators be constructed to ensure accuracy and efficiency in large-scale graph analysis?
• RQ3 What are the theoretical and practical trade-offs between different traversal-based sampling methods like random walk and forest fire?
• RQ4 To what extent can sampled graphs preserve structural properties such as modularity, conductance, and cut values?
• RQ5 How can systematic evaluation frameworks be designed to fairly compare sampling algorithms across diverse graph types and properties?

Key findings

• Uniform vertex sampling preserves the average degree in expectation, but degree distribution is generally not preserved without neighborhood inclusion.
• Edge sampling with contraction preserves minimum cut values with high probability under certain conditions, enabling accurate estimation of sparsest cuts.
• Metropolis-Hastings random walk ensures uniform stationary distribution, making it suitable for unbiased sampling of graph properties.
• Non-uniform edge sampling using effective resistance provides accurate estimation of quadratic forms and spectral properties.
• Traversal-based methods like BFS and random walk can introduce bias in estimating network size and degree distribution, especially in scale-free networks.
• Theoretical results show that certain sampling methods, such as MIRW and MDRW, improve mixing and reduce variance in property estimation compared to simple random walks.