Simplicial Complex Research Articles

Simplicial Complexes with Many Facets are Vertex Decomposable

Suppose $\Delta$ is a pure simplicial complex on $n$ vertices having dimension $d$ and let $c = n-d-1$ be its codimension in the simplex. Terai and Yoshida proved that if the number of facets of $\Delta$ is at least $\binom{n}{c}-2c+1$, then $\Delta$ is Cohen-Macaulay. We improve this result by showing that these hypotheses imply the stronger condition that $\Delta$ is vertex decomposable. We give examples to show that this bound is optimal, and that the conclusion cannot be strengthened to the class of matroids or shifted complexes. We explore an application to Simon's Conjecture and discuss connections to other results from the literature.

Clique Homology is -hard.

We address the long-standing question of the computational complexity of determining homology groups of simplicial complexes, a fundamental task in computational topology, posed by Kaibel and Pfetsch over twenty years ago. We show that decision problem is -hard and the exact counting version is -hard. In fact, we strengthen this by showing that the problems remains hard in the case of clique complexes, a family of simplicial complexes specified by a graph which is relevant to the problem of topological data analysis. The proof combines a number of techniques from Hamiltonian complexity and algebraic topology. As we discuss, a version of the problems satisfying a suitable promise and certain constraints is contained in and , respectively. This suggests that the seemingly classical problem may in fact be quantum mechanical. We discuss potential implications for the problem of quantum advantage in topological data analysis.

Contractibility of the Rips complexes of Integer lattices via local domination

We prove that for each positive integer n n , the Rips complexes of the n n -dimensional integer lattice in the d 1 d_1 metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above n 2 ( 2 n − 1 ) n^2(2n-1) , with the bounds arising from the Jung constants. We introduce a new concept of locally dominated vertices in a simplicial complex, upon which our proof strategy is based. This allows us to deduce the contractibility of the Rips complexes from a local geometric condition called local crushing. In the case of the integer lattices in dimension n n and a fixed scale r r , this condition entails the comparison of finitely many distances to conclude that the corresponding Rips complex is contractible. In particular, we are able to verify that for n = 1 , 2 , 3 n=1,2,3 , the Rips complex of the n n -dimensional integer lattice at scale greater or equal to n n is contractible. We conjecture that the same proof strategy can be used to extend this result to all dimensions n n .

Quantum topological data analysis via the estimation of the density of states

We develop a quantum topological data analysis (QTDA) protocol based on the estimation of the density of states (DOS) of the combinatorial Laplacian. Computing topological features of graphs and simplicial complexes is crucial for analyzing data sets and building explainable artificial intelligence solutions. This task becomes computationally hard for simplicial complexes with over 60 vertices and high-degree topological features due to a combinatorial scaling. We propose to approach the task by embedding underlying hypergraphs as effective quantum Hamiltonians and evaluating their density of states from the time evolution. Specifically, we compose propagators as quantum circuits using the Cartan decomposition of effective Hamiltonians and sample overlaps of time-evolved states using multifidelity protocols. Next we develop various postprocessing routines and implement a Fourier-like transform to recover the rank (and kernel) of Hamiltonians. This enables us to estimate the Betti numbers, revealing the topological features of simplicial complexes. We test our protocol on noiseless and noisy quantum simulators and run examples on IBM quantum processors. We observe the resilience of the proposed QTDA approach to real-hardware noise even in the absence of error mitigation, showing the promise for near-term device implementations and highlighting the utility of global DOS-based estimators. Published by the American Physical Society 2024

Synergistic knowledge

Simplicial complexes are a successful model for distributed computing. They have recently been observed to provide an interesting model for epistemic multi-agent logic where the agents' local states are the main building blocks (instead of the global states). A natural generalization is to study epistemic logic on semi-simplicial sets. However, finding the appropriate modal logic for semi-simplicial models has been an open question. We answer this by introducing the logic of synergistic knowledge and establishing its soundness and completeness.

