Synchronization in networks with delayed coupling are ubiquitous in nature and play a key role in almost all fields of science including physics, biology, ecology, climatology and sociology. In general, the published works on network synchronization are based on data analysis and simulations, with little experimental verification. Here we develop and experimentally demonstrate various multi-cluster phase synchronization scenarios within coupled laser networks. Synchronization is controlled by the network connectivity in accordance to number theory, whereby the number of synchronized clusters equals the greatest common divisor of network loops. This dependence enables remote switching mechanisms to control the optical phase coherence among distant lasers by local network connectivity adjustments. Our results serve as a benchmark for a broad range of coupled oscillators in science and technology, and offer feasible routes to achieve multi-user secure protocols in communication networks and parallel distribution of versatile complex combinatorial tasks in optical computers.
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