Null Model and Community Structure in Multiplex Networks

Xuemeng Zhai, Wanlei Zhou, Gaolei Fei, Weiyi Liu, Zhoujun Xu, Chengbo Jiao, Cai Lu, Guangmin Hu
2018 Scientific Reports  
The multiple relationships among objects in complex systems can be described well by multiplex networks, which contain rich information of the connections between objects. The null model of networks, which can be used to quantify the specific nature of a network, is a powerful tool for analysing the structural characteristics of complex systems. However, the null model for multiplex networks remains largely unexplored. In this paper, we propose a null model for multiplex networks based on the
more » ... de redundancy degree, which is a natural measure for describing the multiple relationships in multiplex networks. Based on this model, we define the modularity of multiplex networks to study the community structures in multiplex networks and demonstrate our theory in practice through community detection in four real-world networks. The results show that our model can reveal the community structures in multiplex networks and indicate that our null model is a useful approach for providing new insights into the specific nature of multiplex networks, which are difficult to quantify. Network science is a fundamental tool for modelling and analysing complex systems 1-3 . The general theories and approaches that have emerged from network science have provided guidelines and resulted in applications for analysis of the objects in the systems 4-6 . Therefore, research on the quantitative and qualitative features of network science has always been a focus for improving the scientific understanding of complex systems 7-10 . Because network models capture the common features of complex systems, many network models have been proposed to study the modelling of real-world systems 11-13 . These single-network models provide a general framework of systems from different fields such as social science 14 , Internet topology 15 , bioscience 16 , engineering 17 , economics 18 , education 19 , and so on. In network science, null models are especially notable because they reveal important network properties that could not be directly quantified due to the complexity of the studied systems 20,21 . The null model concept was proposed by Maslov and Senppen 22 and consists of a network that matches one specific graph in some of its structural features but that is otherwise taken to be a random network instance. The null model is used in comparisons to quantify complex network properties such as community structure 23,24 , assortativity 25,26 , degree correlation 27 , epidemic spreading rate 28 , motif identification 29,30 , routing efficiency 31 , pattern detection 32 , microbial diversification 33 , etc.-all of which have been shown to be significant in various complex networks. Therefore, the null model of single networks has been a powerful tool over the past few decades in analysing the nature of modelling, structures and dynamics of complex networks 34-36 . However, the limitations of single networks have become increasingly evident over the past few years since the mass emergence of complex systems with multiple interaction layers, which are almost impossible to represent using isolated networks. Multiple relationships among objects give rise to multiplex networks in real-world systems that consist of multiple layers 37-39 . In such networks, all the relationship types are constrained by the same objects and are therefore not completely independent. Thus, each type of relationship among nodes can be described in each layer of the multiplex networks, and each network layer contains the same set of nodes. Examples of such multiplex systems include social networks involving multiple relationships from different social platforms such as Twitter, YouTube and Facebook 40 , epidemic networks with multiple diseases 41 , and Internet topologies with multiple levels from the route level to the AS level 42 . Therefore, multiplex networks, including multilayer networks 43,44 , multiscale networks 45,46 , and time-dependent networks 47-49 , are a general framework for modelling and analysing the new phenomena emerging from these multi-layered systems. The research on multiplex networks, including community detection 50 , link prediction 51 , epidemic spreading 41 , controllability 52 , Published: xx xx xxxx OPEN www.nature.com/scientificreports/ 2 SCIEnTIfIC REPORTS | (2018) 8:3245 | synchronization 53 , and network evolution 54 , has illustrated that obvious differences exist between multiplex networks and isolated networks. For example, the synchronization state of the entire system is influenced by each layer; thus, a global unstable state may be caused by the interactions among various stable layers 55 . It is not possible to build the null model of single network for each layer of network separately because each layer is interrelated. However, the null model of multiplex networks remains unexplored, as there are few effective stochastic models that can be used to quantify the specific nature of multiplex networks. In multiplex networks, the rich node connection information leads to redundancies in the networks, meaning that edges between the same pair of nodes could appear repeatedly in different network layers 56 . Nodes with many repeated edges are more likely to belong to the same community. For example, close friends may contact each other using different social networks such as WeChat, Twitter, and Facebook; intuitively such nodes potentially belonging to the same community. Without redundancy, the connection tightness between objects in multiplex networks could not be represented effectively and accurately. Moreover, edge redundancy leads to node redundancy in multiplex networks. The node degree of a single network cannot be used in a multiplex network due to this redundancy. Therefore, a new measure is needed to replace the node degree in multiplex networks to constrain the null model of multiplex networks. In this paper, we propose a new general measure of nodes to fill this gap and generate a novel Null Model with Redundancy (NMR) for multiplex networks. Our goal is to describe the redundant connection relationships among nodes and provide a general framework to quantify the specific nature of multiplex networks. To achieve this, two measures, the Node Redundancy Degree (NRD) and Edge RedundanCy (ERC), are calculated based on the redundancies in multiplex networks. We build the NMR with the same NRD that exists in the original multiplex network through a configuration method. The NMR can also be explained using the traditional random-walk method. The final result is a model with an explicit edge probability under Laplacian dynamics that provides new insight into the specific nature of multiplex networks, which are difficult to quantify. Our model requires no preconditions on the systems and applies to both directed and undirected systems. We demonstrate the performance of our model by building the modularity 57 of multiplex networks to study the community structure. The experimental results show that the community structure of multiplex networks can effectively be exposed through the NMR. Our findings fill the gap in the field of null modelling of multiplex networks and provide a powerful tool for modelling and analysing complex systems with multiple relationships in many general scientific fields.
doi:10.1038/s41598-018-21286-0 pmid:29459696 pmcid:PMC5818485 fatcat:gh6nrgyz5felti7pvy4tjlctvi