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This paper addresses the energy accumulation problem, in terms of the $H_2$norm, of linearly coupled dynamical networks. An interesting outer-couplingrelationship is constructed, under which the $H_2$ norm of the newlyconstructed network with column-input and row-output shaped matrices increasesexponentially fast with the node number $N$: it increases generally much fasterthan $2^N$ 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 $\gamma_2N$, i.e., increasing linearly, when the network is stable, where $\gamma_2$ isthe $H_2$ norm of a single node. And the $H_2$ norm of the network withantisymmetrical coupling also increases, but rather slowly, with the nodenumber $N$. Other networks with block-diagonal-input and block-diagonal-outputmatrices behave similarly. It demonstrates that the changes of $H_2$ norms indifferent networks are very complicated, despite the fact that the networks arelinear. Finally, the influence of the $H_2$ norm of the locally linearizednetwork on the output of a network with Lur'e nodes is discussed.

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