Realizations of Multiassociahedra via Rigidity

AbstractLet $$\Delta _{k}(n)$$ Δ k ( n ) denote the simplicial complex of $$(k+1)$$ ( k + 1 ) -crossing-free subsets of edges in $${\left( {\begin{array}{c}[n]\\ 2\end{array}}\right) }$$ [ n ] 2 . Here $$k,n\in \mathbb {N}$$ k , n ∈ N and $$n\ge 2k+1$$ n ≥ 2 k + 1 . Jonsson (2003) proved that [neglecting the short edges that cannot be part of any $$(k+1)$$ ( k + 1 ) -crossing], $$\Delta _{k}(n)$$ Δ k ( n ) is a shellable sphere of dimension $$k(n-2k-1)-1$$ k ( n - 2 k - 1 ) - 1 , and conjectured it to be polytopal. The same result and question arose in the work of Knutson and Miller (Adv Math 184(1):161-176, 2004) on subword complexes. Despite considerable effort, the only values of (k, n) for which the conjecture is known to hold are $$n\le 2k+3$$ n ≤ 2 k + 3 (Pilaud and Santos, Eur J Comb. 33(4):632–662, 2012. https://doi.org/10.1016/j.ejc.2011.12.003) and (2, 8) (Bokowski and Pilaud, On symmetric realizations of the simplicial complex of 3-crossing-free sets of diagonals of the octagon. In: Proceedings of the 21st annual Canadian conference on computational geometry, 2009). Using ideas from rigidity theory and choosing points along the moment curve we realize $$\Delta _{k}(n)$$ Δ k ( n ) as a polytope for $$(k,n)\in \{(2,9), (2,10) , (3,10)\}$$ ( k , n ) ∈ { ( 2 , 9 ) , ( 2 , 10 ) , ( 3 , 10 ) } . We also realize it as a simplicial fan for all $$n\le 13$$ n ≤ 13 and arbitrary k, except the pairs (3, 12) and (3, 13). Finally, we also show that for $$k\ge 3$$ k ≥ 3 and $$n\ge 2k+6$$ n ≥ 2 k + 6 no choice of points can realize $$\Delta _{k}(n)$$ Δ k ( n ) via bar-and-joint rigidity with points along the moment curve or, more generally, via cofactor rigidity with arbitrary points in convex position.

Multiplayer game dynamics and their applications into simplicial complexes with memory effect

Abstract We investigate the evolutionary dynamics of the multiplayer, snowdrift-like game based on memory effects, in which cooperators and defectors interact with another type of player named as exiter. To avoid being exploited by defectors, the conservative exiters tend to drop out of a game and merely share quite minor returns. Results on the well-mixed populations indicate that the existence of exiters is less conducive to cooperation. Whereas, in networks with simplicial complexes, the memory effect can effectively inhibit evolution of defection and promote cooperation. Other outcomes with respect to the evolution of exiters and defectors, including the co-existence of all three types of players in the network and the favor of exiter, can be obtained by tuning the payoff to exit and the proportion of 1-simplices and 2-simplices. The current results provide insight into how mutual cooperation emerges in real networked populations, influenced by the memory-based exit options.

Wanted dead or alive: epistemic logic for impure simplicial complexes

Abstract We propose a logic of knowledge for impure simplicial complexes. Impure simplicial complexes represent synchronous distributed systems under uncertainty over which processes are still active (are alive) and which processes have failed or crashed (are dead). Our work generalizes the logic of knowledge for pure simplicial complexes, where all processes are alive, by Goubault et al. In our semantics, given a designated face in a complex, a formula can only be true or false there if it is defined. The following are undefined: dead processes cannot know or be ignorant of any proposition, and live processes cannot know or be ignorant of factual propositions involving processes they know to be dead. The semantics are therefore three-valued, with undefined as the third value. We propose an axiomatization that is a version of the modal logic S5. We also show that impure simplicial complexes correspond to certain Kripke models where agents’ accessibility relations are equivalence relations on a subset of the domain only.

