We consider complex-weighted graphs, whose edge weights belong to a sector in the complex plane. We show that the corresponding Dirichlet Laplacian is $m$-sectorial, and, hence, generates a contractive holomorphic $C_0$-semigroup. Further, it is shown that every sectorial complex-weighted graph can be extended to an electrical network, where by electrical networks we mean graphs, whose edge weights are holomorphic functions, arising from physical admittances. This result allows us to establish convergence results for the infinite complex-weighted graphs, e.g. convergence of solutions of Dirichlet problems and convergence of complex-valued capacities on a finite exhaustion. Finally, we define a recurrence for complex-weighted graphs, and, using the convergence results, give its characterizations in terms of functional spaces, capacity, Green's function, resolvents of the Dirichlet Laplacian and properties of the Neumann Laplacian.