  Although most of the real networks contain a mixture of directed and
bidirectional (reciprocal) connections, the reciprocity $r$ has received little
attention as a subject of theoretical understanding. We study the expected
reciprocity of networks with an arbitrary degree sequence and a broad class of
degree correlations by means of statistical ensemble approach. We demonstrate
that degree correlations are crucial to understand the reciprocity in real
networks and a hierarchy of correlation contributions to $r$ is revealed.
Numerical experiments using novel network randomization methods show very good
agreement to our analytical estimations.
