A geometry-induced topological phase transition in random graphs

Abstract

Clustering $\unicode{x2013}$ the tendency for neighbors of nodes to be connected $\unicode{x2013}$ 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-nongeometric 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 nongeometric phase implies that some real networks with relatively high levels of clustering may be better described in this regime.

18 pages, 4 figures (Supplementary: 31 pages)

References in corpus (7)

• Hyperbolic Geometry of Complex Networks
• Self-similarity of complex networks and hidden metric spaces
• An introduction to the Ginzburg-Landau theory of phase transitions and nonequilibrium patterns
• On the scaling of the chemical distance in long-range percolation models
• Emergence of Soft Communities from Geometric Preferential Attachment
• Structural properties of spatially embedded networks
• Percolation in self-similar networks

Cited by in corpus (7)

• Emergence of geometric Turing patterns in complex networks
• Detecting hyperbolic geometry in networks: why triangles are not enough
• Random hyperbolic graphs in $d+1$ dimensions
• Network Renormalization
• Random graphs and real networks with weak geometric coupling
• Effect of clustering on Turing instability in complex networks
• Multiplexity amplifies geometry in networks