Partial remote synchronization in star-like networks with partial connections among leaf nodes

To understand how the long connections of a brain functional network come from the short connections of its corresponding structural network, remote synchronization (RS) was recently studied in star graph networks. However, the motif of the star graph cannot completely characterize the features of brain networks as the leaf nodes of a star graph may also be connected to each other to some extent in real brain networks. Especially, the dynamics of a star motif in a brain network will be seriously influenced by its surrounding nodes, i.e., other parts of the brain network. To study RS of real brain networks, we here present a model of star-like networks by considering both the partial connections among leaf nodes and the influence of other parts of the brain network. We find that RS will not appear in all leaf nodes and instead appears only in the group of indirectly connected leaf nodes when the frequency difference between the hub and leaf nodes is not large enough, resulting in the concept of partial RS (PRS). Further, we find that the partial connections among leaf nodes favor PRS, implying that PRS can more easily appear in real brain networks than RS and thus provides a different way to understand the mechanism of long connections in brain functional networks. Moreover, we find another kind of PRS, i.e., double PRS, and discuss the dependence of PRS on system parameters. Finally, a brief theoretical analysis is provided to explain the results.