The Euler characteristic and topological phase transitions in complex systems

Abstract

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.

The Euler characteristic as a topological marker for outbreaks in vector-borne disease

“…Over the past years, TDA applications across fields have yielded astounding results for science. For instance, topological transitions have been used as bio-markers in protein-protein networks [68] and to identify the emergence of neurological diseases from brain data [32]. There are many ways to compute the Euler characteristic across fields [69][70][71][72][73].…”

“…chords) that best describe the shape of the data. The effectiveness of TDA and GDA has been shown in several applications across several fields [24][25][26][27][28]. For example, Ricci curvature discretization has been useful to predict fragility and risk in stock exchange [29] and cancer diagnostics [30,31], while Betti numbers and the Euler characteristic have been applied to brain activity [32].…”