This article addresses the H2 norm accumulation problem of linearly coupled dynamical networks. An interesting outer-coupling relationship is constructed, under which the H2 norm of the newly constructed network with column-input and row-output matrices increases exponentially fast in terms of the node number N: it increases generally much faster than 2N when N is large while the H2 norm of each node is 1. However, the H2 norm of the network with a diffusive coupling is equal to 2 N, i.e. increasing only linearly, when the network is stable, where 2 is the H2 norm of a single node. And the H2 norm of the network with antisymmetrical coupling also increases, but rather slowly, with respect to the node number N. Other networks with block-diagonal-input and block-diagonal-output matrices behave similarly. It demonstrates that the changes of H2 norms in different networks are very complicated, despite the fact that the network structures are linear. Finally, the influence of the H2 norm of the locally linearised network on the output of a network with Lur'e node dynamics is discussed, demonstrating a significant effect of the H2 norm on the network output synchronisation behaviours.