  We introduce the concept of non-uniform input-to-state stability for
networks. It combines the uniform global stability with the uniform
attractivity of any subnetwork, while it allows for non-uniform convergence of
all components. For an infinite network consisting of input-to-state stable
subsystems, that do not necessarily have a uniform KL-bound on the transient
behaviour, we show: If the gain operator satisfies the uniform small-gain
condition, then the whole network is non-uniformly input-to-state stable and
all its finite subnetworks are input-to-state stable.
