An infinitesimal characterization of nonlinear contracting interference functions

Contractive interference functions are a subclass of the standard interference functions used in the design and analysis of distributed power control algorithms for wireless networks. Their peculiarity is that for the resulting positive system the existence and global asymptotic stability of a unique positive equilibrium point is guaranteed. In this paper we give an infinitesimal characterization of nonlinear contractive interference functions in terms of the spectral radius of the Jacobian linearization at any point in the positive orthant. The condition we obtain, that the spectral radius is always less than 1, extends to the nonlinear case an equivalent property of linear interference functions, and leads to a Jacobian characterization similar to the one commonly used in contraction analysis of nonlinear systems.