Simplicial Complex Research Articles

Robustness of interdependent higher-order networks

In real complex systems, interactions occur not only between a pair of nodes, but also in groups of three or more nodes, which can be abstracted as higher-order structures in the networks. The simplicial complex is one of a model to represent systems with both low-order and higher-order structures. In this paper, we study the robustness of interdependent simplicial complexes under random attacks, where the complementary effects of the higher-order structure are introduced. When a higher-order node in a 2-simplex fails, its dependent node in the other layer survives with a certain probability due to the complementary effects from the 2-simplex. By using the percolation method, we derive the percolation threshold and the size of the giant component when the cascading failure reaches its steady state. The simulation results agree well with analytical predictions. We find that the type of phase transition changes from the first-order to the second-order when the complementary effect of the higher-order structure on the dependent node increases or the number of 2-simplices in the interdependent simplicial complex increases. While the interlayer coupling strength increases, the type of phase transition changes from the second-order to the first-order. In particular, even if the higher-order interactions do not provide complementary effects for dependent nodes, the robustness of the interdependent heterogeneous simplicial complex is higher than that of the ordinary interdependent network with the same average degree due to the existence of 2-simplices in the system. This study furthers our understanding in the robustness of interdependent higher-order networks.

Discrete Differential Calculus on Simplicial Complexes and Constrained Homology

Discrete Differential Calculus on Simplicial Complexes and Constrained Homology

The h-vectors of the edge rings of a special family of graphs

The h-vectors of homogeneous rings are one of the most important invariants that often reflect ring-theoretic properties. On the other hand, there are a few examples of edge rings of graphs whose h-vectors are explicitly computed. In this paper, we compute the h-vector of a special family of graphs, by using the technique of initial ideals and the associated simplicial complex.

A characterization of normal 3-pseudomanifolds with at most two singularities

Characterizing face-number-related invariants of a given class of simplicial complexes has been a central topic in combinatorial topology. In this regard, one of the well-known invariants is g2. Let K be a normal 3-pseudomanifold such that g2(K)≤g2(lk(v))+9 for some vertex v in K. Suppose either K has only one singularity or K has two singularities (at least) one of which is an RP2-singularity. We prove that K is obtained from some boundary complexes of 4-simplices by a sequence of operations of types connected sums, bistellar 1-moves, edge contractions, edge expansions, vertex foldings, and edge foldings. In case K has one singularity, |K| is a handlebody with its boundary coned off. Further, we prove that the above upper bound is sharp for such normal 3-pseudomanifolds.

Rigidity of terminal simplices in persistent homology

Given a filtration function on a finite simplicial complex, stability theorem of persistent homology states that the corresponding barcode is continuous with respect to changes in the filtration function. However, due to the discrete setting of simplicial complexes, the simplices terminating matched bars cannot change continuously for arbitrary perturbations of filtration functions. In this paper we provide a sufficient condition for rigidity of a terminal simplex, i.e., a condition on varepsilon >0 implying that the terminal simplex of a homology class or a bar in persistent homology remains constant through varepsilon -perturbations of filtration function. The condition for a homology class or a bar in dimension n depends only on the barcodes in dimensions n and n+1.

Multiscale Topological Indices for the Quantitative Prediction of SARS CoV-2 Binding Affinity Change upon Mutations.

The Coronavirus disease 2019 (COVID-19) has affected people's lives and the development of the global economy. Biologically, protein-protein interactions between SARS-CoV-2 surface spike (S) protein and human ACE2 protein are the key mechanism behind the COVID-19 disease. In this study, we provide insights into interactions between the SARS-CoV-2 S-protein and ACE2, and propose topological indices to quantitatively characterize the impact of mutations on binding affinity changes (ΔΔG). In our model, a series of nested simplicial complexes and their related adjacency matrices at various different scales are generated from a specially designed filtration process, based on the 3D structures of spike-ACE2 protein complexes. We develop a set of multiscale simplicial complexes-based topological indices, for the first time. Unlike previous graph network models, which give only a qualitative analysis, our topological indices can provide a quantitative prediction of the binding affinity change caused by mutations and achieve great accuracy. In particular, for mutations that happened at specifical amino acids, such as Polar amino acids or Arginine amino acids, the correlation between our topological gravity model index and binding affinity change, in terms of Pearson correlation coefficient, can be higher than 0.8. As far as we know, this is the first time multiscale topological indices have been used in the quantitative analysis of protein-protein interactions.

