A universal assortativity measure for network analysis

Characterizing the connectivity tendency of a network is a fundamental problem in network science. The traditional and well-known assortativity coefficient is calculated on a per-network basis, which is of little use to partial connection tendency of a network. This paper proposes a universal assortativity coefficient(UAC), which is based on the unambiguous definition of each individual edge's contribution to the global assortativity coefficient (GAC). It is able to reveal the connection tendency of microscopic, mesoscopic, macroscopic structures and any given part of a network. Applying UAC to real world networks, we find that, contrary to the popular expectation, most networks (notably the AS-level Internet topology) have markedly more assortative edges/nodes than dissortaive ones despite their global dissortativity. Consequently, networks can be categorized along two dimensions--single global assortativity and local assortativity statistics. Detailed anatomy of the AS-level Internet topology further illustrates how UAC can be used to decipher the hidden patterns of connection tendencies on different scales.