Reciprocity of Networks with Degree Correlations and Arbitrary Degree Sequences

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.