  We consider a canonical ensemble of synchronization-optimized networks of
identical oscillators under external noise. By performing a Markov chain Monte
Carlo (MCMC) simulation using the Kirchhoff index, i.e., the sum of the inverse
eigenvalues of the Laplacian matrix (as a graph Hamiltonian of the network), we
construct more than 1,000 different synchronization-optimized networks. We then
show that the transition from star to core-periphery structure depends on the
connectivity of the network, and is characterized by the node degree variance
of the synchronization-optimized ensemble. We find that thermodynamic
properties such as heat capacity show anomalies for sparse networks.
