Simplicial Complex Research Articles

Influential simplices mining via simplicial convolutional networks

The identification of influential simplices is crucial for understanding higher-order network dynamics. Yet, despite relatively mature research on influential nodes (0-simplices) mining, characterizing simplices’ influence and identifying influential simplices remain challenging due to notable discrepancies in vital nodes and vital simplices mining. In this paper, we propose a higher-order graph learning model, named influential simplices mining neural networks (ISMnet), to identify vital simplices in simplicial complexes. ISMnet leverages novel higher-order representations: hierarchical bipartite graphs and higher-order hierarchical (HoH) Laplacians, where target simplices are grouped into a hub set and can interact with other simplices. It also employs learnable graph convolution operators in each HoH Laplacian domain to capture interactions among simplices and can identify influential simplices of arbitrary order by changing the hub set. Notably, ISMnet addresses the limitations inherent in traditional graph neural networks that struggle with higher-order tasks, while seamlessly retaining the capability to exploit network topology and node features concurrently. Numerical results on empirical and synthetic datasets demonstrate that ISMnet significantly outperforms existing methods by at least 12% and 4%, respectively, in ranking 2-simplices. In general, this novel framework promises to serve as a potent tool in higher-order network analysis.

Topological Learning Approach to Characterizing Biological Membranes.

Biological membranes play key roles in cellular compartmentalization, structure, and its signaling pathways. At varying temperatures, individual membrane lipids sample from different configurations, a process that frequently leads to higher-order phase behavior and phenomena. Here, we present a persistent homology (PH)-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 on lipids from static coordinates. To train our model, we generated multiple molecular dynamics trajectories of dipalmitoyl-phosphatidylcholine membranes at varying temperatures. A fingerprint was then constructed for each set of lipid coordinates by PH filtration, in which interaction 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 using homology, yielding persistence data. Following fingerprint extraction for physiologically relevant temperatures, the persistence data were used to train an attention-based neural network for assignment of effective temperature values to selected membrane regions. Our persistence homology-based method captures the local structural effects, via effective temperature, of lipids adjacent to other membrane constituents, e.g., sterols and proteins. This topological learning approach can predict lipid effective temperatures from static coordinates across multiple spatial resolutions. The tool, called MembTDA, can be accessed at https://github.com/hyunp2/Memb-TDA.

Dancing with math: Using Kleins quartic for music generation

Studying music from a mathematical perspective is often based on the notions of symmetry and topological space. A group G acting on some set S of musical objects in a meaningful way is seen as a symmetry group in music. This can be used in analysing motif development and chord progressions. In neo-Riemannian analysis, one has three principal chord transformations that generate a group GD_12, acting on the set S of 24 major and minor triads. Moreover, this theory is visualized through a simplicial complex whose underlying space is a topological torus. In this paper, we first introduce various symmetry groups and graphic presentations in music theory. We then propose a way of doing neo-Riemannian analysis on Kleins quartic, which is a genus 3 surface instead of a torus, realizing an idea of John Baez. Finally, we use our theory to perform a harmonic analysis of The Imperial March from Star Wars.

Simplicial epidemic model with individual resource

The investment and rational use of medical resources are crucial for the prevention and control of epidemics. Here we propose a coupled dynamics model of individual resource and disease transmission on simplicial complexes that considers the dynamic changes of resource quantity during the spreading of the epidemic. Firstly, we derive the solutions (infection density) of the disease transmission dynamics equations under the mean-field approximation, and analyze their stability. We find that the number of stable solutions is related to the infection rates of both interactions and higher-order interactions, and we obtain the transmission threshold under different initial infection densities. Secondly, in the numerical simulation, we discover a double hysteresis loop of infection density as it evolves with the infection rate of interactions. Moreover, we find a critical value for the cost of disease recovery; when the recovery cost is above this critical value, the system state transitions from partial infection to complete infection. Interestingly, we find a maximum value for infection density when the system is in partial infection state. Finally, we verify the existence of the double hysteresis loop on real-world networks.

