The Euler characteristic and topological phase transitions in complex systems

In this work, we use methods and concepts of applied algebraic topology to comprehensively explore the recent idea of topological phase transitions (TPTs) in complex systems. TPTs are characterized by the emergence of nontrivial homology groups as a function of a threshold parameter. Under certain conditions, one can identify TPTs via the zeros of the Euler characteristic or by singularities of the Euler entropy. Recent works provide strong evidence that TPTs can be interpreted as the intrinsic fingerprint of a complex network. This work illustrates this possibility by investigating various networks from a topological perspective. We first review the concept of TPTs in brain networks and discuss it in the context of high-order interactions in complex systems. We then investigate TPTs in protein–protein interaction networks using methods of topological data analysis for two variants of the duplication–divergence model. We compare our theoretical and computational results to experimental data freely available for gene co-expression networks of S. cerevisiae , also known as baker’s yeast, as well as of the nematode C. elegans . Supporting our theoretical expectations, we can detect TPTs in both networks obtained according to different similarity measures. We then perform numerical simulations of TPTs in four classical network models: the Erdős–Rényi, the Watts–Strogatz, the random geometric, and the Barabasi–Albert models. Finally, we discuss the relevance of these insights for network science. Given the universality and wide use of those network models across disciplines, our work indicates that TPTs permeate a wide range of theoretical and empirical networks, offering promising avenues for further research.