  All types of networks arise as intricate combinations of dyadic building
blocks formed by pairs of vertices. In directed networks, the dyadic patterns
are entirely determined by reciprocity, i.e. the tendency to form, or to avoid,
mutual links. Reciprocity has dramatic effects on every networks dynamical
processes and the emergence of structures like motifs and communities. The
binary reciprocity has been extensively studied: that of weighted networks is
still poorly understood. We introduce a general approach to it, by defining
quantities capturing the observed patterns (from dyad-specific to
vertex-specific and network-wide) and introducing analytically solved models
(Exponential Random Graphs-type). Counter-intuitively, the previous reciprocity
measures based on the similarity of the mutual links-weights are uninformative.
By contrast, our measures can classify different weighted networks, track the
temporal evolution of a networks reciprocity, identify patterns. We show that
in some networks the local reciprocity structure can be inferred from the
global one.
