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Engineering Algorithms for $\ell$-Isolated Maximal Clique Enumeration

Marco D'Elia, Irene Finocchi, Maurizio Patrignani

TL;DR

This work addresses the challenge of enumerating maximal cliques by focusing on $\ell$-isolated maximal cliques, offering a controllable filter to reduce output while preserving meaningful structures. Building on Bron-Kerbosch with pivoting, the authors introduce four pruning heuristics (size, degree, softcore, degeneracy) and a combined approach, with proofs of correctness and analyses of pruning power versus cost. An extensive experimental study on 16 real-world networks and multiple synthetic models demonstrates substantial speedups over baselines, particularly on social-network-like graphs, while clarifying trade-offs between pruning strength and overhead. The results provide practical guidance for selecting $\ell$ and pruning strategies and point to open problems in isolated clique enumeration and its extensions for network analysis tasks.

Abstract

Maximal cliques play a fundamental role in numerous application domains, where their enumeration can prove extremely useful. Yet their sheer number, even in sparse real-world graphs, can make them impractical to be exploited effectively. To address this issue, one approach is to enumerate $\ell$-isolated maximal cliques, whose vertices have (on average) less than $\ell$ edges toward the rest of the graph. By tuning parameter $\ell$, the degree of isolation can be controlled, and cliques that are overly connected to the outside are filtered out. Building on Tomita et al.'s very practical recursive algorithm for maximal clique enumeration, we propose four pruning heuristics, applicable individually or in combination, that discard recursive search branches that are guaranteed not to yield $\ell$-isolated maximal cliques. Besides proving correctness, we characterize both the pruning power and the computational cost of these heuristics, and we conduct an extensive experimental study comparing our methods with Tomita's baseline and with a state-of-the-art approach. Results show that two of our heuristics offer substantial efficiency improvements, especially on real-world graphs with social network properties.

Engineering Algorithms for $\ell$-Isolated Maximal Clique Enumeration

TL;DR

This work addresses the challenge of enumerating maximal cliques by focusing on -isolated maximal cliques, offering a controllable filter to reduce output while preserving meaningful structures. Building on Bron-Kerbosch with pivoting, the authors introduce four pruning heuristics (size, degree, softcore, degeneracy) and a combined approach, with proofs of correctness and analyses of pruning power versus cost. An extensive experimental study on 16 real-world networks and multiple synthetic models demonstrates substantial speedups over baselines, particularly on social-network-like graphs, while clarifying trade-offs between pruning strength and overhead. The results provide practical guidance for selecting and pruning strategies and point to open problems in isolated clique enumeration and its extensions for network analysis tasks.

Abstract

Maximal cliques play a fundamental role in numerous application domains, where their enumeration can prove extremely useful. Yet their sheer number, even in sparse real-world graphs, can make them impractical to be exploited effectively. To address this issue, one approach is to enumerate -isolated maximal cliques, whose vertices have (on average) less than edges toward the rest of the graph. By tuning parameter , the degree of isolation can be controlled, and cliques that are overly connected to the outside are filtered out. Building on Tomita et al.'s very practical recursive algorithm for maximal clique enumeration, we propose four pruning heuristics, applicable individually or in combination, that discard recursive search branches that are guaranteed not to yield -isolated maximal cliques. Besides proving correctness, we characterize both the pruning power and the computational cost of these heuristics, and we conduct an extensive experimental study comparing our methods with Tomita's baseline and with a state-of-the-art approach. Results show that two of our heuristics offer substantial efficiency improvements, especially on real-world graphs with social network properties.

Paper Structure

This paper contains 12 sections, 3 theorems, 5 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\ell$-isolated cliques
  4. Maximal clique enumeration with pivoting
  5. Pruning for $\ell$-isolated maximal clique enumeration
  6. Experimental setup
  7. Experimental results
  8. On the distribution of $\ell$-isolated maximal cliques
  9. Assessment of different pruning strategies
  10. Comparison with state-of-the-art competitors
  11. Experiments with synthetic networks
  12. Conclusions and open problems

Key Result

Lemma 1

Let $r$ be a node in $\mathcal{T}_G$ associated to a candidate set $C$ and a perspective set $P$. Let $P_1$ and $P_2$ be any two subsets of $P$ such that $|P_1|\leq|P_2|$. If $\delta_e(C,P) + |C|(|P| - |P_2|)\geq\ell(|C|+|P_2|)$, then both $C\cup P_2$ and $C\cup P_1$ are not $\ell$-isolated.

Figures (6)

  • Figure 1: A running example of \ref{['alg:bk']} on a small graph. Green, blue, and red vertices belong to $P$, $C$, and $X$, respectively. The current pivot vertex $v_p$ is circled with a dotted curve. The neighbourhood of $v_p$ is circled with a dashed curve, the current vertex $v$ is circled with a red solid curve.
  • Figure 2: $\ell$-isolated clique size distribution for different values of $\ell$.
  • Figure 3: Comparison of the recursive calls (%) and the running time (% with respect to the slowest execution) of different pruning strategies for $\ell=50$.
  • Figure 4: Comparison of size, combo, HKMN, and TTT on real-world graphs.
  • Figure 5: Results on BA$_{n,m}$ synthetic graphs. The dashed red line shows the number of $\ell$-isolated maximal cliques relative to the total number.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Theorem 3