Simplicial Complex Research Articles

Topological representations of simplicial complexes and their applications

Topological representations of simplicial complexes and their applications

Categorifying connected domination via graph überhomology

Überhomology is a recently defined homology theory for simplicial complexes, which yields subtle information on graphs. We prove that bold homology, a certain specialisation of überhomology, is related to dominating sets in graphs. To this end, we interpret überhomology as a poset homology, and investigate its functoriality properties. We then show that the Euler characteristic of the bold homology of a graph coincides with an evaluation of its connected domination polynomial. Even more, the bold chain complex retracts onto a complex generated by connected dominating sets. We conclude with several computations of this homology on families of graphs; these include a vanishing result for trees, and a characterisation result for complete graphs.

Topological data analysis of noise: Uniform unimodal distributions

Random point clouds determine random simplicial complexes which can be characterized via homology. In this work we consider Vietoris–Rips complexes built on top of random point clouds in the D-dimensional Euclidian space in two setups: (i) a homogeneous Poisson point process (HPPP) and (ii) N i.i.d. points sampled from a uniform elliptical distribution. We derive expected topological summaries for such complexes in terms of the average Betti numbers (ABN) and of the newly introduced average integral persistence landscape (AIPL). The latter summary is derived from the average persistence landscape (APL) and preserves its 1-norm. In the HPPP model, we derive general analytical expressions for the density of ABN and AIPL and their 1-norms. For finite samples, we establish the functional form of decline of the 1-norm of the APL with the decrease of the sample size and/or increase of the cloud dimensionality. Furthermore, we derive analytical estimates of boundary effects in a disk, an ellipse, a D-dimensional ball, and 3-dimensional spheroids. We develop asymptotic models accounting for degeneration of a highly eccentric ellipse or a prolate spheroid to a line-segment and of an oblate spheroid to a disk, which describe the decline or growth of the 1-norm of the APL upon degeneration of an elliptic or spheroidal point cloud to a one- or two-dimensional cloud, respectively. Our results can serve as a “null model” for selection of the hyperparameters of the method D and N, as well as for the interpretation of observations in a broad range of real-world problems.

Enhanced explosive synchronization in heterogeneous oscillator populations with higher-order interactions

Explosive synchronization underlying many realistic dynamic processes has attracted great attention in the various fields. Here, we generalize the Kuramoto model with heterogeneous coupling incorporating the higher-order interactions encoded with the simplicial complexes. We show that the introduced nonlinear couplings can significantly enhance the emergence of explosive synchronization in addition to the pairwise interactions. In particular, we uncover that the critical thresholds manifesting the onset or vanishing of the abrupt synchronization are enlarged as the fraction of the higher-order interactions increases. As a consequence, the backward critical order parameter characterizing the desynchronization transition and the width of hysteresis supporting the bistability are remarkably improved. More importantly, we provide an analytical treatment for untangling the stability properties of the equilibrium states at different levels, which allows us to understand the bifurcation mechanism and locate the associated critical points. This study is a step forward in highlighting the importance of higher-order interactions among dynamical units, which might provide control strategies for the induced abrupt emergent phenomena in networked system.

Simplicial Complex-Enhanced Manifold Embedding of Spatiotemporal Data for Structural Health Monitoring

Structural Health Monitoring requires the continuous assessment of a structure’s operational conditions, which involves the collection and analysis of a large amount of data in both spatial and temporal domains. Conventionally, both data-driven and physics-based models for structural damage detection have relied on handcrafted features, which are susceptible to the practitioner’s expertise and experience in feature selection. The limitations of handcrafted features stem from the potential for information loss during the extraction of high-dimensional spatiotemporal data collected from the sensing system. To address this challenge, this paper proposes a novel, automated structural damage detection technique called Simplicial Complex Enhanced Manifold Embedding (SCEME). The key innovation of SCEME is the reduction of dimensions in both the temporal and spatial domains for efficient and information-preserving feature extraction. This is achieved by constructing a simplicial complex for each signal and using the resulting topological invariants as key features in the temporal domain. Subsequently, curvature-enhanced topological manifold embedding is performed for spatial dimension reduction. The proposed methodology effectively represents both intra-series and inter-series correlations in the low-dimensional embeddings, making it useful for classification and visualization. Numerical simulations and two benchmark experimental datasets validate the high accuracy of the proposed method in classifying different damage scenarios and preserving useful information for structural identification. It is especially beneficial for structural damage detection using complex data with high spatial and temporal dimensions and large uncertainties in reality.

