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Advancing Local Clustering on Graphs via Compressive Sensing: Semi-supervised and Unsupervised Methods

Zhaiming Shen, Sung Ha Kang

TL;DR

This paper reframes local clustering on graphs as recovering sparse cluster indicators from a diffusion-like process tied to the graph Laplacian, enabling targeted discovery of small subgraphs with minimal labeling. It introduces semi-supervised Local Clustering (SSLC) and unsupervised Local Clustering (USLC), each with theoretical guarantees under mild perturbation and RIP-type conditions, and demonstrates that multiple clusters can be identified efficiently. Empirically, SSLC/USLC achieve state-of-the-art results in low-label settings across synthetic SBM benchmarks and real datasets (FashionMNIST, CIFAR-10, Planetoid graphs), while maintaining robustness to outliers and favorable runtimes. The methods expand practical applicability of local clustering to large graphs where full labeling is impractical, though they are best suited for sparse graphs and low-label scenarios.

Abstract

Local clustering aims to identify specific substructures within a large graph without any additional structural information of the graph. These substructures are typically small compared to the overall graph, enabling the problem to be approached by finding a sparse solution to a linear system associated with the graph Laplacian. In this work, we first propose a method for identifying specific local clusters when very few labeled data are given, which we term semi-supervised local clustering. We then extend this approach to the unsupervised setting when no prior information on labels is available. The proposed methods involve randomly sampling the graph, applying diffusion through local cluster extraction, then examining the overlap among the results to find each cluster. We establish the co-membership conditions for any pair of nodes, and rigorously prove the correctness of our methods. Additionally, we conduct extensive experiments to demonstrate that the proposed methods achieve state of the art results in the low-label rates regime.

Advancing Local Clustering on Graphs via Compressive Sensing: Semi-supervised and Unsupervised Methods

TL;DR

This paper reframes local clustering on graphs as recovering sparse cluster indicators from a diffusion-like process tied to the graph Laplacian, enabling targeted discovery of small subgraphs with minimal labeling. It introduces semi-supervised Local Clustering (SSLC) and unsupervised Local Clustering (USLC), each with theoretical guarantees under mild perturbation and RIP-type conditions, and demonstrates that multiple clusters can be identified efficiently. Empirically, SSLC/USLC achieve state-of-the-art results in low-label settings across synthetic SBM benchmarks and real datasets (FashionMNIST, CIFAR-10, Planetoid graphs), while maintaining robustness to outliers and favorable runtimes. The methods expand practical applicability of local clustering to large graphs where full labeling is impractical, though they are best suited for sparse graphs and low-label scenarios.

Abstract

Local clustering aims to identify specific substructures within a large graph without any additional structural information of the graph. These substructures are typically small compared to the overall graph, enabling the problem to be approached by finding a sparse solution to a linear system associated with the graph Laplacian. In this work, we first propose a method for identifying specific local clusters when very few labeled data are given, which we term semi-supervised local clustering. We then extend this approach to the unsupervised setting when no prior information on labels is available. The proposed methods involve randomly sampling the graph, applying diffusion through local cluster extraction, then examining the overlap among the results to find each cluster. We establish the co-membership conditions for any pair of nodes, and rigorously prove the correctness of our methods. Additionally, we conduct extensive experiments to demonstrate that the proposed methods achieve state of the art results in the low-label rates regime.
Paper Structure (26 sections, 10 theorems, 22 equations, 5 figures, 7 tables, 4 algorithms)

This paper contains 26 sections, 10 theorems, 22 equations, 5 figures, 7 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Model Assumptions
  3. Semi-supervised Local Clustering
  4. Unsupervised Local Clustering
  5. Experiments
  6. Synthetic Data
  7. Real Data
  8. Conclusion and Discussion
  9. Lemmas and deferred proofs
  10. Proof of Lemma \ref{['IndAna']}
  11. Proof of Theorem \ref{['thmSSLC']}
  12. Proof of Lemma \ref{['CCMIV']}
  13. Proof of Proposition \ref{['Mmatrix']}
  14. Proof of Theorem \ref{['thmUSLC']}
  15. Additional Algorithms
  16. ...and 11 more sections

Key Result

Theorem 1

Suppose $G$ satisfies Assumptions assump1 - assump3. Then when $n$ (the size of $G$) gets large, for each $s=1,\cdots,k$, we have

Figures (5)

  • Figure 1: Illustration of semi-supervised local clustering (SSLC) for a single cluster. Each subplot indicates one iteration (Blue dots: seeds in $\Gamma_1$. Brown dots: randomly sampled node in each iteration).
  • Figure 2: Illustration of USLC procedure. Top row: each dashed circle represents one iteration of LCE generated from a randomly sampled node, different colors indicate local clusters generated from nodes of different underlying clusters. Bottom row: Aggregated co-membership matrix.
  • Figure 3: Plots of Jaccard index (over 100 trials) and logarithm of running time of SSLC for stochastic block model. (a) The first column shows experiment with three equal cluster size and single cluster $C_1$ extraction. (b) The middle column shows experiment with three equal cluster size and all clusters $C_s$ extraction. (c) The right column shows experiment with unequal cluster sizes, focusing on the most dominent cluster.
  • Figure 4: 2D Visulization of Geometric Dataset
  • Figure 5: Affinity matrix of FashionMNIST after adding the outlier images (the last block consists of $10\%$ outliers compared to the size of original dataset)

Theorems & Definitions (13)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Remark 1
  • Remark 2
  • Lemma 3
  • Theorem 3
  • Lemma 4
  • ...and 3 more