The structure of communities in scale-free networks

Scale-free networks are often used to model a wide range of real-world networks, such as social, technological, and biological networks. Understanding the structure of scale-free networks evolves into a big data problem for business, management, and protein function prediction. In the past decade, there has been a surge of interest in exploring the properties of scale-free networks. Two interesting properties have attracted much attention: the assortative mixing and community structure. However, these two properties have been studied separately in either theoretical models or real-world networks. In this paper, we show that the structural features of communities are highly related with the assortative mixing in scale-free networks. According to the value of assortativity coefficient, scale-free networks can be categorized into assortative, disassortative, and neutral networks, respectively. We systematically analyze the community structure in these three types of scale-free networks through six metrics: node embeddedness, link density, hub dominance, community compactness, the distribution of community sizes, and the presence of hierarchical communities. We find that the three types of scale-free networks exhibit significant differences in these six metrics of community structures. First, assortative networks present high embeddedness, meaning that many links lying within communities but few links lying between communities. This leads to the high link density of communities. Second, disassortative networks exhibit great hubs in communities, which results in the high compactness of communities that nodes can reach each other via short paths. Third, in neutral networks, a big portion of links act as community bridges, so they display sparse and less compact communities. In addition, we find that (dis)assortative networks show hierarchical community structure with power-law-distributed community sizes, while neutral networks present no hierarchy. Understanding the structure of communities from the angle of assortative mixing patterns of nodes can provide insights into the network structure and guide us in modeling information propagation in different categories of scale-free networks.