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Efficient Maintenance of Leiden Communities in Large Dynamic Graphs

Chunxu Lin, Yumao Xie, Yixiang Fang, Yongmin Hu, Yingqian Hu, Chen Cheng

TL;DR

A novel efficient maintenance algorithm is developed, called Hierarchical Incremental Tree Leiden (HIT-Leiden), which effectively reduces the range of affected vertices by maintaining the connected components and hierarchical community structures.

Abstract

As a well-known community detection algorithm, Leiden has been widely used in various scenarios such as large language model generation (e.g., Graph-RAG), anomaly detection, and biological analysis. In these scenarios, the graphs are often large and dynamic, where vertices and edges are inserted and deleted frequently, so it is costly to obtain the updated communities by Leiden from scratch when the graph has changed. Recently, one work has attempted to study how to maintain Leiden communities in the dynamic graph, but it lacks a detailed theoretical analysis, and its algorithms are inefficient for large graphs. To address these issues, in this paper, we first theoretically show that the existing algorithms are relatively unbounded via the boundedness analysis (a powerful tool for analyzing incremental algorithms on dynamic graphs), and also analyze the memberships of vertices in communities when the graph changes. Based on theoretical analysis, we develop a novel efficient maintenance algorithm, called Hierarchical Incremental Tree Leiden (HIT-Leiden), which effectively reduces the range of affected vertices by maintaining the connected components and hierarchical community structures. Comprehensive experiments in various datasets demonstrate the superior performance of HIT-Leiden. In particular, it achieves speedups of up to five orders of magnitude over existing methods.

Efficient Maintenance of Leiden Communities in Large Dynamic Graphs

TL;DR

A novel efficient maintenance algorithm is developed, called Hierarchical Incremental Tree Leiden (HIT-Leiden), which effectively reduces the range of affected vertices by maintaining the connected components and hierarchical community structures.

Abstract

As a well-known community detection algorithm, Leiden has been widely used in various scenarios such as large language model generation (e.g., Graph-RAG), anomaly detection, and biological analysis. In these scenarios, the graphs are often large and dynamic, where vertices and edges are inserted and deleted frequently, so it is costly to obtain the updated communities by Leiden from scratch when the graph has changed. Recently, one work has attempted to study how to maintain Leiden communities in the dynamic graph, but it lacks a detailed theoretical analysis, and its algorithms are inefficient for large graphs. To address these issues, in this paper, we first theoretically show that the existing algorithms are relatively unbounded via the boundedness analysis (a powerful tool for analyzing incremental algorithms on dynamic graphs), and also analyze the memberships of vertices in communities when the graph changes. Based on theoretical analysis, we develop a novel efficient maintenance algorithm, called Hierarchical Incremental Tree Leiden (HIT-Leiden), which effectively reduces the range of affected vertices by maintaining the connected components and hierarchical community structures. Comprehensive experiments in various datasets demonstrate the superior performance of HIT-Leiden. In particular, it achieves speedups of up to five orders of magnitude over existing methods.
Paper Structure (23 sections, 5 theorems, 2 equations, 19 figures, 3 tables, 6 algorithms)

This paper contains 23 sections, 5 theorems, 2 equations, 19 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Problem definition
  5. Leiden algorithm
  6. Theoretical Analysis of Leiden
  7. Boundedness analysis
  8. Vertex optimality and subpartition $\gamma$-density
  9. Our HIT-Leiden algorithm
  10. Inc-movement
  11. Inc-refinement
  12. Inc-aggregation
  13. Overall HIT-Leiden algorithm
  14. Experiments
  15. Setup
  16. ...and 8 more sections

Key Result

Theorem 1

When processing an edge deletion or insertion, the incremental Leiden algorithms proposed in sahu2024starting all cost $O(P\cdot(|V| + |E|))$.

Figures (19)

  • Figure 1: Illustrating community maintenance, where ($v_1$, $v_3$) is a newly inserted edge and ($v_3$, $v_5$) is a newly deleted edge.
  • Figure 2: Illustrating the Louvain and Leiden algorithms.
  • Figure 3: Algorithms for maintaining Leiden communities.
  • Figure 4: The process of Leiden for the graph $G$ in Figure \ref{['fig:mod']}(\ref{['fig:mod:org_graph']}).
  • Figure 5: The process of hierarchical partitions of Figure \ref{['fig:hie_structural']} at level-1 with the Leiden algorithm.
  • ...and 14 more figures

Theorems & Definitions (21)

  • Definition 1: Modularity blondel2008fast
  • Definition 2: Modularity gain blondel2008fast
  • Example 1
  • Example 2
  • Definition 3: Boundedness ramalingam1996computationalfan2017incremental
  • Definition 4: ${\tt AFF}$ fan2017incremental
  • Definition 5: Relative boundedness fan2017incremental
  • Theorem 1
  • Definition 6: Vertex optimality blondel2008fast
  • Lemma 1
  • ...and 11 more