Simplicial complexes and complex systems

Abstract

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.

8 pages, 4 figures

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Cited by in corpus (52)

• Networks beyond pairwise interactions: structure and dynamics
• Simplicial models of social contagion
• Explosive higher-order Kuramoto dynamics on simplicial complexes
• Higher-order interactions in complex networks of phase oscillators promote abrupt synchronization switching
• Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplices
• Abrupt phase transition of epidemic spreading in simplicial complexes
• Multi-body Interactions and Non-Linear Consensus Dynamics on Networked Systems
• Higher-order simplicial synchronization of coupled topological signals
• Beyond the clustering coefficient: A topological analysis of node neighbourhoods in complex networks
• Simplicial complexes: higher-order spectral dimension and dynamics
• Higher-order interactions improve optimal collective dynamics on networks
• Weighted simplicial complexes and their representation power of higher-order network data and topology
• Higher-order interactions promote chimera states
• Synchronization in Network Geometries with Finite Spectral Dimension
• Growing scale-free simplices
• The dynamic nature of percolation on networks with triadic interactions
• Clustering in graphs and hypergraphs with categorical edge labels
• Topological Percolation on Hyperbolic Simplicial Complexes
• Mapping flows on hypergraphs
• Bifurcation analysis and structural stability of simplicial oscillator populations
• D-dimensional oscillators in simplicial structures: odd and even dimensions display different synchronization scenarios
• Spectrum of extensive multiclusters in the Kuramoto model with higher-order interactions
• Modelling Non-Linear Consensus Dynamics on Hypergraphs
• The distance backbone of complex networks
• Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian
• Spectral Detection of Simplicial Communities via Hodge Laplacians
• Percolation on branching simplicial and cell complexes and its relation to interdependent percolation
• Diffusion-driven instability of topological signals coupled by the Dirac operator
• Memory selection and information switching in oscillator networks with higher-order interactions
• The nature of hypergraph $k$-core percolation problems
• Spectral properties of hyperbolic nano-networks with tunable aggregation of simplexes
• Renormalization group for link percolation on planar hyperbolic manifolds
• Stochastic epidemic model on a simplicial complex
• Hypergraph reconstruction from noisy pairwise observations
• Global topological synchronization of weighted simplicial complexes
• Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes
• Tiered synchronization in coupled oscillator populations with interaction delays and higher-order interactions
• Higher-order Connection Laplacians for Directed Simplicial Complexes
• Persistent Homology of Fractional Gaussian Noise
• Zoo Guide to Network Embedding
• FreSCo: Mining Frequent Patterns in Simplicial Complexes
• Higher-order triadic percolation on random hypergraphs
• Influence of Asymmetric Parameters in Higher-Order Coupling With Bimodal Frequency Distribution
• Persistent homology analysis of type 2 diabetes genome-wide association studies in protein-protein interaction networks
• Community detection in hypergraphs through hyperedge percolation
• Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning
• Local topological moves determine global diffusion properties of hyperbolic higher-order networks
• Flow-based Community Detection in Hypergraphs
• Emergent synchrony in oscillator networks with adaptive arbitrary-order interactions
• Combinatorial Properties for a Class of Simplicial Complexes Extended from Pseudo-fractal Scale-free Web
• A Perspective on the Ubiquity of Interaction Streams in Human Realm
• Dynamical systems defined on simplicial complexes: symmetries, conjugacies, and invariant subspaces