On the performance of damaged linear networks

Abstract

The conductivity of a structured or unstructured, finite or infinite linear network consisting of nodes connected by links of varying conductance is discussed. The rate of transport of mass, heat, electricity, analogue signal, or digitized information across each link is given by the product of the link conductance and the difference in the corresponding nodal values of an appropriate driving potential. Balance equations at each node provide us with a system of linear equations whose coefficients depend on the link conductances. Link damage or disruption causes a perturbation nodal field that affects the performance of the entire network. An expression for the perturbation field relative to that established over an unperturbed network in the absence of link damage or disruption is derived based on a generalization of the Sherman–Morrison–Woodbury formula. The generalized formula provides us with the inverse of a perturbed matrix that differs from an unperturbed matrix by a sum of tensor vector products. The theoretical formulation suggests a venue for computing the effective conductance of an imperfect network and assessing critical conditions for global disruption.