Reconstructing higher-order interactions in coupled dynamical systems

Higher-order interactions play a key role for the operation and function of a complex system. However, how to identify them is still an open problem. Here, we propose a method to fully reconstruct the structural connectivity of a system of coupled dynamical units, identifying both pairwise and higher-order interactions from the system time evolution. Our method works for any dynamics, and allows the reconstruction of both hypergraphs and simplicial complexes, either undirected or directed, unweighted or weighted. With two concrete applications, we show how the method can help understanding the complexity of bacterial systems, or the microscopic mechanisms of interaction underlying coupled chaotic oscillators.

Zigzag persistence for image processing: New software and applications

Topological image analysis is a powerful tool for understanding the structure and topology of images, being persistent homology one of its most popular methods. However, persistent homology requires a chain of inclusions of topological spaces, which can be challenging for digital images. In this article, we explore the use of zigzag persistence, a recent variant of traditional persistence, for digital image processing. To this end, new algorithms are developed to build a simplicial complex associated to a digital image and to compute the relationships between homology classes of a sequence of binary images via zigzag persistence. Additionally, we provide a simple software to use them. We demonstrate its effectiveness by applying it to a real-world problem of analyzing honey bee sperm videos.

Higher-order non-Markovian social contagions in simplicial complexes

Higher-order structures such as simplicial complexes are ubiquitous in numerous real-world networks. Empirical evidence reveals that interactions among nodes occur not only through edges but also through higher-dimensional simplicial structures such as triangles. Nevertheless, classic models such as the threshold model fail to capture group interactions within these higher-order structures. In this paper, we propose a higher-order non-Markovian social contagion model, considering both higher-order interactions and the non-Markovian characteristics of real-world spreading processes. We develop a mean-field theory to describe its evolutionary dynamics. Simulation results reveal that the theory is capable of predicting the steady state of the model. Our theoretical analyses indicate that there is an equivalence between the higher-order non-Markovian and the higher-order Markovian social contagions. Besides, we find that non-Markovian recovery can boost the system resilience to withstand a large-scale infection or a small-scale infection under different conditions. This work deepens our understanding of the behaviors of higher-order non-Markovian social contagions in the real world.

Evolutionary dynamics of memory-one extortion and generosity on scale-free simplices

Both extortionate and generous strategies within the framework of zero-determinant (ZD) strategy can be linearly related to the opponent's payoffs. Here we explore their evolutionary performances for both iterated two-player and multi-player games. We mainly investigate two scenarios on simplicial complexes: one is the evolutionary scenario with cooperation (C), defection (D) and extortion (E), the other is the cooperation, defection and generosity (G). We find that both extortion and generosity can help cooperators resist the invasion of defectors. Intriguingly, the extortioner, who always enforces higher payoff than co-players, is more beneficial to promote cooperation than generosity. Compared with kindness and indulgence, being strict with the co-players can actually maintain cooperative behaviors in the long run. Simulations on the simplicial complexes revel that the catalytic effect of extortion on the evolution of cooperation even be more obvious in iterated multi-player social dilemmas than the two-player case, hence cooperation becomes more popular when the networks include more 2-simplex interactions. Our results can help to illustrate the role of higher-order interaction in the evolution of altruistic behaviors.

The SIQRS Propagation Model With Quarantine on Simplicial Complexes

The SIQRS Propagation Model With Quarantine on Simplicial Complexes

Linear codes from simplicial complexes over $${\mathbb {F}}_{2^n}$$

Linear codes from simplicial complexes over $${\mathbb {F}}_{2^n}$$

Quantum walk on simplicial complexes for simplicial community detection

Quantum walk on simplicial complexes for simplicial community detection

Not your private tête-à-tête: leveraging the power of higher-order networks to study animal communication.

Animal communication is frequently studied with conventional network representations that link pairs of individuals who interact, for example, through vocalization. However, acoustic signals often have multiple simultaneous receivers, or receivers integrate information from multiple signallers, meaning these interactions are not dyadic. Additionally, non-dyadic social structures often shape an individual's behavioural response to vocal communication. Recently, major advances have been made in the study of these non-dyadic, higher-order networks (e.g. hypergraphs and simplicial complexes). Here, we show how these approaches can provide new insights into vocal communication through three case studies that illustrate how higher-order network models can: (i) alter predictions made about the outcome of vocally coordinated group departures; (ii) generate different patterns of song synchronization from models that only include dyadic interactions; and (iii) inform models of cultural evolution of vocal communication. Together, our examples highlight the potential power of higher-order networks to study animal vocal communication. We then build on our case studies to identify key challenges in applying higher-order network approaches in this context and outline important research questions that these techniques could help answer. This article is part of the theme issue 'The power of sound: unravelling how acoustic communication shapes group dynamics'.

