We show that a general context of weighted graphs G induces a dual pair of Hermitian representations of algebras constructed from G, acting on Hilbert spaces. In particular, we are interested in two Hilbert spaces: There is a Hilbert space of finite energy functions on the vertices of G, and a second Hilbert space modeling dissipation in systems of resistors in current network flows, electric resistance networks (ERNs). To realize these algebras, we make use of ideas from both the theory of ERNs and free probability theory. For the algebras, we give realizations of the representations. We show that there are regular representations in each of the two Hilbert spaces, exhibiting an intrinsic duality. We study their connection to free random variables, and we establish freeness properties for some of the representations.