<p>Community structure detection is a key ingredient for studying real-world networks across fields like physical sciences, medical sciences, social sciences, and technology. There are numerous algorithms for this task. Many quality measures have been developed to assess different types of community structures, whether disjoint, overlapping, or hierarchical. Unfortunately, many of these metrics face limitations, such as resolution issues, unable to consider edge weights or being applicable only to disjoint community structures. Despite this, there are not much efforts to study the relative performance or behaviour of these measures. In this paper, we analyse and rank 12 existing quality measures based on their performance. A critical examination reveals that the widely used weighted overlapping modularity, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41060_2025_791_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$Q_{wo}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mrow> <mi mathvariant="italic">wo</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, achieves the first rank. However, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41060_2025_791_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$Q_{wo}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mrow> <mi mathvariant="italic">wo</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> also suffers with some limitations. Consequently, we introduce a novel quality measure, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41060_2025_791_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$Q_{wo}^{*}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>Q</mi> <mrow> <mi mathvariant="italic">wo</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation>, specifically designed to evaluate overlapping community structures in undirected weighted networks. Experimental results across a wide range of artificial and real-world networks demonstrate that the proposed measure outperforms current metrics, providing a more accurate and robust evaluation of overlapping community structures.</p>

A new modularity metric for disjoint and overlapping community structure evaluation in weighted complex networks

期刊论文
发表日期：

2025年5月17日

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Anjali Kumari,
Abhinav Kumar,
Pawan Kumar

摘要

Community structure detection is a key ingredient for studying real-world networks across fields like physical sciences, medical sciences, social sciences, and technology. There are numerous algorithms for this task. Many quality measures have been developed to assess different types of community structures, whether disjoint, overlapping, or hierarchical. Unfortunately, many of these metrics face limitations, such as resolution issues, unable to consider edge weights or being applicable only to disjoint community structures. Despite this, there are not much efforts to study the relative performance or behaviour of these measures. In this paper, we analyse and rank 12 existing quality measures based on their performance. A critical examination reveals that the widely used weighted overlapping modularity, $Q_{wo}$ $Q_{wo}$ , achieves the first rank. However, $Q_{wo}$ $Q_{wo}$ also suffers with some limitations. Consequently, we introduce a novel quality measure, $Q_{wo}^{*}$ $Q_{wo}^{*}$ , specifically designed to evaluate overlapping community structures in undirected weighted networks. Experimental results across a wide range of artificial and real-world networks demonstrate that the proposed measure outperforms current metrics, providing a more accurate and robust evaluation of overlapping community structures.