Temporal network analysis using zigzag persistence

This work presents a framework for studying temporal networks using zigzag persistence, a tool from the field of Topological Data Analysis (TDA). The resulting approach is general and applicable to a wide variety of time-varying graphs. For example, these graphs may correspond to a system modeled as a network with edges whose weights are functions of time, or they may represent a time series of a complex dynamical system. We use simplicial complexes to represent snapshots of the temporal networks that can then be analyzed using zigzag persistence. We show two applications of our method to dynamic networks: an analysis of commuting trends on multiple temporal scales, e.g., daily and weekly, in the Great Britain transportation network, and the detection of periodic/chaotic transitions due to intermittency in dynamical systems represented by temporal ordinal partition networks. Our findings show that the resulting zero- and one-dimensional zigzag persistence diagrams can detect changes in the networks’ shapes that are missed by traditional connectivity and centrality graph statistics.

Canonical sphere bases for simplicial and cubical complexes

Sphere-bases are constructed for the [Formula: see text] vector space formed by the [Formula: see text]-dimensional subcomplexes, of [Formula: see text]-simplex (or [Formula: see text]-cube), for which every [Formula: see text]-face is contained in a positive even number of [Formula: see text]-cells; addition is symmetric difference of the corresponding sets of [Formula: see text]-cells. The bases consist of the boundaries of an algorithmically-specified family of [Formula: see text]-simplexes or [Formula: see text]-cubes. Properties of the bases are investigated.

Gromov’s Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper Volume and bounded cohomology. In order to study the main properties of simplicial volume, Gromov himself initiated the dual theory of bounded cohomology, which then developed into a very active and independent research field. Gromov’s theory of bounded cohomology of topological spaces was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets. The first aim of this paper is to lay the foundation of the theory of multicomplexes. After setting the main definitions, we construct the singular multicomplex K ( X ) \mathcal {K}(X) associated to a topological space X X , and we prove that the geometric realization of K ( X ) \mathcal {K}(X) is homotopy equivalent to X X for every CW complex X X . Following Gromov, we introduce the notion of completeness, which, roughly speaking, translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes, and we show that K ( X ) \mathcal {K}(X) is complete for every CW complex X X . In the second part of this work we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity. The third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the ℓ 1 \ell ^1 -invisibility of closed manifolds in terms of amenable covers. As an application, we give the first detailed proof of the vanishing of the simplicial volume of the product of three open manifolds.

On Elser's conjecture and the topology of U-nucleus complex

Dorpalen-Barry et al. proved Elser's conjecture about signs of Elser's numbers by interpreting them as certain sums of reduced Euler characteristic of an abstract simplicial complex known as U-nucleus complex. We prove a conjecture posed by them regarding the homology of U-nucleus complex.

Systolic inequalities for the number of vertices

Inspired by the classical Riemannian systolic inequality of Gromov, we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of “essentiality”, our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results quantitatively. We also illustrate our methods in the continuous setting, generalizing and improving quantitatively the Minkowski principle of Balacheff and Karam; a corollary of this result is the extension of the Guth–Nakamura cup-length systolic bound from manifolds to complexes.

A Characterization of Two-Dimensional Buchsbaum Matching Complexes

The matching complex $M(G)$ of a graph G is the set of all matchings in G. A Buchsbaum simplicial complex is a generalization of both a homology manifold and a Cohen–Macaulay complex. We give a complete characterization of the graphs G for which $M(G)$ is a two-dimensional Buchsbaum complex. As an intermediate step, we determine which graphs have matching complexes that are themselves connected graphs.

Non-linear consensus dynamics on temporal hypergraphs with random noisy higher-order interactions

AbstractComplex networks encoding the topological architecture of real-world complex systems have recently been undergoing a fundamental transition beyond pairwise interactions described by dyadic connections among nodes. Higher-order structures such as hypergraphs and simplicial complexes have been utilized to model group interactions for varied networked systems from brain, society, to biological and physical systems. In this article, we investigate the consensus dynamics over temporal hypergraphs featuring non-linear modulating functions, time-dependent topology and random perturbations. Based upon analytical tools in matrix, hypergraph, stochastic process and real analysis, we establish the sufficient conditions for all nodes in the network to reach consensus in the sense of almost sure convergence and $\mathscr{L}^2$ convergence. The rate of consensus and the moments of the equilibrium have been determined. Our results offer a theoretical foundation for the recent series of numerical studies and physical observations in the multi-body non-linear dynamical systems.

Relating CAT(0) cubical complexes and flag simplicial complexes

Given a finite CAT(0) cubical complex, we define a flag simplicial complex associated to it, called the crossing complex. We show that the crossing complex holds much of the combinatorial information of the original cubical complex: for example, hyperplanes in the cubical complex correspond to vertex links in the crossing complex, and the crossing complex is balanced if and only if the cubical complex is cubically balanced. The most significant result is that the sets of f-vectors of CAT(0) cubical complexes and flag simplicial complexes are equal, up to an invertible linear transformation.

