Optimizing the coherence of composite networks

We consider how to connect a set of disjoint networks to optimize the performance of the resulting composite network. We quantify this performance by the coherence of the composite network, which is defined by an H
₂
norm of the system. Two dynamics are considered: noisy consensus dynamics with and without stubborn agents. For noisy consensus dynamics without stubborn agents, we derive analytical expressions for the coherence of composite networks in terms of the coherence of the individual networks and the structure of their interconnections. We also identify optimal interconnection topologies and give bounds on coherence for general composite graphs. For noisy consensus dynamics with stubborn agents, we show that the coherence of a composite network is a submodular function over the set of potential edges between the disjoint networks. We leverage this submodularity to develop a non-combinatorial algorithm that identifies connecting edges such that the composite network coherence is within a provable bound of optimal.