Distant remote synchronization in multiple connected star graphs

Remote synchronization (RS) has recently attracted a lot of attention, but it is mainly focused on the synchronization among leaf nodes of a star graph and thus cannot be applied to the synchronization between distant nodes across different regions of a brain network. To understand how the distant connections of the brain functional network come from its structural network, we extend our previous bridge-connected model of two star graphs to the case of multiple connected star graphs. We surprisingly find that this extended model can show not only the RS among all the leaf nodes, but also a new form of RS between distant leaf nodes without the involvement of any middle nodes. We call the former as relay RS and the latter as distant RS. To the best of our knowledge, this is the first implementation of a distant RS in star-like graph models, and thus may open a window for applying RS to discuss the mechanisms of long connections in brain structural networks. Further, we show that there are two necessary conditions for the emergence of distant RS: (i) heterogeneous morphology of the network and (ii) asymmetric couplings between the bridge links and non-bridge links. Finally, we present a brief theoretical analysis to explain the distant RS and find that the key element is the dynamical symmetry, in contrast to the previous morphology symmetry.