  We study a process of pattern formation for a generic model of species
anchored to the nodes of a network where local reactions take place, and that
experience non-reciprocal long-range interactions, encoded by the network
directed links. By assuming the system to exhibit a stable homogeneous
equilibrium whenever only local interactions are considered, we prove that such
equilibrium can turn unstable once suitable non-reciprocal long-range
interactions are allowed for. Stated differently we propose sufficient
conditions allowing for patterns to emerge using a non-symmetric coupling,
while initial perturbations about the homogenous equilibrium fade away assuming
reciprocal coupling. The instability, precursor of the emerging spatio-temporal
patterns, can be traced back, via a linear stability analysis, to the complex
spectrum of an interaction non-symmetric Laplace operator. Taken together, our
results pave the way for the understanding of the many and heterogeneous
patterns of complexity found in ecological, chemical or physical systems
composed by interacting parts, once no diffusion takes place.
