  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 between 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 induces a uniformly globally
asymptotically stable (UGAS) system, a Lyapunov function for the infinite
network can be constructed. We analyze necessary and sufficient conditions for
UGAS and relate them to small-gain conditions used in the stability analysis of
finite networks.
