Multipolarization versus unification in community networks

Highlights

• We explore the core–periphery structures of communities in complex networks.

• A well performed method to find core–periphery structure in a community is proposed.

• We find two communities’ relationships in complex networks: unitive or multipolar.

• Generalized Girvan–Newman(GGN) model is proposed to generate community networks.

• We can generate both multipolar and unitive community networks separately by GGN.

Abstract

Community structure and core–periphery structure are two natural properties of complex networks. Both structures have been studied separately for decades. However, few researchers focus on the combination of these two important structures in complex networks. In this paper, we explore the core–periphery structures of communities in complex networks especially community networks, more precisely, we propose a linear algorithm to divide each community into a densely interconnected core and a periphery where the nodes are rarely linked to each other. Based on core–periphery structures, we perform quantitative analysis of the edges between different communities and find two relationships of two communities in real networks: unitive and multipolar. Communities are called unitive if edges between different cores are more than edges between different peripheries. Otherwise, communities are called multipolar. Furthermore, we propose a random model called Generalized Girvan–Newman(GGN) model, which can generate community networks where communities are either unitive or multipolar. The model sheds some new light on community formation and core–periphery structures in complex systems.

Introduction

Complex systems are not random [1], [2], but rather have structures that can be categorically referred to communities, where the nodes within a community are more densely connected than the nodes across two different communities [3]. Communities are one of the important network properties [3], [4] since they typically constitute functional units of a network. Identification of communities in complex systems has attracted much attention [5]. Recent studies have extended to discovering internal, fine structures of network communities. Two types of internal community structures have recently been proposed, the leader communities and self-organizing communities [6]. A network is called a community network if it has a fine community structure.

A distinct type of large-scale structure is core–periphery structure. Many networks are observed to divide into a densely interconnected core surrounded by a sparser halo or periphery [7]. The research of core–periphery structure can date back to 1990s, scientists found it in social networks [8] and many other different types of networks [9], [10], [11]. The role of core nodes may be different from that of the periphery ones [12], which inspires a lot of researchers to distinguish core from periphery in real networks [10].

A number of researchers have made lots of algorithms based on comparing a network to a block model. Plenty of models are used in these ways, for example, a model used by Borgatti and Everett that consists of a fully connected core and a periphery that has no internal edges but is fully connected to the core [8]. Rombach et al. follow on the same idea, but using a more flexible model [10]. In order to choosing nodes to be in cores, Holme defined a core–periphery coefficient [9], Da Silva, Ma, and Zeng introduced a measure of connectivity known as network capacity [13], and Lee et al. made use of centrality measures based on notions of local density and transport in networks [14]. Zhang et al. proposed an statistically principled method using a maximum-likelihood fit to a generative network model [7].

However, few researchers focus on the combination of community structure and core–periphery structure in a study simultaneously. In our viewpoint, a community is a subnetwork, which should also have a core–periphery structure. For example, several communities may exist in a friendship network. In each of them, active persons are more likely to cluster together and form the core, while the rest nodes of the community (inactive persons) rarely trend to be friends of each other and constitute the periphery. What interests us is the relationship between different communities, are they unitive or multipolar? If edges between two different cores of communities in a friendship network are more than edges between two different peripheries, the two communities must be friendly to each other because many active members of both communities are friends. Otherwise, the two communities may be polar, because inactive persons from different communities are more likely to be friends rather than cores persons. Therefore, for two communities, if edges between two cores are more than edges between two peripheries, we call that these two communities are unitive. Otherwise, they are called multipolar.

In this paper, we propose a linear algorithm to identify core–periphery structures in communities. It is applied to three real-world networks, the Zachary’s karate club network, the Lusseau’s bottlenose dolphins network and the Political blogs network. Then we discover two relationships between different communities: unitive and multipolar. For two communities, if edges between two cores are more than edges between two peripheries, then the two communities are called unitive. Otherwise, they are called multipolar. Both kinds of relationships between two communities are found in real-world networks. Furthermore, we propose a Generalized Girvan–Newman(GGN) model based on the original Girvan–Newman(GN) [3] model. A new parameter is introduced in our GGN model, which makes the model not only generate networks in which communities are unitive but also generate networks in which communities are multipolar.

The paper is organized as follows. Section 2 gives the linear algorithm of identifying core–periphery structure in a community, along with the application on real-world networks. Section 3 introduces our Generalized Girvan–Newman(GGN) model and the corresponding analysis. Section 4 closes this paper with conclusions.