Topological phase transitions in functional brain networks

Functional brain networks are often constructed by quantifying correlations among brain regions. Their topological structure includes nodes, edges, triangles and even higher-dimensional objects. Topological data analysis (TDA) is the emerging framework to process datasets under this perspective. In parallel, topology has proven essential for understanding fundamental questions in physics. Here we report the discovery of topological phase transitions in functional brain networks by merging concepts from TDA, topology, geometry, physics, and network theory. We show that topological phase transitions occur when the Euler entropy has a singularity, which remarkably coincides with the emergence of multidimensional topological holes in the brain network. Our results suggest that a major alteration in the pattern of brain correlations can modify the signature of such transitions, and may point to suboptimal brain functioning. Due to the universal character of phase transitions and noise robustness of TDA, our findings open perspectives towards establishing reliable topological and geometrical biomarkers of individual and group differences in functional brain network organization.