Simplicial Complex Research Articles

Brain chains as topological signatures for Alzheimer’s disease

AbstractTopology is providing new insights for neuroscience. For instance, graphs, simplicial complexes, directed graphs, flag complexes, persistent homology and convex covers have been used to study functional brain networks, synaptic connectivity, and hippocampal place cell codes. We propose a topological framework to study the evolution of Alzheimer’s disease, the most common neurodegenerative disease. The modeling of this disease starts with the representation of the brain connectivity as a graph and the seeding of a toxic protein in a specific region represented by a vertex. Over time, the accumulation of toxic proteins at vertices and their propagation along edges are modeled by a dynamical system on this graph. These dynamics provide an order on the edges of the graph according to the damage created by high concentrations of proteins. This sequence of edges defines a filtration of the graph. We consider different filtrations given by different disease seeding locations. To study these filtrations we propose a new combinatorial and topological method. A filtration defines a maximal chain in the partially ordered set of spanning subgraphs ordered by inclusion. To identify similar graphs, and define a topological signature, we quotient this poset by graph homotopy equivalence, which gives maximal chains in a smaller poset. We provide an algorithm to compute this direct quotient without computing all subgraphs and then propose bounds on the total number of graphs up to homotopy equivalence. To compare the maximal chains generated by this method, we extend Kendall’s $$d_K$$ d K metric for permutations to more general graded posets and establish bounds for this metric. We then demonstrate the utility of this framework on actual brain graphs by studying the dynamics of tau proteins on the structural connectome. We show that the proposed topological brain chain equivalence classes distinguish different simulated subtypes of Alzheimer’s disease.

Bifurcation and chaos analysis of a fractional-order delay financial risk system using dynamic system approach and persistent homology

A comprehensive theoretical and numerical analysis of the dynamical features of a fractional-order delay financial risk system(FDRS) is presented in this paper. Applying the linearization method and Laplace transform, the critical value of delay when Hopf bifurcation first appears near the equilibrium is firstly derived in an explicit formula. Comparison simulations clarify the reasonableness of fractional-order derivative and delay in describing the financial risk management processes. Then we employ persistent homology and six topological indicators to reveal the geometric and topological structures of FDRS in delay interval. Persistence barcodes, diagrams, and landscapes are utilized for visualizing the simplicial complex’s information. The approximate values of delay when FDRS undergoes different periodic oscillations and even chaos are determined. The existence of periodic windows within the chaotic interval is correctly decided. The results of this paper contribute to capturing intricate information of underlying financial activities and detecting the critical transition of FDRS, which has promising and reliable implications for a deeper comprehension of complex behaviors in financial markets.

Algorithmic reconstruction of the fiber of persistent homology on cell complexes

Let K be a finite simplicial, cubical, delta or CW complex. The persistence map PH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}$$\\end{document} takes a filter f:K→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:K\\rightarrow \\mathbb {R}$$\\end{document} as input and returns the barcodes of the sublevel set persistent homology of f in each dimension. We address the inverse problem: given target barcodes D, computing the fiber PH-1(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}^{-1}(D)$$\\end{document}. For this, we use the fact that PH-1(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}^{-1}(D)$$\\end{document} decomposes as a polyhedral complex when K is a simplicial complex, and we generalise this result to arbitrary based chain complexes. We then design and implement a depth-first search that recovers the polytopes forming the fiber PH-1(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}^{-1}(D)$$\\end{document}. As an application, we solve a corpus of 120 sample problems, providing a first insight into the statistical structure of these fibers, for general CW complexes.

Enhancing the Vietoris–Rips simplicial complex for topological data analysis: applications in cancer gene expression datasets

AbstractThe aim of this study is to enhance the extraction of informative features from complex data through the application of topological data analysis (TDA) using novel topological overlapping measures. Topological data analysis has emerged as a promising methodology for extracting meaningful insights from complex datasets. Existing approaches in TDA often involve extrapolating data points using distance correlation measures, which subsequently constrain downstream predictive tasks. Our objective is to improve the construction of the Vietoris–Rips simplicial complex by introducing topological overlapping measures. These measures take into account the interplay of direct connection strengths and shared neighbours, leading to the identification of persistent topological features. We propose the utilisation of topological overlapping measures to optimise the construction of the Vietoris–Rips simplicial complex, offering a more refined representation of complex data structures. The application of topological overlapping measures results in the identification of plentiful persistent topological features. This enhancement contributes to an improvement of up to 20% in cancer phenotype prediction across different cancer types. Our study demonstrates the effectiveness of utilising topological overlapping measures in optimising the construction of the Vietoris–Rips simplicial complex. The identified persistent topological features significantly enhance the predictive accuracy of cancer phenotypes. This novel approach has the potential to advance the field of topological data analysis and improve our understanding of complex data structures, particularly in the context of cancer research and predictive modelling. Further exploration and application of these measures may yield valuable insights in various domains dealing with intricate datasets.

