Complexity in linearly coupled dynamical networks: Some unusual phenomena in energy accumulation

This paper addresses the energy accumulation problem, in terms of the $H_2$ norm, of linearly coupled dynamical networks. An interesting outer-coupling relationship is constructed, under which the $H_2$ norm of the newly constructed network with column-input and row-output shaped matrices increases exponentially fast with the node number $N$: it increases generally much faster than $2^N$ when $N$ is large while the $H_2$ norm of each node is 1. However, the $H_2$ norm of the network with a diffusive coupling is equal to $\gamma_2 N$, i.e., increasing linearly, when the network is stable, where $\gamma_2$ is the $H_2$ norm of a single node. And the $H_2$ norm of the network with antisymmetrical coupling also increases, but rather slowly, with the node number $N$. Other networks with block-diagonal-input and block-diagonal-output matrices behave similarly. It demonstrates that the changes of $H_2$ norms in different networks are very complicated, despite the fact that the networks are linear. Finally, the influence of the $H_2$ norm of the locally linearized network on the output of a network with Lur'e nodes is discussed.