Normalized Hodge Laplacian Matrix and Application to Random Walk on Simplicial Complexes

Normalized Hodge Laplacian Matrix and Application to Random Walk on Simplicial Complexes

Coupled propagation between one communicable disease and related two types of information on multiplex networks with simplicial complexes

Currently, regarding the coupling transmission of information and disease, most researches only focus on the impact of one type of information on disease transmission, and some works may be correlated with two types of information, but they are usually based upon pairwise interactions, which ignored the complexity and diversity of information transmission. In real life, the characteristics of interpersonal networks often involve both group interactions and pairwise interactions. Aiming to explore the influence of active or passive information diffusion on the transmission of epidemics on a higher-order network, the coupled two-layered multiplex network model between communicable disease and information transmission is presented. Here, the upper layer characterizes the information transmission network constructed from random simplicial complexes, while the lower layer provides the underlying network that can transmit communicable diseases. First, on the basis of micro Markov chain (MMC) method, the epidemic threshold is analytically derived for the proposed model, indicating that the diffusion of positive and negative preventive information can exert a substantial influence on the critical threshold. Then, by comparing the results acquired by MMC and Monte Carlo simulations (MCS), it can be explicitly expressed that the differences between them are minimal, which implies that the model can predict the transmission of epidemics in the network well. Finally, through extensive MCS, it is also indicated that introducing a 2-simplex into the information layer helps to suppress the propagation or outbreak of communicable diseases, which offers some new ideas or routes for formulating strategies to inhibit communicable diseases.

COMPLEX CONTAGION IN SOCIAL SYSTEMS WITH DISTRUST

Social systems are characterized by the presence of group interactions and by the existence of both trust and distrust relations. Although there is a wide literature on signed social networks, where positive signs associated to the links indicate trust, friendship, agreement, while negative signs represent distrust, antagonism, and disagreement, very little is known about the effect that signed interactions can have on the spreading of social behaviors when, not only pairwise, but also higher-order interactions are taken into account. In this paper, we introduce a model of complex contagion on signed simplicial complexes, and we investigate the role played by trust and distrust on the dynamics of a social contagion process, where exposure to multiple sources is needed for the contagion to occur. The presence of higher-order signed structures in our model naturally induces new infection and recovery mechanisms, thus increasing the richness of the contagion dynamics. Through numerical simulations and analytical results in the mean-field approximation, we show how distrust determines the way the system moves from a state where no individuals adopt the social behavior, to a state where a finite fraction of the population actively spreads it. Interestingly, we observe that the fraction of spreading individuals displays a non-monotonic dependence with respect to the average number of connections between individuals. We then investigate how social balance affects social contagion, finding that balanced triads have an ambivalent impact on the spreading process, either promoting or impeding contagion based on the relative abundance of fully trusted relations. Our results shed light on the nontrivial effect of trust on the spreading of social behaviors in systems with group interactions, paving the way to further investigations of spreading phenomena in structured populations.

An approach to extract topological information from Intuitionistic Fuzzy Sets and their application in obtaining a Natural Hierarchical Clustering Algorithm

