  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.
