Harmonic Functions and Inverse Conductivity Problems on Networks

Abstract

In this paper, we discuss the inverse problem of identifying the connectivity and the conductivity of the links between adjacent pair of nodes in a network, in terms of an input-output map. To do this we introduce an elliptic operator Dw and an w-harmonic function on thegraph, with its physical interpretation been the diffusion equation on the graph, which models an electric network. After deriving the basic properties of w-harmonic functions, we prove the solvability of (direct) problems such as the Dirichlet and Neumann boundary value problems.Our main result is the global uniqueness of the inverse conductivity problem for a network under a suitable monotonicity condition.