Directed simplicial complexes in brain real-world networks

Directed simplicial complexes in brain real-world networks

Pinning control of simplicial complexes

In this paper, pinning control of simplicial complexes is investigated. The D-dimensional simplicial complex can be successfully pinned to a desired state (i.e., an equilibrium of the intrinsic dynamics) under the proposed control scheme. The local stability of synchronization in the pinned simplicial complex is analyzed and discussed. The relationship between pinning synchronization and the individual dynamics, higher-order structures, coupling strengths, as well as the control strategy is established. Moreover, the effects of control gain and the number of controlled nodes on synchronizability are examined in detail. The obtained results can provide valuable references for designing optimal higher-order networks in practical applications.

Discrete Morse Sandwich: Fast Computation of Persistence Diagrams for Scalar Data - An Algorithm and A Benchmark.

This paper introduces an efficient algorithm for persistence diagram computation, given an input piecewise linear scalar field f defined on a d-dimensional simplicial complex K, with d ≤ 3. Our work revisits the seminal algorithm "PairSimplices" [31], [103] with discrete Morse theory (DMT) [34], [80], which greatly reduces the number of input simplices to consider. Further, we also extend to DMT and accelerate the stratification strategy described in "PairSimplices" [31], [103] for the fast computation of the 0th and (d-1)th diagrams, noted D0(f) and Dd-1(f). Minima-saddle persistence pairs ( D0(f)) and saddle-maximum persistence pairs ( Dd-1(f)) are efficiently computed by processing , with a Union-Find , the unstable sets of 1-saddles and the stable sets of (d-1)-saddles. We provide a detailed description of the (optional) handling of the boundary component of K when processing (d-1)-saddles. This fast pre-computation for the dimensions 0 and (d-1) enables an aggressive specialization of [4] to the 3D case, which results in a drastic reduction of the number of input simplices for the computation of D1(f), the intermediate layer of the sandwich. Finally, we document several performance improvements via shared-memory parallelism. We provide an open-source implementation of our algorithm for reproducibility purposes. We also contribute a reproducible benchmark package, which exploits three-dimensional data from a public repository and compares our algorithm to a variety of publicly available implementations. Extensive experiments indicate that our algorithm improves by two orders of magnitude the time performance of the seminal "PairSimplices" algorithm it extends. Moreover, it also improves memory footprint and time performance over a selection of 14 competing approaches, with a substantial gain over the fastest available approaches, while producing a strictly identical output. We illustrate the utility of our contributions with an application to the fast and robust extraction of persistent 1-dimensional generators on surfaces, volume data and high-dimensional point clouds.

Higher-Order Graph Convolutional Network with Flower-Petals Laplacians on Simplicial Complexes

Despite the recent successes of vanilla Graph Neural Networks (GNNs) on various tasks, their foundation on pairwise networks inherently limits their capacity to discern latent higher-order interactions in complex systems. To bridge this capability gap, we propose a novel approach exploiting the rich mathematical theory of simplicial complexes (SCs) - a robust tool for modeling higher-order interactions. Current SC-based GNNs are burdened by high complexity and rigidity, and quantifying higher-order interaction strengths remains challenging. Innovatively, we present a higher-order Flower-Petals (FP) model, incorporating FP Laplacians into SCs. Further, we introduce a Higher-order Graph Convolutional Network (HiGCN) grounded in FP Laplacians, capable of discerning intrinsic features across varying topological scales. By employing learnable graph filters, a parameter group within each FP Laplacian domain, we can identify diverse patterns where the filters' weights serve as a quantifiable measure of higher-order interaction strengths. The theoretical underpinnings of HiGCN's advanced expressiveness are rigorously demonstrated. Additionally, our empirical investigations reveal that the proposed model accomplishes state-of-the-art performance on a range of graph tasks and provides a scalable and flexible solution to explore higher-order interactions in graphs. Codes and datasets are available at https://github.com/Yiminghh/HiGCN.

