Geometry-free renormalization of directed networks: scale-invariance and reciprocity

Recent research has tried to extend the concept of renormalization, which is naturally defined for geometric objects, to more general networks with arbitrary topology. The current attempts do not naturally apply to directed networks, for instance because they are based on the identification of (necessarily symmetric) inter-node distances arising from geometric embeddings or on the definition of Hermitian Laplacian operators requiring symmetric adjacency matrices in spectral approaches. Here we show that the Scale-Invariant Model, recently proposed as an approach to consistently model undirected networks at arbitrary (and possibly multi-scale) resolution levels, can be extended coherently to directed networks without the requirement of an embedding geometry or Laplacian structure. Moreover, it can account for nontrivial reciprocity, i.e. the empirically well-documented tendency of links to occur in mutual pairs more or less often than predicted by chance. After deriving the renormalization rules for networks with arbitrary reciprocity, we consider various examples. In particular we propose the first multiscale model of the international trade network with nontrivial reciprocity and an annealed model where positive reciprocity emerges spontaneously from coarse-graining. In the latter case, the renormalization process defines a group, not only a semigroup, and therefore allows to fine-grain networks to arbitrarily small scales. These results strengthen the notion that network renormalization can be defined in a much more general way than required by geometric or spectral approaches, because it needs only node-specific (metric-free) features and can coexist with the asymmetry of directed networks.