  Consensus networks are usually understood as arithmetic mean driven dynamical
averaging systems. In applications, however, network dynamics often describe
inherently non-arithmetic and non-linear consensus processes. In this paper, we
propose and study three novel consensus protocols driven by geometric mean
averaging: a polynomial, an entropic, and a scaling-invariant protocol, where
terminology characterizes the particular non-linearity appearing in the
respective differential protocol equation. We prove exponential convergence to
consensus for positive initial conditions. For the novel protocols we highlight
connections to applied network problems: The polynomial consensus system is
structured like a system of chemical kinetics on a graph. The entropic
consensus system converges to the weighted geometric mean of the initial
condition, which is an immediate extension of the (weighted) average consensus
problem. We find that all three protocols generate gradient flows of free
energy on the simplex of constant mass distribution vectors albeit in different
metrics. On this basis, we propose a novel variational characterization of the
geometric mean as the solution of a non-linear constrained optimization problem
involving free energy as cost function. We illustrate our findings in numerical
simulations.
