Reciprocity of weighted networks
  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. 
Tiziano Squartini, Francesco Picciolo, Franco Ruzzenenti, Diego   Garlaschelli
physics.data-an cs.SI physics.soc-ph