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 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.

Strain‐Driven Mixed‐Phase Domain Architectures and Topological Transitions in Pb1−xSrxTiO3 Thin Films

“…[41] In these cases, as is seen herein for the ferroelectric domain structures, the result is that not only does the phase fraction of the secondary phase change, but also how that phase is arranged within the primary phase. In turn, taking lessons from these other fields of study, we have applied analysis approaches developed therein [42][43][44] to the domain structures observed herein. In particular, we translate ideas of topology (a branch of mathematics that offers a means to characterize the shape of data objects) [44] including the so-called Euler characteristic (EC) which is a topologically invariant number that describes a topological space's shape or structure regardless of the way it is bent or stretch without breaking, [45] to explore these domain structures.…”

A hands-on tutorial on network and topological neuroscience

“…topological phase transitions in complex networks (Amorim et al 2019;Santos et al 2019) can also be identified between the changes in the dimensionality of the birth/death graphs mentioned above (Fig. 3e).…”

“…This strategy has already been applied for investigating group differences between controls and glioma brain networks (Santos et al 2019) and typically developing children and children with attentiondeficit/hyperactivity disorder (Gracia-Tabuenca et al 2020). In other fields, topological phase transitions were also investigated in the S. cerevisiae and C. elegans protein interaction networks, reionisation processes, and evolving coauthorship networks (Amorim et al 2019;Giri and Mellema 2021;Lee et al 2021).…”

A hands-on tutorial on network and topological neuroscience

“…This strategy has already been applied for investigating group differences between controls and glioma brain networks [64], and typically developing children and children with attention-deficit/hyperactivity disorder [26]. In other fields, topological phase transitions were also investigated in the S. cerevisiae and C. elegans protein interaction networks, reionization processes, and evolving coauthorship networks [70][71][72].…”

“…Here we used the Gudhi package for the implementation of those steps [83]. The topological phase transitions in complex networks [64,70], which can also be identified between the changes in the dimensionality of the birth/death graphs mentioned above ( Fig 3E).…”