Farzam, Amirhossein; Areejit Sanal and Jurgen Jost

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 present a simple, elegant yet unexplored measure, ‘degree difference’ (DD) between vertices of an edge, to quantify the local network geometry. We 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 atomic root of assortativity. We provide an explanation why DD can characterize structural heterogeneity in mixing patterns unlike global assortativity and local node assortativity. By analyzing model 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.