  Models describing transport and diffusion processes occurring along the edges
of a graph and interlinked by its vertices have been recently receiving a
considerable attention. In this paper we generalize such models and consider a
network of transport or diffusion operators defined on one dimensional domains
and connected through boundary conditions linking the end-points of these
domains in an arbitrary way (not necessarily as the edges of a graph are
connected). We prove the existence of $C_0$-semigroups solving such problems
and provide conditions fully characterizing when they are positive.
