Vector Calculus on Weighted Networks

We present here a vector calculus on weighted networks following the guidelines of Differential Geometry. The key to develop an efficient calculus on weighted networks which mimetizes the calculus in the smooth case is an adequate construction of the tangent space at each vertex. This allows to consider discrete vector fields, inner products and general metrics. Then, we obtain discrete versions of derivative, gradient, divergence, curl and Laplace-Beltrami operators, satisfying analogous properties to those verified by their continuous counterparts. Also we construct the De Rham cohomology of a weighted networks, obtaining in particular a Hodge decomposition theorem type. On the other hand we develop the corresponding integral calculus that includes the discrete versions of the Integration by Parts technique and Green’s Identities. As an application we study the variational formulation for general boundary value problems on weighted networks, obtaining in particular the discrete version of the Dirichlet Principle.