The inverse problem of Q-analysis of complex systems structure in cyber security


 
 
 It is considered the inverse problem of Q-analysis. In the course of the research, an algorithm for the recovery of simplicial complexes from elementary simplex using local maps and a structural tree was developed. This algorithm will reduce the amount of data stored and improve the management process if the simplicial complex describes a real big complex system wich it can imagine cyber security system.
 
 

Large deviations for subcomplex counts and Betti numbers in multiparameter simplicial complexes

We consider the multiparameter random simplicial complex as a higher dimensional extension of the classical Erdős–Rényi graph. We investigate appearance of “unusual” topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices of the multiparameter simplicial complexes. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.

Positive cluster complexes and τ-tilting simplicial complexes of cluster-tilted algebras of finite type

In this study, we consider the positive cluster complex, a full subcomplex of a cluster complex the vertices of which are all non-initial cluster variables. In particular, we provide a formula for the difference in face vectors of positive cluster complexes caused by a mutation for finite type. Moreover, we explicitly describe specific positive cluster complexes of finite type and calculate their face vectors. We also provide a method to compute the face vector of an arbitrary positive cluster complex of finite type using these results. Furthermore, we apply our results to the -tilting theory of cluster-tilted algebras of finite representation type using the correspondence between clusters and support -tilting modules.

A stochastic simplicial SIS model for complex networks

We propose a stochastic epidemiological model for simplicial complex networks by means of a stochastic differential equation (SDE) that extends the mean field approach of the simplicial social contagion model. We show that, under appropriate conditions, if the stochastic basic reproductive number is smaller than one, then the disease dies out with probability one; otherwise the solution of the SDE oscillates infinitely often around a point which can be explicitly computed. We perform numerical experiments which illustrate the theoretical results. In addition, we carry out simulations on a real simplicial network and on a synthetic network, which show good agreement with the theoretical and numerical predictions of the SDE.

Expected invariants of simplicial complexes obtained from random point samples

Expected invariants of simplicial complexes obtained from random point samples

Persistence of Weighted Ordinal Partition Networks for Dynamic State Detection

One of the most important problems arising in time series analysis is that of bifurcation, or change point detection. That is, given a collection of time series over a varying parameter, when has the structure of the underlying dynamical system changed? For this task, we turn to the field of topological data analysis (TDA), which encodes information about the shape and structure of data. The idea of utilizing tools from TDA for signal processing tasks, known as topological signal processing (TSP), has gained much attention in recent years, largely through a standard pipeline that computes the persistent homology of the point cloud generated by the Takens' embedding. However, this procedure is limited by computation time since the simplicial complex generated in this case is large, but also has a great deal of redundant data. For this reason, we turn to a more recent method for encoding the structure of the attractor, which constructs an ordinal partition network (OPN) representing information about when the dynamical system has passed between certain regions of state space. The result is a weighted graph whose structure encodes information about the underlying attractor. Our previous work began to find ways to package the information of the OPN in a manner that is amenable to TDA; however, that work only used the network structure and did nothing to encode the additional weighting information. In this paper, we take the next step: building a pipeline to analyze the weighted OPN with TDA and showing that this framework provides more resilience to noise or perturbations in the system and improves the accuracy of the dynamic state detection.

Topological learning approach to characterize lipids.

Biological membranes play a key role in cellular compartmentalization, structure, and its signaling pathways. At varying temperatures, individual membrane lipids sample from different configurations, frequently leading to higher order phase phenomena. Order parameters of lipid tails are often used as a metric for quantifying phase behavior, however typically only representing bulk ensembles. Here we present a persistent homology-based method for quantifying the structural features of individual and bulk lipids, providing local and contextual information on lipid tail organization. Our method leverages the mathematical machinery of algebraic topology and machine learning to infer temperature dependent structural information of lipids from static coordinates. To train our model, we generated multiple molecular dynamic trajectories of DPPC membranes at varying temperatures. A fingerprint was then constructed for each set of lipid coordinates by a persistent homology filtration, in which spheres were grown around the lipid atoms while tracking their intersections. The sphere filtration formed a simplicial complex that captures enduring key topological features of the configuration landscape. Following fingerprint extraction for physiologically relevant temperatures, the persistence data were used to train an Attention-based neural network for assignment of temperature values to selected membrane regions. Attention is a deep learning algorithm traditionally used for natural language processing, having shown great promise in biological problems as demonstrated by Alphafold2. The Attention algorithm uses global context of the data to predict a desired property; in this case pair-wise interactions between atoms, the global context, and lipid coordinates are used to predict a structural score, indicative of temperature. Our persistence homology-based method captures the local structural effects of lipids adjacent to other membrane constituents, e.g., sterols and proteins, quantifying structural features that facilitate entropically driven membrane organization. This topological learning approach predicts membrane temperature values from static coordinates across multiple spatial resolutions.