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Performance of modularity maximization in practical contexts
Open accessAbstract
Although widely used in practice, the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts. Here, we present a broad characterization of its performance in such situations. First, we revisit and clarify the resolution limit phenomenon for modularity maximization. Second, we show that the modularity function Q exhibits extreme degeneracies: it typically admits an exponential number of distinct high-scoring solutions and typically lacks a clear global maximum. Third, we derive the limiting behavior of the maximum modularity Qmax for one model of infinitely modular networks, showing that it depends strongly both on the size of the network and on the number of modules it contains. Finally, using three real-world metabolic networks as examples, we show that the degenerate solutions can fundamentally disagree on many, but not all, partition properties such as the composition of the largest modules and the distribution of module sizes. These results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts. They also explain why many heuristics are often successful at finding high-scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network. We conclude by discussing avenues for mitigating some of these behaviors, such as combining information from many degenerate solutions or using generative models.
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“One recent study has shown that, even in relatively small networks, a large number of good solutions exist for the standard modularity criterion [51], [52].”
From Spatial Guilds in the Serengeti Food Web Revealed by a Bayesian Group Model (2011) · cites this paper as [52]
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“However, as of today, a common method for dealing with this degeneracy is to examine statistics averaged over the 100 repetitions of the optimization, with equal weight being placed on each solution, rather than just on the unique solutions.”
From Dynamic reconfiguration of frontal brain networks during executive cognition in humans (2015) · cites this paper as [51]
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Metadata
- Authors
- Good, B. H., de Montjoye, Y.-A., & Clauset, A.
- Affiliations
- Swarthmore College (US)
- Santa Fe Institute (US)
- Venue
- Physical Review E
- Publisher
- American Physical Society (APS)
- Published
- 2010-04-15
- Volume
- 81
- Issue
- 4
- Pages
- 046106-046106
- ISSN
- 1539-3755 (ISSN-L)
- 1550-2376
- OpenAlex
- W2061099285
- Cited by
- 937
- References
- 66
- Citations by year
- 2026: 18 · 2025: 38 · 2024: 36 · 2023: 36 · 2022: 35 · 2021: 45 · 2020: 58 · 2019: 57 · 2018: 94 · 2017: 76 · 2016: 59 · 2015: 71 · 2014: 80 · 2013: 79 · 2012: 70
- FWCI
- 52.48
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review E (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Digital Access to Libraries (Université catholique de Louvain (UCL), l'Université de Namur (UNamur) and the Université Saint-Louis (USL-B)) (submittedVersion, other-oa)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Bioinformatics and Genomic Networks
- Physical Sciences › Computer Science › Artificial Intelligence › Advanced Clustering Algorithms Research
- Keywords
- modularity optimization, resolution limit, network module identification, degenerate solutions, modularity function, metabolic networks, community detection
- MeSH terms
- Models, Theoretical
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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