  Directed networks are ubiquitous and are necessary to represent complex
systems with asymmetric interactions---from food webs to the World Wide Web.
Despite the importance of edge direction for detecting local and community
structure, it has been disregarded in studying a basic type of global diversity
in networks: the tendency of nodes with similar numbers of edges to connect.
This tendency, called assortativity, affects crucial structural and dynamic
properties of real-world networks, such as error tolerance or epidemic
spreading. Here we demonstrate that edge direction has profound effects on
assortativity. We define a set of four directed assortativity measures and
assign statistical significance by comparison to randomized networks. We apply
these measures to three network classes---online/social networks, food webs,
and word-adjacency networks. Our measures (i) reveal patterns common to each
class, (ii) separate networks that have been previously classified together,
and (iii) expose limitations of several existing theoretical models. We reject
the standard classification of directed networks as purely assortative or
disassortative. Many display a class-specific mixture, likely reflecting
functional or historical constraints, contingencies, and forces guiding the
system's evolution.
