Source-linked AI summary
Simplicial complexes and complex systems
Vsevolod Salnikov, Daniele Cassese, Renaud Lambiotte
TL;DR
Complex-systems research needs tools that characterize data shape and represent interactions beyond pairwise networks. This paper introduces TDA through simplicial complexes and persistent homology, surveys applications, and identifies methodological challenges for studying collective phenomena.
Problem
Network approaches may not properly represent the complexity of real-world interacting systems because they emphasize pairwise relations rather than many-body interactions.
Method
The paper provides an introductory synthesis of TDA, focusing on simplicial complexes, homology, persistent homology, and applications to complex systems.
Results
The review presents applications of TDA across complex systems, including brain, biological, financial, social, dynamical, and scientometric data.
Takeaways & Limitations
Simplicial complexes offer a candidate framework for analyzing systems composed of many elements interacting through many-body relations while retaining geometric information.
Takeaways & Limitations
The construction of a simplicial complex depends on parameters whose appropriate values cannot be chosen a priori, and those choices affect the resulting analysis.
Abstract
from arXiv · showhide
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of the presence of holes or cavities between the points. The methods, based on notion of simplicial complexes, generalise standard network tools by naturally allowing for many-body interactions and providing results robust under continuous deformations of the data. We present strengths and weaknesses of current methods, as well as a range of empirical studies relevant to the field of complex systems, before identifying future methodological challenges to help understand the emergence of collective phenomena.
I. INTRODUCTION
The paper introduces topological data analysis (TDA) as a way to characterize the shape of point-cloud data and examines its relevance to complex systems. It combines an introductory treatment with applications and future methodological challenges.
- TDA analyzes point clouds from diverse domains to reveal structure hidden in noisy patterns and reduce system dimensionality.
- The paper explains simplicial complexes and persistent homology for complex-systems researchers, while directing readers to introductory references.
- It surveys salient TDA applications to examples of complex systems.
- The paper identifies situations where TDA may be relevant and methodological steps toward a science of complex topology.
II. TOPOLOGICAL DATA ANALYSIS IN A NUTSHELL
TDA converts data into simplicial complexes whose homology describes topological shape through interpretable counts of holes. Persistent homology tracks these features across construction parameters, reducing dependence on any single parameter choice while retaining important persistence information.
- Topology studies properties invariant under continuous deformations, and TDA translates raw data into simplicial complexes for topological analysis.
- Čech complexes connect points when radius-τ balls intersect, while Vietoris–Rips complexes include simplices whose points are pairwise within a distance threshold.
- Simplicial complexes contain every face of each simplex, and non-empty intersections of simplices are faces of both.
- A k-chain is a formal sum of k-simplices, commonly with coefficients in Z2, forming the vector space C_k.
- The boundary operator maps a simplex to a formal sum of its faces, while cycles have zero boundary and boundaries are generated by higher-dimensional chains.
- Homology groups H_k = Z_k/B_k characterize shape, with their ranks giving Betti numbers equal to the numbers of k-dimensional holes.
- Persistent homology tracks hole births and deaths across filtrations, representing their parameter lifetimes with barcodes and persistence lengths.
- Choosing complex-construction parameters can change the data’s shape and resulting analysis, so the appropriate value cannot be fixed a priori.
III. FROM NETWORKS TO SIMPLICIAL COMPLEXES FOR COMPLEX SYSTEMS
Network methods emphasize pairwise connectivity, but real interacting systems may require representations of many-body interactions. Simplicial complexes extend this perspective while preserving the geometric structure of data and support applications such as detecting recurrent dynamics.
- Geometric graphs construct pairwise networks from metric-space points, while embedding methods map network nodes into geometric spaces for analysis.
- Network science focuses largely on pairwise edges and indirect paths, despite the complexity of real-world interacting systems.
- Simplicial complexes encode higher-order interactions involving more than two nodes and emphasize the systems’ underlying geometry.
- TDA can detect and quantify recurrent motion in noisy observations of dynamical systems, including through voids in sampled data.
- Data can be encoded into simplicial complexes through metric-space constructions such as Čech and Vietoris–Rips complexes or through hypergraph-like non-binary interactions.
IV. IMPORTANT TOOLS
Persistent homology is computed by reducing boundary-operator matrices across a filtration, with established algorithms and software implementations supporting practical analysis.
- Matrix-based computation: The boundary operator is represented by an integer matrix whose columns are k-simplices and rows are (k−1)-simplices.Its entries lie in {−1, 0, 1}.
- Matrix-based computation: Smith normal form provides sufficient information to characterize homology in all dimensions and compute Betti numbers.The ranks of the reduced matrices determine the ranks of cycles and boundaries.
- Persistent computation: Persistent homology requires a basis compatible across all complexes in a filtration, rather than a basis for only one complex.The simplified reduction algorithm has worst-case cubic complexity in the number of simplices.
- Software: Public software packages implement reduction and persistent homology, including javaPlex, Perseus, Dionysus, PHAT, DIPHA, Gudhi, ripser, and jHoles.A performance comparison identified ripser, Gudhi, and DIPHA as fastest overall, while javaPlex was easiest for beginners on small complexes.
V. APPLICATION OF TDA
TDA is increasingly applied across complex-systems domains, especially neuroscience and machine learning, while broader applications remain more sparse and fragmented. These studies use topological structure to distinguish states, classify data, and characterize complex networks and signals.
- Neuroscience: TDA adoption is increasing across disciplines studying complex systems, with neuroscience described as a leading area of application.The use of simplicial complexes relaxes the assumption that all relevant interactions are pairwise.
- Constructing complexes: Figure 4 summarizes constructions of simplicial complexes from point clouds, networks, and hypergraphs.The depicted simplicial complex corresponds to the hypergraph shown on its left.
- Neuroscience: Homological features distinguished psychedelic-drug and placebo brain states, while cycle-based edge measures complemented standard graph metrics.Other studies linked structural brain cycles to regions of different evolutionary origin and used persistence landscape distances to detect functional changes.
- Machine learning: Persistent convolutional neural networks combine persistence landscapes with convolutional networks, using piecewise linearity to make backpropagation straightforward.Applications reported improved music classification and strong classification results for 2D-object shapes.
- Other fields: Beyond neuroscience and machine learning, TDA applications remain more sparse and fragmented but span biology, finance, economics, social science, dynamical systems, and scientometrics.Examples include identifying high-survival breast cancers, financial-market crises, RNA-virus evolutionary patterns, and international communities in mobile-phone data.
VI. PERSPECTIVES
TDA has not yet developed the interconnected body of algorithms, patterns, mechanisms, and behavioral explanations that established network science as a science of complex systems.
- Future perspectives: For TDA to become a building block of complex-systems science, further development is needed across algorithms, recurring patterns, emergence mechanisms, and effects on system behavior.Network science is presented as a model because these facets reinforce one another in explaining complex systems.