IT University of Copenhagen, Copenhagen, 2300, Denmark.
InfoLab, Department of Information Technology, Uppsala University, Uppsala, Sweden.
Sci Rep. 2020 Jul 6;10(1):11053. doi: 10.1038/s41598-020-66956-0.
A principled approach to recover communities in social networks is to find a clustering of the network nodes into modules (i.e groups of nodes) for which the modularity over the network is maximal. This guarantees partitioning the network nodes into sparsely connected groups of densely connected nodes. A popular extension of modularity has been proposed in the literature so it applies to multi-layer networks, that is, networks that model different types/aspects of interactions among a set of actors. In this extension, a new parameter, the coupling strength ω, has been introduced to couple different copies (i.e nodes) of the same actor with specific weights across different layers. This allows two nodes that refer to the same actor to reward the modularity score with an amount proportional to ω when they appear in the same community. While this extension seems to provide an effective tool to detect communities in multi-layer networks, it is not always clear what kind of communities maximising the generalised modularity can identify in multi-layer networks and whether these communities are inclusive to all possible community structures possible to exist in multi-layer networks. In addition, it has not been thoroughly investigated yet how to interpret ω in real-world scenarios, and whether a proper tuning of ω, if exists, is enough to guarantee an accurate recoverability for different types of multi-layer community structures. In this article, we report the different ways used in the literature to tune ω. We analyse different community structures that can be recovered by maximising the generalised modularity in relation to ω. We propose different models for multi-layer communities in multiplex and time-dependent networks and test if they are recoverable by modularity-maximization community detection methods under any assignment of ω. Our main finding is that only few simple models of multi-layer communities in multiplex and time-dependent networks are recoverable by modularity maximisation methods while more complex models are not accurately recoverable under any assignment of ω.
一种从社交网络中恢复社区的原则方法是找到网络节点的聚类成模块（即节点组），其中网络的模块度最大化。这保证了将网络节点分割成稀疏连接的密集连接节点组。文献中提出了一种流行的模块度扩展，因此它适用于多层网络，即模型化一组参与者之间不同类型/方面的相互作用的网络。在这个扩展中，引入了一个新的参数，即耦合强度ω，用于用特定的权重在不同的层之间连接相同参与者的不同副本（即节点）。这允许当两个节点出现在同一社区中时，通过它们在同一社区中的出现，以与ω成比例的量来奖励模块度得分。虽然这种扩展似乎提供了一种有效的工具来检测多层网络中的社区，但并不总是清楚在多层网络中最大化广义模块度可以识别什么样的社区，以及这些社区是否包括多层网络中可能存在的所有可能的社区结构。此外，尚未彻底研究如何在实际场景中解释ω，以及如果存在适当的ω调整，是否足以保证不同类型的多层社区结构的准确可恢复性。在本文中，我们报告了文献中用于调整ω的不同方法。我们分析了与ω相关的最大化广义模块度可以恢复的不同社区结构。我们提出了多层社区的不同模型在多路复用和时变网络中，并测试它们是否可以通过模块度最大化社区检测方法在任何ω分配下恢复。我们的主要发现是，只有少数简单的多层社区模型在多路复用和时变网络中可以通过模块度最大化方法恢复，而更复杂的模型在任何ω分配下都无法准确恢复。
