Network composition for optimal disturbance rejection

Abstract

This paper investigates how the topology of a dynamical network affects its robustness against exogenous disturbances. We consider Laplacian-based network dynamics, and we adopt the H2 system norm to measure the robustness of the network against disturbances. For networks arising from the composition of atomic structures, we provide a closed-form expression of the robustness against disturbances, and we identify optimal composition rules. Specifically, we show that networks consisting of multiple atomic structures are less robust than each isolated part, and that robust structures arise by interconnecting nodes of the atomic components with highest degree. Finally, we describe an algorithm for the design of robust composite networks

Optimizing the coherence of composite networks

“…Then, we divide use (7) to replace Ω i with 2n i H C (G i ) and divide (14) by 2N to finally obtain (10).…”

“…The coherence of a composite graph with a ring backbone graph |V B | = n is derived from (10) by noting that in a ring graph with n nodes, r(l i , l j ) = (j−i)(n−(j−i)) n . We then get the following formula:…”

“…3 . We then use (10) to calculate the coherence of G to be: H C (G) = 8 3 . We also use the above pair of graphs to illustrate the results of Algorithm 1.…”

“…A system of interacting networks is considered in the works [8], [9], where the performance is based on robustness to cascading and random failures. In [10], the authors study the performance of a composite network in terms of the H 2 norm of the system, but they consider different dynamics than those presented in this paper.…”

Diffusion and Consensus in a Weakly Coupled Network of Networks

“…More recent work [22] considers an NoN model with noisy consensus dynamics and proposes methods to identify the optimal interconnection topology. And, in [23], the authors consider a similar NoN model, but with slightly different dynamics. They show that interconnection between the nodes of subgraphs with the highest degree maximizes the robustness of the NoN.…”