Community detection, a fundamental problem in computational sciences, finds
applications in various domains. Heuristics are often employed to detect
communities through maximizing an objective function, modularity, over
partitions of network nodes. Our research delves into the performance of
different modularity maximization algorithms in achieving optimal partitions.
We use 104 networks, comprising real-world instances from diverse contexts and
synthetic graphs with modular structures. We analyze ten inexact
modularity-based algorithms against an exact baseline which is an exact integer
programming method that globally optimizes modularity. The ten algorithms
analyzed include eight heuristics, two variations of a graph neural network
algorithm, and several variations of the Bayan approximation algorithm. Our
analysis uncovers substantial dissimilarities between the partitions obtained
by most commonly used modularity-based methods and any optimal partition of the
networks, as indicated by both adjusted and reduced mutual information metrics.
Importantly, our results show that near-optimal partitions are often
disproportionately dissimilar to any optimal partition. Taken together, our
analysis points to a crucial limitation of the commonly used unguaranteed
modularity-based methods for discovering communities: they rarely produce an
optimal partition or a partition resembling an optimal partition even on
networks with modular structures. If modularity is to be used for detecting
communities, approximate optimization algorithms are recommendable for a more
methodologically sound usage of modularity within its applicability limits.

分析表明，常用的基于模块性的方法在发现社区结构时很少产生最优划分或最接近最优划分的划分，因此，推荐使用近似优化算法以在方法适用范围内更加方法论严谨地使用模块性。