An anomalous topological phase transition in spatial random graphs

The Berezinskii, Kosterlitz, and Thouless (BKT) model shows that there are finite temperature phase transitions driven not by symmetry breaking, but rather by topological defects such as vortices. The authors show that a transition occurring in a general class of sparse space random networks model is topological in nature with no broken symmetry Clustering-the tendency for neighbors of nodes to be connected-quantifies the coupling of a complex network to its latent metric space. In random geometric graphs, clustering undergoes a continuous phase transition, separating a phase with finite clustering from a regime where clustering vanishes in the thermodynamic limit. We prove this geometric to non-geometric phase transition to be topological in nature, with anomalous features such as diverging entropy as well as atypical finite-size scaling behavior of clustering. Moreover, a slow decay of clustering in the non-geometric phase implies that some real networks with relatively high levels of clustering may be better described in this regime.