Privacy-Preserving Representation Learning for Text-Attributed Networks with Simplicial Complexes

Although recent network representation learning (NRL) works in text-attributed networks demonstrated superior performance for various graph inference tasks, learning network representations could always raise privacy concerns when nodes represent people or human-related variables. Moreover, standard NRLs that leverage structural information from a graph proceed by first encoding pairwise relationships into learned representations and then analysing its properties. This approach is fundamentally misaligned with problems where the relationships involve multiple points, and topological structure must be encoded beyond pairwise interactions. Fortunately, the machinery of topological data analysis (TDA) and, in particular, simplicial neural networks (SNNs) offer a mathematically rigorous framework to evaluate not only higher-order interactions, but also global invariant features of the observed graph to systematically learn topological structures. It is critical to investigate if the representation outputs from SNNs are more vulnerable compared to regular representation outputs from graph neural networks (GNNs) via pairwise interactions. In my dissertation, I will first study learning the representations with text attributes for simplicial complexes (RT4SC) via SNNs. Then, I will conduct research on two potential attacks on the representation outputs from SNNs: (1) membership inference attack, which infers whether a certain node of a graph is inside the training data of the GNN model; and (2) graph reconstruction attacks, which infer the confidential edges of a text-attributed network. Finally, I will study a privacy-preserving deterministic differentially private alternating direction method of multiplier to learn secure representation outputs from SNNs that capture multi-scale relationships and facilitate the passage from local structure to global invariant features on text-attributed networks.

A framework for topological music analysis (TMA)

In the present article we describe and discuss a framework for applying different topological data analysis (TDA) techniques to a music fragment given as a score in traditional Western notation. We first consider different sets of points in Euclidean spaces of different dimensions that correspond to musical events in the score, and obtain their persistent homology features. Then we introduce two families of simplicial complexes that can be associated with chord sequences, and leverage homology to compute their salient features. Finally, we show the results of applying the described methods to the analysis and stylistic comparison of fragments from three Brandenburg Concertos by J.S. Bach and two pieces from the Graffiti series by Mexican composer Armando Luna.

The Impact of Higher-Order Interactions on the Synchronization of Hindmarsh–Rose Neuron Maps under Different Coupling Functions

In network analysis, links depict the connections between each pair of network nodes. However, such pairwise connections fail to consider the interactions among more agents, which may be indirectly connected. Such non-pairwise or higher-order connections can be signified by involving simplicial complexes. The higher-order connections become even more noteworthy when it comes to neuronal network synchronization, an emerging phenomenon responsible for the many biological processes in real-world phenomena. However, involving higher-order interactions may considerably increase the computational costs. To confound this issue, map-based models are more suitable since they are faster, simpler, more flexible, and computationally more optimal. Therefore, this paper addresses the impact of pairwise and non-pairwise neuronal interactions on the synchronization state of 10 coupled memristive Hindmarsh–Rose neuron maps. To this aim, electrical, inner linking, and chemical synaptic functions are considered as two- and three-body interactions in three homogeneous and two heterogeneous cases. The results show that through chemical pairwise and non-pairwise synapses, the neurons achieve synchrony with the weakest coupling strengths.

Perfect Matching Complexes of Honeycomb Graphs

The perfect matching complex of a graph is the simplicial complex on the edge set of the graph with facets corresponding to perfect matchings of the graph. This paper studies the perfect matching complexes, $\mathcal{M}_p(H_{k \times m\times n})$, of honeycomb graphs. For $k = 1$, $\mathcal{M}_p(H_{1\times m\times n})$ is contractible unless $n\geq m=2$, in which case it is homotopy equivalent to the $(n-1)$-sphere. Also, $\mathcal{M}_p(H_{2\times 2\times 2})$ is homotopy equivalent to the wedge of two 3-spheres. The proofs use discrete Morse theory.

Closed-loop control of higher-order complex networks: Finite-time and pinning strategies

Increasing evidences have demonstrated the essentiality of many-body interactions in modeling various systems in physics, biology, and social sciences. Control strategies are useful tools to steer real-world complex systems to desired targets. Existing works focus on open-loop control, which relies on predefined control signals, and closed-loop control, which requires an infinite-time duration, on conventional networks expressed by graphs. This work designs closed-loop controllers for higher-order complex networks characterized by simplicial complexes. Protocols including the linear feedback controller and a switching controller, which is a combination of a linear controller and a finite-time controller, are first adapted to higher-order networks to realize successful control tasks, with the rigorous upper bound of the control time theoretically derived and compared. Extensive numerical simulations on benchmark and real-world higher-order complex networks demonstrate the effectiveness of the control protocols and further provide insights on pinning control strategy to higher-order networks. These results shed light on a comprehensive discovery of controlling higher-order complex networks and have also applied values.

