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DIST: Efficient k-Clique Listing via Induced Subgraph Trie

Yehyun Nam, Jihoon Jang, Kunsoo Park, Jianye Yang, Cheng Long

TL;DR

This work tackles the challenging problem of listing all $k$-cliques in large graphs. It introduces the Induced Subgraph Trie to memoize and efficiently retrieve cliques, coupled with a pruning mechanism based on soft embeddings of $l$-trees and a density-aware ListingDense routine for dense subgraphs. Empirical results on 16 real networks show DIST substantially outperforms state-of-the-art methods in both running time and memory usage, including notable gains on graphs with large maximum clique sizes and in parallel execution. The approach enables scalable, exact enumeration of $k$-cliques and offers a practical memory management strategy, suggesting broad applicability to cohesive subgraph mining tasks.

Abstract

Listing k-cliques plays a fundamental role in various data mining tasks, such as community detection and mining of cohesive substructures. Existing algorithms for the k-clique listing problem are built upon a general framework, which finds k-cliques by recursively finding (k-1)-cliques within subgraphs induced by the out-neighbors of each vertex. However, this framework has inherent inefficiency of finding smaller cliques within certain subgraphs repeatedly. In this paper, we propose an algorithm DIST for the k-clique listing problem. In contrast to existing works, the main idea in our approach is to compute each clique in the given graph only once and store it into a data structure called Induced Subgraph Trie, which allows us to retrieve the cliques efficiently. Furthermore, we propose a method to prune search space based on a novel concept called soft embedding of an l-tree, which further improves the running time. We show the superiority of our approach in terms of time and space usage through comprehensive experiments conducted on real-world networks; DIST outperforms the state-of-the-art algorithm by up to two orders of magnitude in both single-threaded and parallel experiments.

DIST: Efficient k-Clique Listing via Induced Subgraph Trie

TL;DR

This work tackles the challenging problem of listing all -cliques in large graphs. It introduces the Induced Subgraph Trie to memoize and efficiently retrieve cliques, coupled with a pruning mechanism based on soft embeddings of -trees and a density-aware ListingDense routine for dense subgraphs. Empirical results on 16 real networks show DIST substantially outperforms state-of-the-art methods in both running time and memory usage, including notable gains on graphs with large maximum clique sizes and in parallel execution. The approach enables scalable, exact enumeration of -cliques and offers a practical memory management strategy, suggesting broad applicability to cohesive subgraph mining tasks.

Abstract

Listing k-cliques plays a fundamental role in various data mining tasks, such as community detection and mining of cohesive substructures. Existing algorithms for the k-clique listing problem are built upon a general framework, which finds k-cliques by recursively finding (k-1)-cliques within subgraphs induced by the out-neighbors of each vertex. However, this framework has inherent inefficiency of finding smaller cliques within certain subgraphs repeatedly. In this paper, we propose an algorithm DIST for the k-clique listing problem. In contrast to existing works, the main idea in our approach is to compute each clique in the given graph only once and store it into a data structure called Induced Subgraph Trie, which allows us to retrieve the cliques efficiently. Furthermore, we propose a method to prune search space based on a novel concept called soft embedding of an l-tree, which further improves the running time. We show the superiority of our approach in terms of time and space usage through comprehensive experiments conducted on real-world networks; DIST outperforms the state-of-the-art algorithm by up to two orders of magnitude in both single-threaded and parallel experiments.

Paper Structure

This paper contains 14 sections, 3 theorems, 2 equations, 11 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Statement
  4. Related Works
  5. Overview of Algorithm
  6. Induced Subgraph Trie
  7. Pruning by Soft Embedding
  8. Space Usage of DIST
  9. Performance Evaluation
  10. Experimental Setup
  11. Single-Threaded
  12. Evaluation of Techniques
  13. Parallelization
  14. Conclusion

Key Result

Theorem 1

Algorithm 2 lists all $k$-cliques in $\vec{G}$.

Figures (11)

  • Figure 1: Two $k$-clique communities at $k\space=\space4$. A $k$-clique community is a union of $k$-cliques adjacent to each other, where adjacency means sharing $k\space-\space1$ vertices.
  • Figure 2: Graph $G$ and DAG $\vec{G}$ based on degeneracy ordering.
  • Figure 3: Induced Subgraph Trie for the DAG $\vec{G}$ in Figure \ref{['fig:DAG']}. Red arc (resp. blue arc) labeled $l$ outgoing from a node $t$ is the $l$-child link (resp. $l$-sibling link) of $t$.
  • Figure 4: DAG $\vec{G}$, $4$-tree, and two soft embeddings $f$ and $g$.
  • Figure 5: Running time of algorithms on small-$\omega$ graphs.
  • ...and 6 more figures

Theorems & Definitions (19)

  • Definition 1: $k$-clique listing
  • Example 1
  • Definition 2
  • Example 2
  • Example 3
  • Definition 3
  • Definition 4
  • Example 4
  • Definition 5
  • Theorem 1
  • ...and 9 more