Topological data analysis (TDA) is a powerful mathematical framework that extracts valuable insights about the shape and structure of complex datasets by identifying and analyzing underlying topological features, including connected components, holes, voids, and higher-dimensional counterparts. TDA achieves this by constructing a simplicial complex from the data, comprising simple shapes like points, lines, triangles, and higher-dimensional counterparts. The Vietoris–Rips complex (VRC), a widely used method, constructs simplicial complexes by measuring the distances between data points in a metric space. While the link between Intuitionistic Fuzzy Sets (IFSs) and TDA is clear through intuitionistic fuzzy distance measures, the literature has not explored the application of TDA for extracting topological features from uncertain and vague datasets using IFSs. In this paper, we propose a novel approach that leverages Intuitionistic Fuzzy Distance Measures (IFDMs) to generate the Intuitionistic Fuzzy Vietoris–Rips Complex (IFVRC) of IFSs. Specifically, we investigate the persistence of invariant relations of topological features between IFVRCs constructed using different distances, including Atanassov’s Euclidean distance (l2IFS), Boran and Akay’s distance (D), and Liu’s distance measure (dL), over artificially generated IFSs. Furthermore, we demonstrate that the generation of IFVRC can be viewed as a Natural Hierarchical Clustering Algorithm (NHCA) for IFSs. Finally, we apply the proposed IFVRC as a Natural Hierarchical Clustering Algorithm to cluster intuitionistic fuzzy datasets related to Car and Building materials. Moreover, we incorporate a relatively large dataset, the GTZAN dataset from the Kaggle datasets repository, to classify music genres based on their topological information.

Khovanov homology, wedges of spheres and complexity

Our main result has topological, combinatorial and computational flavor. It is motivated by a fundamental conjecture stating that computing Khovanov homology of a closed braid of fixed number of strands has polynomial time complexity. We show that the independence simplicial complex I(w) associated to the 4-braid diagram w (and therefore its Khovanov spectrum at extreme quantum degree) is contractible or homotopy equivalent to either a sphere, or a wedge of two spheres (possibly of different dimensions), or a wedge of three spheres (at least two of them of the same dimension), or a wedge of four spheres (at least three of them of the same dimension). On the algorithmic side we prove that finding the homotopy type of I(w) can be done in polynomial time with respect to the number of crossings in w. In particular, we prove the wedge of spheres conjecture for circle graphs obtained from 4-braid diagrams. We also introduce the concept of Khovanov adequate diagram and discuss criteria for a link to have a Khovanov adequate braid diagram with at most 4 strands.

Synchronizability in randomized weighted simplicial complexes.

We present a formula for determining synchronizability in large, randomized, and weighted simplicial complexes. This formula leverages eigenratios and costs to assess complete synchronizability under diverse network topologies and intensity distributions. We systematically vary coupling strengths (pairwise and three body), degree, and intensity distributions to identify the synchronizability of these simplicial complexes of the identical oscillators with natural coupling. We focus on randomized weighted connections with diffusive couplings and check synchronizability for different cases. For all these scenarios, eigenratios and costs reliably gauge synchronizability, eliminating the need for explicit connectivity matrices and eigenvalue calculations. This efficient approach offers a general formula for manipulating synchronizability in diffusively coupled identical systems with higher-order interactions simply by manipulating degrees, weights, and coupling strengths. We validate our findings with simplicial complexes of Rössler oscillators and confirm that the results are independent of the number of oscillators, connectivity components, and distributions of degrees and intensities. Finally, we validate the theory by considering a real-world connection topology using chaotic Rössler oscillators.

A unified framework for simplicial Kuramoto models

Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models, categorized into three main groups: “simple” models, “Hodge-coupled” models, and “order-coupled” (Dirac) models. Our framework is based on topology and discrete differential geometry, as well as gradient systems and frustrations, and permits a systematic analysis of their properties. We establish an equivalence between the simple simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of manifoldness of the simplicial complex. Then, starting from simple models, we describe the notion of simplicial synchronization and derive bounds on the coupling strength necessary or sufficient for achieving it. For some variants, we generalize these results and provide new ones, such as the controllability of equilibrium solutions. Finally, we explore a potential application in the reconstruction of brain functional connectivity from structural connectomes and find that simple edge-based Kuramoto models perform competitively or even outperform complex extensions of node-based models.

The parameterized complexity of finding minimum bounded chains

Finding the smallest d-chain with a specific (d−1)-boundary in a simplicial complex is known as the Minimum Bounded Chain problem (MBCd). MBCd is NP-hard for all d≥2. In this paper, we prove that it is also W[1]-hard for all d≥2, if we parameterize the problem by solution size. We also give an algorithm solving MBC1 in polynomial time and introduce and implement two fixed parameter tractable (FPT) algorithms solving MBCd for all d. The first algorithm uses a shortest path approach and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.