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Recovering Fairness Directly from Modularity: a New Way for Fair Community Partitioning

Yufeng Wang, Yiguang Bai, Tianqing Zhu, Ismail Ben Ayed, Jing Yuan

TL;DR

This work addresses fairness in modularity-based community detection by introducing a protected group network and a fairness-modularity metric $Q^P$, which quantifies deviation from global protected-group distributions. Minimizing $Q^P$ yields fair partitions, enabling a multi-objective optimization that preserves the standard modularity $Q$ while promoting fairness. The authors propose FairFN, an efficient extension of the Fast Newman algorithm with a simple fairness constraint, achieving superior fairness (higher FR, lower AWD) and competitive modularity on synthetic and real datasets, including unbalanced cases, and capable of automatic stopping without pre-specifying the number of communities. The approach has practical impact for applications ranging from social networks to brain networks by reducing bias in detected communities while maintaining high-quality partitions.

Abstract

Community partitioning is crucial in network analysis, with modularity optimization being the prevailing technique. However, traditional modularity-based methods often overlook fairness, a critical aspect in real-world applications. To address this, we introduce protected group networks and propose a novel fairness-modularity metric. This metric extends traditional modularity by explicitly incorporating fairness, and we prove that minimizing it yields naturally fair partitions for protected groups while maintaining theoretical soundness. We develop a general optimization framework for fairness partitioning and design the efficient Fair Fast Newman (FairFN) algorithm, enhancing the Fast Newman (FN) method to optimize both modularity and fairness. Experiments show FairFN achieves significantly improved fairness and high-quality partitions compared to state-of-the-art methods, especially on unbalanced datasets.

Recovering Fairness Directly from Modularity: a New Way for Fair Community Partitioning

TL;DR

This work addresses fairness in modularity-based community detection by introducing a protected group network and a fairness-modularity metric , which quantifies deviation from global protected-group distributions. Minimizing yields fair partitions, enabling a multi-objective optimization that preserves the standard modularity while promoting fairness. The authors propose FairFN, an efficient extension of the Fast Newman algorithm with a simple fairness constraint, achieving superior fairness (higher FR, lower AWD) and competitive modularity on synthetic and real datasets, including unbalanced cases, and capable of automatic stopping without pre-specifying the number of communities. The approach has practical impact for applications ranging from social networks to brain networks by reducing bias in detected communities while maintaining high-quality partitions.

Abstract

Community partitioning is crucial in network analysis, with modularity optimization being the prevailing technique. However, traditional modularity-based methods often overlook fairness, a critical aspect in real-world applications. To address this, we introduce protected group networks and propose a novel fairness-modularity metric. This metric extends traditional modularity by explicitly incorporating fairness, and we prove that minimizing it yields naturally fair partitions for protected groups while maintaining theoretical soundness. We develop a general optimization framework for fairness partitioning and design the efficient Fair Fast Newman (FairFN) algorithm, enhancing the Fast Newman (FN) method to optimize both modularity and fairness. Experiments show FairFN achieves significantly improved fairness and high-quality partitions compared to state-of-the-art methods, especially on unbalanced datasets.

Paper Structure

This paper contains 20 sections, 4 theorems, 30 equations, 5 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Method
  4. Experiments
  5. Datasets and Validation Measures
  6. Validation Parameters
  7. Experiments over LFR Networks
  8. Comparisons with Baseline Algorithms
  9. Discussions and Conclusions
  10. Proof of Lemma \ref{['thm_corr']}
  11. Proof of Theorem \ref{['thm-fm']}
  12. Properties of fairness-modularity
  13. Datasets details
  14. Parameters of algorithms
  15. More results in the LFR network
  16. ...and 5 more sections

Key Result

Lemma 3.2

Let $G_d(V,E)$ be a directed graph such that $(v_j, v_i) \in E$ only when $(v_i, v_j) \in E$. And define $k_i=k^{in}_i=k^{out}_i$ for any node $v_i$ and $m=\sum_i{k^{in}_i}$. The modularity of its community partition $\mathbb{C} =\{C_1,C_2,\cdots,C_k\}$ can be calculated by where $S$ is a community indicator matrix and $A$ is the adjacency matrix of $G_d$.

Figures (5)

  • Figure 1: Partitioning examples of protected group network. There are three protected groups, with the vertices in the same group represented by the same color. The vertices are divided into two communities represented by the dashed boxes.
  • Figure 2: Communities detected and protected groups of each community partition in the LFR network by different algorithms. The network is divided into five communities. Figure \ref{['fig_lfr_fn_net']} and \ref{['fig_lfr_fair_fn_net']} show the community partitioning results in different colors. Figure \ref{['fig_lfr_fn_pgnet']} and \ref{['fig_lfr_fair_fn_pgnet']} show two protected groups of each community detected by FN or FairFN, and each cluster is a community. Clearly, the proposed FairFN performs much better to achieve community fairness.
  • Figure 3: Experiment results in the given LFR network. The number of communities ranges from 5 to 1000 which is decreasing due to the merging process. The thresholds of $\alpha$ are calculated as $2m\Delta Q_{max}$, and $\alpha$ should exceed the thresholds if one wants to allow community merging to continue. The results show that FairFN obtained better fairness with a low impact on modularity.
  • Figure 4: The proportion of protected groups in different communities. The X-axis represents different protected groups in a community. The results show that the proportion of the protected group in communities detected by FairFN is nearly consistent with one in the whole network, which indicates that the fairness of FairFN is better than FN.
  • Figure 5: The threshold of $\alpha$ in synthetic clustering datasets and real-world datasets. The number of communities ranges from 100 to finish due to the threshold in the beginning stage being nearly zero.

Theorems & Definitions (8)

  • Remark 3.1
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • proof
  • Corollary C.1
  • proof
  • Proposition C.2