  Motivated by a paradigm shift towards a hyper-connected world, we develop a
computationally tractable small-gain theorem for a network of infinitely many
systems, termed as infinite networks. The proposed small-gain theorem addresses
exponential input-to-state stability with respect to closed sets, which enables
us to analyze diverse stability problems in a unified manner. The small-gain
condition, expressed in terms of the spectral radius of a gain operator
collecting all the information about the internal Lyapunov gains, can be
numerically computed for a large class of systems in an efficient way. To
demonstrate broad applicability of our small-gain theorem, we apply it to the
stability analysis of infinite time-varying networks, to consensus in
infinite-agent systems, as well as to the design of distributed observers for
infinite networks.
