Topologically protected synchronization in networks

Abstract

In a graph, we say that two nodes are topologically equivalent if their sets of first neighbors, excluding the two nodes, coincide. We prove that nonlinearly coupled heterogeneous oscillators located on a group of topologically equivalent nodes can get easily synchronized when the group forms a fully connected subgraph (or combinations thereof), regardless of the status of all the other oscillators. More generally, any change occurring in the remainder of the graph will not alter the synchronization status of the group. Typically, the group can synchronize when $k^{(\mathrm{OUT})}\leq k^{(\mathrm{IN})}$, $k^{(\mathrm{IN})}$ and $k^{(\mathrm{OUT})}$ being the common internal and outgoing degree of each node in the group, respectively. Simulations confirm our rigorous analysis and suggest that groups of topologically equivalent nodes act as independent pacemakers.