Weisfeiler and Lehman Go Paths: Learning Topological Features via Path Complexes

Graph Neural Networks (GNNs), despite achieving remarkable performance across different tasks, are theoretically bounded by the 1-Weisfeiler-Lehman test, resulting in limitations in terms of graph expressivity. Even though prior works on topological higher-order GNNs overcome that boundary, these models often depend on assumptions about sub-structures of graphs. Specifically, topological GNNs leverage the prevalence of cliques, cycles, and rings to enhance the message-passing procedure. Our study presents a novel perspective by focusing on simple paths within graphs during the topological message-passing process, thus liberating the model from restrictive inductive biases. We prove that by lifting graphs to path complexes, our model can generalize the existing works on topology while inheriting several theoretical results on simplicial complexes and regular cell complexes. Without making prior assumptions about graph sub-structures, our method outperforms earlier works in other topological domains and achieves state-of-the-art results on various benchmarks.

Boolean Function Analysis on High-Dimensional Expanders

We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut–Kalai–Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only |X(k-1)|=O(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|X(k-1)|=O(n)$$\\end{document} points in contrast to nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( {\\begin{array}{c}n\\\\ k\\end{array}}\\right) $$\\end{document} points in the (k)-slice (which consists of all n-bit strings with exactly k ones).

Rotating clusters in phase-lagged Kuramoto oscillators with higher-order interactions.

The effect of phase-lag in pairwise interactions has been a topic of great interest for a while. However, real-world systems often have interactions that are beyond pairwise and can be modeled using simplicial complexes. We show that the inclusion of higher-order interactions in phase-lagged coupled Kuramoto oscillators shifts the critical point at which first-order transition from a cluster synchronized state to an incoherent state takes place. Considering the polar coordinates, we obtain the rotation frequency of the clusters, which turns out to be a function of the phase-lag parameter. In turn, the phase- lag can be used as a control parameter to achieve a desired cluster frequency. Moreover, in the thermodynamic limit, by employing the Ott-Antonsen approach we derive a reduced equation for the order parameter measuring cluster synchronization and progress further through the self-consistency method to obtain a closed form of the order parameter measuring global synchronization which was lacking in the Ott-Antonsen approach.

Ranicki–Weiss assembly and the Steenrod construction

It is shown that the Ranicki–Weiss assembly functor, going from chain complex valued presheaves on a simplicial complex to comodules over its Alexander–Whitney coalgebra, factors fully faithfully through the category of comodules over its Steenrod cup- i i coalgebra.

A value for cooperative games on simplicial complexes with a filtration

The classical Shapley value for cooperative games determines a payoff vector considering that the formation of the grand coalition is made by incorporating players one by one. Later, this method was generalized for games with restricted cooperation by several known mathematical structures: partitions, graphs, convex geometries, antimatroids, matroids or simplicial complexes. In this paper we consider games over simplicial complex with an extra information about the relationships of the agents, a filtration of the complex. Filtrations are very known simplicial structures in the social and neuronal networks. We propose a Shapley value for these situations and an axiomatization.

Higher-order homophily on simplicial complexes

Higher-order network models are becoming increasingly relevant for their ability to explicitly capture interactions between three or more entities in a complex system at once. In this paper, we study homophily, the tendency for alike individuals to form connections, as it pertains to higher-order interactions. We find that straightforward extensions of classical homophily measures to interactions of size 3 and larger are often inflated by homophily present in pairwise interactions. This inflation can even hide the presence of anti-homophily in higher-order interactions. Hence, we develop a structural measure of homophily, simplicial homophily, which decouples homophily in pairwise interactions from that of higher-order interactions. The definition applies when the network can be modeled as a simplicial complex, a mathematical abstraction which makes a closure assumption that for any higher-order relationship in the network, all corresponding subsets of that relationship occur in the data. Whereas previous work has used this closure assumption to develop a rich theory in algebraic topology, here we use the assumption to make empirical comparisons between interactions of different sizes. The simplicial homophily measure is validated theoretically using an extension of a stochastic block model for simplicial complexes and empirically in large-scale experiments across 16 datasets. We further find that simplicial homophily can be used to identify when node features are valuable for higher-order link prediction. Ultimately, this highlights a subtlety in studying node features in higher-order networks, as measures defined on groups of size k can inherit features described by interactions of size [Formula: see text].

The simpliciality of higher-order networks

Higher-order networks are widely used to describe complex systems in which interactions can involve more than two entities at once. In this paper, we focus on inclusion within higher-order networks, referring to situations where specific entities participate in an interaction, and subsets of those entities also interact with each other. Traditional modeling approaches to higher-order networks tend to either not consider inclusion at all (e.g., hypergraph models) or explicitly assume perfect and complete inclusion (e.g., simplicial complex models). To allow for a more nuanced assessment of inclusion in higher-order networks, we introduce the concept of “simpliciality” and several corresponding measures. Contrary to current modeling practice, we show that empirically observed systems rarely lie at either end of the simpliciality spectrum. In addition, we show that generative models fitted to these datasets struggle to capture their inclusion structure. These findings suggest new modeling directions for the field of higher-order network science.

