Comparison of Inter-Layer Couplings of Multilayer Networks

Abstract

There are many real-world interactions that can be naturally represented as temporal networks, such as communications in social media and transportation with urban transport facilities. As one of the approaches for analyzing temporal networks, they are often represented as multilayer networks composed of many single-layer networks sharing the same set of nodes. Edges in multilayer networks are divided into two category: intra-layer networks within each layer and inter-layer edges between vertices of different layers. One of the advantages of representing temporal networks as multilayer networks is that each layer is regarded as a snapshot of dynamic interactions and is useful for visualization. Another advantage is that there are some tools for analyzing multilayer networks (such as MuxViz or GenLouvain). On the other hand, many previous research on multilayer networks focus on collection of intra-layer networks, so more discussions are required for inter-layer connections. For example, Mucha et al. propose extended modularity for multilayer networks. Their modularity basically focus on two types of inter-layer connections: ordinal coupling and categorical coupling. Only the corresponding nodes of adjacent layers are connected in ordinal coupling, and all corresponding nodes in each layers are connected in categorical coupling. In order to detect communities, optimization of Mucha's modularity is often employed. However, the modularity is based on an assumption that inter-layer coupling are either ordinal or categorical, which are not enough for representing of impacts of interactions, which is important for representing temporal networks with multilayer networks. This paper proposes generalization of inter-layer couplings of multilayer networks. The following inter-layer connections are attempted for detecting communities in multilayer networks: (i) rectangular function, (ii) constant function, (iii) exponential decay function and (iv) Gaussian function. Based on our framework, community detection of multilayer networks of several inter-layer connections is performed. Experimental results show that the properties of inter-layer couplings are crucial for the stability of detected communities.