Dynamics on a graph that are governed by power law rate equations are discussed by means of a prototypical polynomial network protocol, which is derived from a simple allometric scaling relation as it arises e.g. in biology or chemical processes. We prove exponential convergence and stability results for the proposed nonlinear system class by transforming the nonlinear equation system into linear time-varying consensus form. Necessary and sufficient conditions for fixed points to be consensus points are provided and they are equivalent to the Wegscheider equilibrium relation known in chemistry. We illustrate our results in numerical simulations. The methods we introduce to transform the nonlinear problem to linear Laplacian form result in a novel conductance formulation for chemical reaction networks. This conductance model has the functional form of a heat exchanger. It allows to write chemical network dynamics as Gibbs free energy gradient system, with dissipation metric given by heat exchanger conductances and chemical potentials, resp. complex thermodynamic affinities, acting as driving forces.