  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.
