Co-community Structure in Time-varying Networks

Networks consisting of vertices and edges connecting some pairs of vertices are powerful abstractions of relational data, hence have become very popular tools in many fields including sociology, biology and physics [1]. The characteristic of community structure in networks, i.e., networks are naturally divided into modules or communities, has attracted huge attention in the past decade which can provide insights into the structure and dynamic formation of networks. Many methods for community detection in one network have been developed and studied even including the fuzzy community structure identification problem [2] and the more challenging community detection problem in directed networks [3] (see Ref. [4] for recent comprehensive reviews). However, previous studies have concentrated on uncovering community structure in a static network which only represents a summarized picture of all possible relations. A typical example is the protein interaction network in biology which represent all proteins of an organism and all interactions regardless of the conditions and time under which interactions may take place [5]. In reality, most of relationships modeled by networks evolve with time or conditions [6]. Several recent studies have touched on the analysis of dynamic networks including analyzing changes of global properties, detecting anomalous changes, mining dynamic frequent subnets, and discovering similar evolving regions in evolving networks [7] and even the dynamic communities by combining the information of communities in each network using traditional community detection methods. However, the community structure in two or more slices of a series of time-varying networks has not been well addressed directly [8, 9]. In this report, we propose the concept of co-community structure in two or more networks of a series of timevarying networks. The basic assumption is that an essential and common community structure may exist in two or more networks, and local dynamic changes may happen. This is very realistic in time-varying networks of many robust systems. Suppose that we are given the structure of two or