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 (TPT) 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 a complex networks intrinsic fingerprint. This work illustrates this possibility by investigating some classic network and empirical protein interaction networks under a topological perspective. We first investigate TPT in protein-protein interaction networks (PPIN) using methods of topological data analysis for two variants of the Duplication-Divergence model, namely, the totally asymmetric model and the heterodimerization model. We compare our theoretical and computational results to experimental data freely available for gene co-expression networks (GCN) of Saccharomyces cerevisiae, also known as bakers yeast, as well as of the nematode Caenorhabditis elegans. Supporting our theoretical expectations, we can detect topological phase transitions in both networks obtained according to different similarity measures. Later, we perform numerical simulations of TPTs in four classical network models: the Erd[o]s-Renyi model, the Watts-Strogatz model, the Random Geometric model, and the Barabasi-Albert. Finally, we discuss some perspectives and insights on the topic. Given the universality and wide use of those models across disciplines, our work indicates that TPT permeates a wide range of theoretical and empirical networks.