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Scalable Approximate Biclique Counting over Large Bipartite Graphs

Jingbang Chen, Weinuo Li, Yingli Zhou, Hangrui Zhou, Qiuyang Mang, Can Wang, Yixiang Fang, Chenhao Ma

TL;DR

This work tackles scalable counting of (p,q)-bicliques in large bipartite graphs by introducing Colored Broom-based Sampling (CBS), which combines graph coloring, counting of a special (p,q)-broom motif via dynamic programming, and principled sampling to obtain unbiased biclique counts with provable error bounds. The core ideas are a novel broom-based motif that closely mirrors bicliques, a coloring scheme that enables efficient DP counting, and a sampling scheme that leverages DP-derived probabilities for accurate, fast estimates. Theoretical guarantees (unbiasedness and Hoeffding-based error bounds) accompany extensive empirical evaluation on nine real-world datasets, where CBS achieves up to 8x improvement in accuracy and up to 50x speedups over state-of-the-art approximate methods, demonstrating strong scalability for practical applications. The approach offers a practical pathway for kernel methods and cohesive subgraph analysis where exact counts are infeasible but reliable approximate counts are valuable.

Abstract

Counting $(p,q)$-bicliques in bipartite graphs is crucial for a variety of applications, from recommendation systems to cohesive subgraph analysis. Yet, it remains computationally challenging due to the combinatorial explosion to exactly count the $(p,q)$-bicliques. In many scenarios, e.g., graph kernel methods, however, exact counts are not strictly required. To design a scalable and high-quality approximate solution, we novelly resort to $(p,q)$-broom, a special spanning tree of the $(p,q)$-biclique, which can be counted via graph coloring and efficient dynamic programming. Based on the intermediate results of the dynamic programming, we propose an efficient sampling algorithm to derive the approximate $(p,q)$-biclique count from the $(p,q)$-broom counts. Theoretically, our method offers unbiased estimates with provable error guarantees. Empirically, our solution outperforms existing approximation techniques in both accuracy (up to 8$\times$ error reduction) and runtime (up to 50$\times$ speedup) on nine real-world bipartite networks, providing a scalable solution for large-scale $(p,q)$-biclique counting.

Scalable Approximate Biclique Counting over Large Bipartite Graphs

TL;DR

This work tackles scalable counting of (p,q)-bicliques in large bipartite graphs by introducing Colored Broom-based Sampling (CBS), which combines graph coloring, counting of a special (p,q)-broom motif via dynamic programming, and principled sampling to obtain unbiased biclique counts with provable error bounds. The core ideas are a novel broom-based motif that closely mirrors bicliques, a coloring scheme that enables efficient DP counting, and a sampling scheme that leverages DP-derived probabilities for accurate, fast estimates. Theoretical guarantees (unbiasedness and Hoeffding-based error bounds) accompany extensive empirical evaluation on nine real-world datasets, where CBS achieves up to 8x improvement in accuracy and up to 50x speedups over state-of-the-art approximate methods, demonstrating strong scalability for practical applications. The approach offers a practical pathway for kernel methods and cohesive subgraph analysis where exact counts are infeasible but reliable approximate counts are valuable.

Abstract

Counting -bicliques in bipartite graphs is crucial for a variety of applications, from recommendation systems to cohesive subgraph analysis. Yet, it remains computationally challenging due to the combinatorial explosion to exactly count the -bicliques. In many scenarios, e.g., graph kernel methods, however, exact counts are not strictly required. To design a scalable and high-quality approximate solution, we novelly resort to -broom, a special spanning tree of the -biclique, which can be counted via graph coloring and efficient dynamic programming. Based on the intermediate results of the dynamic programming, we propose an efficient sampling algorithm to derive the approximate -biclique count from the -broom counts. Theoretically, our method offers unbiased estimates with provable error guarantees. Empirically, our solution outperforms existing approximation techniques in both accuracy (up to 8 error reduction) and runtime (up to 50 speedup) on nine real-world bipartite networks, providing a scalable solution for large-scale -biclique counting.
Paper Structure (32 sections, 7 theorems, 2 equations, 12 figures, 5 tables, 4 algorithms)

This paper contains 32 sections, 7 theorems, 2 equations, 12 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. ALGORITHM
  5. 1. Coloring.
  6. 2. Broom Pattern Counting.
  7. 3. Counting by Sampling Brooms.
  8. Coloring
  9. Pattern Counting
  10. Approximate Counting via Sampling
  11. Overall Algorithm
  12. Unbiasedness.
  13. Error Analysis.
  14. Time and Space Complexity.
  15. EVALUATION
  16. ...and 17 more sections

Key Result

lemma 1

After executing $\texttt{Coloring}$$(G,p,q)$, for any $(p,q)$-clique $H$ in $G$, there are no two vertices $x,y$ either both in $U(H)$ or in $V(H)$ that share the same color, i.e. $c(x)=c(y)$.

Figures (12)

  • Figure 1: An illustrative example of two $(3,2)$-bicliques.
  • Figure 2: An illustrative example of a $(6,3)$-broom.
  • Figure 3: An example of edges' directions in a $(6, 3)$-broom, where edges with different directions are represented by $\texttt{Up}$ and $\texttt{Down}$ in $Dir$ and colored in red and blue, respectively.
  • Figure 4: Average runtime of different biclique counting algorithms for all $3 \leq p,q \leq 9$.
  • Figure 5: Average error of EP/Zz++ and CBS for all $3 \leq p,q \leq 9$.
  • ...and 7 more figures

Theorems & Definitions (9)

  • definition 1: Biclique
  • definition 2: Broom
  • lemma 1
  • lemma 2
  • theorem 1
  • lemma 3: Hoeffding’s inequality, chernoff1952measurehoeffding1994probability
  • theorem 2
  • theorem 3
  • theorem 4