A novel simplicial complex representation of social media texts: The case of Twitter

With the rise in technological advances in the last decades, there has been an increase in the complexity of the many systems that we are members of. The emergence and development of web-based platforms have particularly revolutionized the way we express our opinions and feelings, communicate with one another, and in general, the manner we gain and share information. This constant rise in the scale and variety of information and opinions and the overall complexity of online social networks have motivated the search for alternative methods that can capture the complex and higher-order characteristics of the interactions within these systems.For this purpose, this paper has attempted to showcase and introduce a novel method for studying the various dimensions of the characteristics and interactions present in online social networks. Using the concept of simplicial complexes, a novel framework is defined and presented to analyze the text contexts of opinions shared by users of these complex systems. The spread and propagation of information is then researched in the representations using epidemic models and then applied to real-life datasets obtained through Twitter.

Epidemic trajectories and awareness diffusion among unequals in simplicial complexes

The interplay between awareness diffusion and epidemic spreading has been an active topic of research in recent years. Studies have shown that group interactions are an important consideration in contagion processes, and that thus higher-order interactions should be introduced into epidemic modeling. Research has also shown that individual responses to an unfolding epidemic are often strongly heterogeneous. We therefore present a two-layer network model, where the diffusion of awareness unfolds over 2-simplicial complexes in one layer, and the actual epidemic spreading unfolds over pairwise physical contacts in the other layer. The model takes into account individual differences in the degree of acceptance of information and self-protection measures once the epidemic is perceived. We use the micro Markov chain approach to determine the epidemic threshold of the model, which agrees well with the results obtained by Monte Carlo simulations. We show that the synergistic reinforcement due to 2-simplicial complexes in the virtual layer can restrain epidemic spreading by facilitating awareness diffusion, and moreover, that individual heterogeneity in the physical layer can increase the epidemic threshold and decrease the size of epidemic transmission. However, heterogeneity in the perception can also have the opposite effect because it inhibits the diffusion of awareness. Our results reveal the intricate interplay between awareness diffusion and epidemic spreading, and we hope they can help determine effective control measures.

Minimal graphs for contractible and dismantlable properties

The notion of a contractible transformation on a graph was introduced by Ivashchenko as a means to study molecular spaces arising from digital topology and computer image analysis, and more recently has been applied to topological data analysis. Contractible transformations involve a list of four elementary moves that can be performed on the vertices and edges of a graph, and it has been shown by Chen, Yau, and Yeh that these moves preserve the simple homotopy type of the underlying clique complex. A graph is said to be I-contractible if one can reduce it to a single isolated vertex via a sequence of contractible transformations. Inspired by the notions of collapsible and non-evasive simplicial complexes, in this paper we study certain subclasses of I-contractible graphs where one can collapse to a vertex using only a subset of these moves. Our main results involve constructions of minimal examples of graphs for which the resulting classes differ. We also relate these classes of graphs to the notion of k-dismantlable graphs and k-collapsible complexes, which also leads to a minimal counterexample to an erroneous claim of Ivashchenko from the literature. We end with some open questions.

On the dimension of dual modules of local cohomology and the Serre's condition for the unmixed Stanley–Reisner ideals of small height

In this paper, we focus on the dimension of dual modules of local cohomology of Stanley–Reisner rings to obtain a new vector. This vector contains important information on the Serre's condition (Sr) and the CMt property as well as the depth of Stanley–Reisner rings. We prove some results in this regard including lower bounds for the depth of Stanley–Reisner rings. Further, we give a characterization of (d−1)-dimensional simplicial complexes with codimension two which are (Sd−3) but they are not Cohen–Macaulay. By using this characterization, we obtain a condition to equality of projective dimension of the Stanley–Reisner rings and the arithmetical rank of their Stanley–Reisner ideals. Moreover, our characterization allows us to compute the h-vectors and give a negative answer to a known question regarding these vectors.

