Synchronization in network geometries with finite spectral dimension

Recently there is a surge of interest in network geometry and topology. Here we show that the spectral dimension plays a fundamental role in establishing a clear relation between the topological and geometrical properties of a network and its dynamics. Specifically we explore the role of the spectral dimension in determining the synchronization properties of the Kuramoto model. We show that the synchronized phase can only be thermodynamically stable for spectral dimensions above four and that phase entrainment of the oscillators can only be found for spectral dimensions greater than two. We numerically test our analytical predictions on the recently introduced model of network geometry called complex network manifolds, which displays a tunable spectral dimension.We acknowledge interesting discussions with Z. Burda, R. Burioni, R. Loll, D. Mulder, L. Smolin, R. Sorkin, G. Vidal, and P. Jizba. We are grateful for financial support from the Spanish Ministry of Science and the "Agencia Espanola de Investigacion" (AEI) under Grant No. FIS2017-84256-P (FEDER funds) and from "Obra Social La Caixa" (ID 100010434, with code LCF/BQ/ES15/10360004). G.B. was partially supported by the Perimeter Institute for Theoretical Physics (PI). The PI is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation