In this work, we use methods and concepts of applied algebraic topology to comprehensively explore topological phase transitions in complex systems. Topological phase transitions are characterized by the zeros of the Euler characteristic (EC) or by singularities of the Euler entropy and also indicate signal changes in the mean node curvature of networks. Here, we provide strong evidence that the zeros of the Euler characteristic can be interpreted as a complex network's intrinsic fingerprint. We theoretically and empirically illustrate this across different biological networks: We first target our investigation to protein-protein interaction networks (PPIN). To do so, we used methods of topological data analysis to compute the Euler characteristic analytically, and the Betti numbers numerically as a function of the attachment probability for two variants of the Duplication Divergence model, namely the totally asymmetric model and the heterodimerization model. We contrast our theoretical results with experimental data freely available for gene co-expression networks (GCN) of Saccharomyces cerevisiae, also known as baker's yeast, as well as of the nematode Caenorhabditis elegans. Supporting our theoretical expectations, we are able to detect topological phase transitions in both networks obtained according to different similarity measures. Later, we theoretically illustrate the emergence of topological phase transitions in three classical network models, namely the Watts-Strogratz model, the Random Geometric Graph, and the Barabasi-Albert model. Given the universality and wide use of those models across disciplines, our results indicate that topological phase transitions may permeate across a wide range of theoretical and empirical networks. Hereby, our paper reinforces the idea of using topological phase transitions to advance the understanding of complex systems more generally.

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 TPT's 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 network's 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 baker's 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 Erdos-Renyi, the Watts-Strogatz model, the Random Geometric Graph, 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.