Leray Numbers of Tolerance Complexes

Let K be a simplicial complex on vertex set V. K is called d-Leray if the homology groups of any induced subcomplex of K are trivial in dimensions d and higher. K is called d-collapsible if it can be reduced to the void complex by sequentially removing a simplex of size at most d that is contained in a unique maximal face. Motivated by results of Eckhoff and of Montejano and Oliveros on “tolerant” versions of Helly’s theorem, we define the t-tolerance complex of K, {mathcal {T}}_{t}(K), as the simplicial complex on vertex set V whose simplices are formed as the union of a simplex in K and a set of size at most t. We prove that for any d and t there exists a positive integer h(t, d) such that, for every d-collapsible complex K, the t-tolerance complex {mathcal {T}}_t(K) is h(t, d)-Leray. As an application, we present some new tolerant versions of the colorful Helly theorem.

A directed persistent homology theory for dissimilarity functions

We develop a theory of persistent homology for directed simplicial complexes which detects persistent directed cycles in odd dimensions. We relate directed persistent homology to classical persistent homology, prove some stability results, and discuss the computational challenges of our approach. Our directed persistent homology theory is motivated by homology with semiring coefficients: by explicitly removing additive inverses, we are able to detect directed cycles algebraically.

Impact of simplicial complexes on epidemic spreading in partially mapping activity-driven multiplex networks.

Over the past decade, the coupled spread of information and epidemic on multiplex networks has become an active and interesting topic. Recently, it has been shown that stationary and pairwise interactions have limitations in describing inter-individual interactions , and thus, the introduction of higher-order representation is significant. To this end, we present a new two-layer activity-driven network epidemic model, which considers the partial mapping relationship among nodes across two layers and simultaneously introduces simplicial complexes into one layer, to investigate the effect of 2-simplex and inter-layer mapping rate on epidemic transmission. In this model, the top network, called the virtual information layer, characterizes information dissemination in online social networks, where information can be diffused through simplicial complexes and/or pairwise interactions. The bottom network, named as the physical contact layer, denotes the spread of infectious diseases in real-world social networks. It is noteworthy that the correspondence among nodes between two networks is not one-to-one but partial mapping. Then, a theoretical analysis using the microscopic Markov chain (MMC) method is performed to obtain the outbreak threshold of epidemics, and extensive Monte Carlo (MC) simulations are also carried out to validate the theoretical predictions. It is obviously shown that MMC method can be used to estimate the epidemic threshold; meanwhile, the inclusion of simplicial complexes in the virtual layer or introductory partial mapping relationship between layers can inhibit the spread of epidemics. Current results are conducive to understanding the coupling behaviors between epidemics and disease-related information.

Algebraic Characterization of SSC of Uni-Cyclic Multigraphs

We introduce first the spanning simplicial complex (SSC) of a multigraph [Formula: see text], which gives a generalization of the SSC associated with a simple graph [Formula: see text]. Combinatorial properties are discussed for the SSC of a family of uni-cyclic multigraphs [Formula: see text] with [Formula: see text] edges including [Formula: see text] multiple edges within and outside the cycle of length [Formula: see text], which are then used to compute the [Formula: see text]-vector and Hilbert series of face ring [Formula: see text] for the SSC[Formula: see text]. Moreover, we find the associated primes of the facet ideal [Formula: see text]. Finally, we device a formula for homology groups of [Formula: see text] and prove that the SSC of a family of uni-cyclic multigraphs is Cohen-Macaulay.

Role of second-order reputation evaluation in the multi-player snowdrift game on scale-free simplicial complexes

Over the past few decades, the network topology is often characterized via pairwise interactions in the field of indirect reciprocity. However, from human society to ecosystem, interactions usually occur in a group with three or more agents and cannot be fully explained in term of dyadic interactions. In this paper, we improve the multi-player snowdrift game based on scale-free simplicial complexes, and introduce four representative second-order assessment norms to analyze the impact of higher-order topology and reputation evaluation on collective cooperation behaviors. In the model, a focal player i’s general payoff will be regulated by the weight (λ) of player i’s payoff gained in 1-simplices and 2-simplices. Through plenty of Monte Carlo simulations, the results of numerical simulation show that cooperation level can be elevated under certain parameters by introducing non-pairwise interactions. The frequency of cooperation increases (decreases) with the growth of percentage of 2-simplex (ρ) when the cost-to-benefit (r) is smaller (larger). Meanwhile, cooperation behavior can be greatly changed by different reputation evaluation norms (e.g., Shunning rule presents the worst case in terms of the stationary cooperation level, while Image scoring norm creates the highest one). Current results provide some insightful clues for us to comprehend the emergence of mutually beneficial symbiosis in the realistic networked population.