发表机构
Delft University of Technology; KairosHelix, Inc.（代尔夫特理工大学; KairosHelix公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对多重网络中的模块度社区检测，分析比较了早期融合、同时融合和晚期融合三种策略的目标函数，揭示了它们之间的显式关系，并通过实验验证了理论结果。
AI 中文摘要
多重网络中的基于模块度的社区检测通常通过三种策略之一来处理：早期融合（EF），即首先将各层聚合为单个网络，然后应用社区检测；同时融合（SF），即在模块度优化过程中结合各层的信息；以及晚期融合（LF），即结合在每一层中分别检测社区所获得的社区分配结果。尽管近期的分类学和综述在概念上区分了这些策略，但它们的客观函数很少被分析性地比较。我们针对具有归一化非负边权重和层权重、一个公共分辨率参数以及无层间边的共享节点多重网络提供了这样的比较。节点属性网络由拓扑层和属性相似性层表示。我们对这三种表述使用统一的符号，并推导出它们的目标函数之间的显式关系。特别是，我们表明EF目标等于SF目标加上一个非负的异质性项。我们证明了SF目标在单纯形顶点处存在最优层权重向量，尽管在并列情况下也可能出现非顶点最优解，而EF目标对于每个固定划分关于层参数是凹的，并且其联合最优解可能出现在单纯形的内部。对于LF方法，所得的目标函数和结论在很大程度上取决于瞬态图是如何构建的。在本研究中，我们仅关注这种LF方法中的一种。我们通过合成多重网络上的暴力实验和真实世界网络（包括表示为多重网络的节点属性网络）上的启发式实验来补充分析结果。这些实验说明了分析结果以及启发式优化器对观察到的比较的影响。
英文摘要
Modularity-based community detection in multiplex networks is commonly approached through one of three strategies: early fusion (EF), which first aggregates the layers into a single network and then applies community detection; simultaneous fusion (SF), which combines information from the layers during modularity optimization; and late fusion (LF), which combines the community assignments obtained by detecting communities separately in each layer. Although recent taxonomies and surveys distinguish these strategies conceptually, their objective functions have rarely been compared analytically. We provide such a comparison for shared-node multiplex networks with normalized non-negative edge and layer weights, a common resolution parameter, and no interlayer edges. Node-attributed networks are represented by topology and attribute-similarity layers. We use a common notation for the three formulations and derive explicit relationships between their objective functions. In particular, we show that the EF objective equals the SF objective plus a non-negative heterogeneity term. We prove that the SF objective admits an optimal layer-weight vector at a simplex vertex, although nonvertex optima may also occur under ties, whereas the EF objective is concave with respect to the layer parameter for each fixed partition, and its joint optimum may occur in the interior of the simplex. For LF methods, the resulting objective and conclusions depend heavily on how the transient graph is constructed. In this study, we focus only on one of such LF methods. We complement the analytical results with brute-force experiments on synthetic multiplexes and heuristic experiments on real-world networks, including node-attributed networks represented as multiplexes. The experiments illustrate the analytical results and the influence of heuristic optimizers on the observed comparisons.
CommentsCode and reproduction materials: https://gitlab.com/aipetr/modularity_objective_functions
