The subject of the paper is the network of coupled identical oscillators
where is the parameter which controls the overall coupling, is the phase variable of -th oscillator, is the relative strength of the coupling, and is the output function.
The following optimization problems are considered: 1. Maximization of synchronizability, i.e. the maximization of the region of parameter , for which stable synchronization occurs. 2. Minimization of the synchronization cost, which is defined as the sum of the total input strength of all nodes at the lower synchronization threshold : .
The main result shows that the solution sets for these two optimization problems coincide and are characterized by the following condition on the Laplacian (i.e. the matrix ) eigenvalues:
The authors study such optimal networks and drive some useful conclusions about their structure.