Degree difference: a simple measure to characterize structural heterogeneity in complex networks

Amirhossein Farzam,Jürgen Jost,Areejit Samal

doi:10.1038/s41598-020-78336-9

Abstract

Despite the growing interest in characterizing the local geometry leading to the global topology of networks, our understanding of the local structure of complex networks, especially real-world networks, is still incomplete. Here, we analyze a simple, elegant yet underexplored measure, ‘degree difference’ (DD) between vertices of an edge, to understand the local network geometry. We describe the connection between DD and global assortativity of the network from both formal and conceptual perspective, and show that DD can reveal structural properties that are not obtained from other such measures in network science. Typically, edges with different DD play different structural roles and the DD distribution is an important network signature. Notably, DD is the basic unit of assortativity. We provide an explanation as to why DD can characterize structural heterogeneity in mixing patterns unlike global assortativity and local node assortativity. By analyzing synthetic and real networks, we show that DD distribution can be used to distinguish between different types of networks including those networks that cannot be easily distinguished using degree sequence and global assortativity. Moreover, we show DD to be an indicator for topological robustness of scale-free networks. Overall, DD is a local measure that is simple to define, easy to evaluate, and that reveals structural properties of networks not readily seen from other measures.

Highlights

Despite the growing interest in characterizing the local geometry leading to the global topology of networks, our understanding of the local structure of complex networks, especially real-world networks, is still incomplete
In addition to explaining the connection between degree difference’ (DD) and global assortativity (GA), Eq (16) further clarifies that we can compute GA and local node assortativity (LNA) using DD and excess degrees, while DD cannot be deduced from GA and LNA. We demonstrate this remark in supplementary information (SI) Figure S1 where we show DD distribution in an ensemble of BA networks as the network is rewired to increase its GA, and in SI Figure S2 where we set a constraint on GA of ensembles of ER and BA networks and show DD distribution after two random independent rewirings
Unravelling the structure of complex networks is a key interest since the rise of network science

Summary

Introduction

Despite the growing interest in characterizing the local geometry leading to the global topology of networks, our understanding of the local structure of complex networks, especially real-world networks, is still incomplete. It is important to investigate local measures and their distributions in complex networks.

Results

Conclusion