Although widely used in practice, the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood  . Here, we present a broad and systematic characterization of its performance in practical situations. First, we generalize and clarify the recently identified resolution limit phenomenon  . Second, we show that the modularity function Q exhibits extreme degeneracies: that is, the modularity landscape admits an exponential number of distinct high-scoring solutions and does not typically exhibit a clear global maximum. Third, we derive the limiting behavior of the maximum modularity Q_max for  infinitely modular networks, showing that it depends strongly  on the size of the network and  the number of module-like subgraphs it contains. Finally, using three real-world examples of metabolic networks , we show that the degenerate solutions can fundamentally disagree on the composition of even the largest modules. Together, these results significantly extend and clarify our understanding of this popular method. In particular, they explain why so manyheuristics perform well in practice at finding high-scoring partitions, why these heuristics can disagree on the composition of the identified modules , and how the estimated value of Q_max should be interpreted. Further, they imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts.  We conclude by discussing avenues for mitigating  these behaviors, such as combining information from many degenerate solutions or using generative models.
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 Q_max 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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in practical contexts
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and systematic
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practical
such
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generalize
revisit
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recently identified
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for modularity maximization
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that is, the modularity landscape
it typically
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does not typically exhibit
typically lacks
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one model of
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both
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on
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module-like subgraphs
modules
R
examples of metabolic networks
metabolic networks as examples
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the composition of even the largest modules. Together, these results significantly extend and clarify our understanding of this popular method. In particular, they explain why so manyheuristics perform well in practice at finding high-scoring partitions, why these heuristics can disagree on
many, but not all, partition properties such as
R
identified modules , and how the estimated value of Q_max should be interpreted. Further, they
largest modules and the distribution of module sizes. These results
A
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.
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some of