Higher-order connection Laplacians for directed simplicial complexes

Higher-order networks encode the many-body interactions existing in complex systems, such as the brain, protein complexes, and social interactions. Simplicial complexes are higher-order networks that allow a comprehensive investigation of the interplay between topology and dynamics. However, simplicial complexes have the limitation that they only capture undirected higher-order interactions while in real-world scenarios, often there is a need to introduce the direction of simplices, extending the popular notion of direction of edges. On graphs and networks the Magnetic Laplacian, a special case of connection Laplacian, is becoming a popular operator to address edge directionality. Here we tackle the challenge of handling directionality in simplicial complexes by formulating higher-order connection Laplacians taking into account the configurations induced by the simplices’ directions. Specifically, we define all the connection Laplacians of directed simplicial complexes of dimension two and we discuss the induced higher-order diffusion dynamics by considering instructive synthetic examples of simplicial complexes. The proposed higher-order diffusion processes can be adopted in real scenarios when we want to consider higher-order diffusion displaying non-trivial frustration effects due to conflicting directionalities of the incident simplices.

The interaction of information diffusion and epidemic transmission in time-varying multiplex networks with simplicial complexes and asymmetric activity levels

Information diffusion among individuals occurs in various ways, mainly involving pairwise and higher-order interactions, and their coexistence can be characterized by simplicial complexes. This paper introduces a novel two-layer model that investigates coupled information-epidemic propagation. Specifically, the upper layer represents the virtual layer that depicts information diffusion, where the interaction process among individuals is not only limited to pairwise interactions but also influenced by higher-order interactions. The lower layer denotes the physical contact layer to portray epidemic transmission, where the interaction process among individuals is only considered in pairwise interactions. In particular, the emergence of asymmetric activity levels in two-layer networks reshapes the propagation mechanism. We then employ the micro-Marko chain approach (MMCA) to establish the probabilistic transfer equation for each state, deduce the epidemic outbreak threshold, and investigate the equilibrium and stability of the proposed model. Furthermore, we perform extensive Monte Carlo (MC) simulations to validate the theoretical predictions. The results demonstrate that the higher-order interaction generates synergistic reinforcement mechanisms that both facilitate information diffusion and inhibit epidemic transmission. Moreover, this study suggests that the activity level of individuals at the physical contact level has a greater impact on epidemic transmission. In addition, we utilize two different networks to explore the influence of network structural features on the transmission and scale of epidemics.

New Constructions of Optimal Linear Codes From Simplicial Complexes

New Constructions of Optimal Linear Codes From Simplicial Complexes

Global synchronization on time-varying higher-order structures

Synchronization has received a lot of attention from the scientific community for systems evolving on static networks or higher-order structures, such as hypergraphs and simplicial complexes. In many relevant real-world applications, the latter are not static but do evolve in time, in this work we thus discuss the impact of the time-varying nature of higher-order structures in the emergence of global synchronization. To achieve this goal, we extend the master stability formalism to account, in a general way, for the additional contributions arising from the time evolution of the higher-order structure supporting the dynamical systems. The theory is successfully challenged against two illustrative examples, the Stuart–Landau nonlinear oscillator and the Lorenz chaotic oscillator.

Higher-order structures of local collaboration networks are associated with individual scientific productivity

The prevalence of teamwork in contemporary science has raised new questions about collaboration networks and the potential impact on research outcomes. Previous studies primarily focused on pairwise interactions between scientists when constructing collaboration networks, potentially overlooking group interactions among scientists. In this study, we introduce a higher-order network representation using algebraic topology to capture multi-agent interactions, i.e., simplicial complexes. Our main objective is to investigate the influence of higher-order structures in local collaboration networks on the productivity of the focal scientist. Leveraging a dataset comprising more than 3.7 million scientists from the Microsoft Academic Graph, we uncover several intriguing findings. Firstly, we observe an inverted U-shaped relationship between the number of disconnected components in the local collaboration network and scientific productivity. Secondly, there is a positive association between the presence of higher-order loops and individual scientific productivity, indicating the intriguing role of higher-order structures in advancing science. Thirdly, these effects hold across various scientific domains and scientists with different impacts, suggesting strong generalizability of our findings. The findings highlight the role of higher-order loops in shaping the development of individual scientists, thus may have implications for nurturing scientific talent and promoting innovative breakthroughs.