  We introduce a program aimed to studying problems arising from the theory of
complex networks with differential geometric means. We study the propagation of
influences on manifolds assuming that at each point only a finite number of
propagation velocities are allowed. This leads to the computation of the volume
of the moduli spaces of directed paths, i.e. paths satisfying the imposed
tangential restrictions. The proposed settings provide a fertile ground for
research with potential applications in geometry, mathematical physics,
differential equations, and combinatorics. We establish the general framework,
develop its structural properties, and consider a few basic examples of
relevance. The interaction between differential geometry and complex networks
is a new and promising field of study.
