We derive scalable decentralized conditions that can guarantee robust stability for networks of linearly interconnected, stable, linear time invariant dynamical systems. Unlike previous results of this kind, we allow for heterogeneous dynamics on arbitrary interconnection topologies (i.e., linear systems on arbitrary underlying graphs). Robust stability of the entire network is guaranteed by satisfying local rules that involve only an agent and its neighboring dynamics; each new agent will introduce only one additional condition of this kind and hence the stability certificates scale with the network size. An application of this theory is given, where robustness analysis of Internet congestion control protocols is carried out in the general case of dynamics at both users and resources without any global bounds on the dynamics.