  In this paper, we show that an infinite network of input-to-state stable
(ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS
Lyapunov function, provided that the couplings of the subsystems are
sufficiently weak. The strength of the couplings is described in terms of the
properties of an infinite-dimensional nonlinear positive operator, built from
the interconnection gains. If this operator satisfies the so-called robust
small-gain condition and induces a uniformly globally asymptotically stable
discrete-time system, a Lyapunov function for the infinite network can be
constructed. In the case of linear interconnection gains, these conditions are
equivalent to a spectral small-gain condition, expressed in terms of a
generalized spectral radius.
