<i>H₂ </i> norm accumulation and its impact on synchronisation of complex dynamical networks

This article addresses the H 2 norm accumulation problem 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 matrices increases exponentially fast in terms of 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 γ2 N, i.e. increasing only linearly, when the network is stable, where γ2 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 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 H 2 norms in different networks are very complicated, despite the fact that the network structures are linear. Finally, the influence of the H 2 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 H 2 norm on the network output synchronisation behaviours.