Non-uniform ISS small-gain theorem for infinite networks

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, which do not necessarily have a uniform $\mathscr{K}\mathscr{L}$-bound on the transient behaviour, we show the following: 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.