Higher-order network information propagation model based on social impact theory

User relationships are becoming increasingly diverse in current online social networks. This paper utilizes simplicial complexes in higher-order networks to depict multiple user relationships and characterize the topology of social networks. By employing social impact theory and factors such as user interest and information aging to define state transition rules among users, we establish a UAPR information propagation model based on simplicial complexes to simulate the information propagation process in higher-order social networks. The proposed model is simulated on three types of synthetic networks and four real-world simplicial complexes, demonstrating its ability to accurately describe the dynamics and influences of information propagation. Furthermore, our results indicate that different network structures, user interest values, information aging speeds, and intimacy levels significantly influence the information transmission process.

Growing simplicial complex with face dimension selection and preferential attachment.

When simplicial complexes are used to represent higher-order systems, information regarding when and how interactions happen may be lost. In this paper, we propose the concept of temporal simplicial complexes, in which simplices with timestamps (or temporal simplices) are used to represent interactions, and faces with weights are used to represent relations. Then, we propose a growing model with two rules, face dimension selection (FDS), and preferential attachment. By properly setting the probability parameter vector q in the FDS rule, one can balance network diameter expansion and network centrality, thus attaining more flexibility in the growing process. Our theoretical analysis and simulations that followed show the generalized degree of faces of any dimension follows a power-law distribution, with a scaling component controlled by q. Our work provides a flexible growing model and can be used to study higher-order systems with temporal properties.

Equilibrium seeking of higher-order networks under facet cover constraints.

Agent intelligence involves specific requirements for social attributes. Intelligent agents make their decisions based on the groups they are part of, tend to satisfy co-members, and enlarge their own benefits. A fundamental question is whether this form of subgroup decision-making accommodate each individual's preferences. In this paper, we examine the evolution of an anticoordination game on a higher-order network in the form of a simplicial complex in relation to the facet cover problem, which ensures that each subgroup yields a positive benefit. We introduce and apply the facet update rule to regulate nodes' group-based interactions. We identify the payoff parameter condition that a strict Nash equilibrium (SNE) satisfies a facet cover. The proposed facet update rule enables the activated facet to reach a facet equilibrium, and all nodes would converge to an SNE with no more than 2m strategy switches, where m is the number of nodes in the simplicial complex. Additionally, we analyze the convergence of the asynchronous update rule, which can be seen as a special case of the facet update rule. Our simulations and extensive examples reveal that the SNE achieved by the facet update rule, on average, covers fewer nodes compared to the asynchronous update rule.

Pattern dynamics of networked epidemic model with higher-order infections.

Current research on pattern formations in networked reaction-diffusion (RD) systems predominantly focuses on the impacts of diffusion heterogeneity between nodes, often overlooking the contact heterogeneity between individuals within nodes in the reaction terms. In this paper, we establish a networked RD model incorporating infection through higher-order interaction in simplicial complexes in the reaction terms. Through theoretical and numerical analysis, we find that these higher-order interactions may induce Turing instability in the system. Notably, the relationship between the size of the Turing instability range and the average 2-simplices degree within nodes can be approximated by a quadratic function. Additionally, as the average 2-simplices degree increases, the amplitude of the patterns exhibits three distinct trends: increasing, decreasing, and initially increasing then decreasing, while the average infection density increases consistently. We then provide a possible explanation for these observations. Our findings offer new insights into the effects of contact heterogeneity within nodes on networked pattern formations, thereby informing the development of epidemic prevention and control measures.

Robustness of multilayer interdependent higher-order network

In real-world complex systems, most networks are interconnected with other networks through interlayer dependencies, forming multilayer interdependent networks. In each system, the interactions between nodes are not limited to pairwise but also exist in a higher-order interaction composed of three or more individuals, thus inducing a multilayer interdependent higher-order network (MIHN). First, we build four types of artificial MIHN models (i.e., chain-like, tree-like, star-like and ring-like), in which the higher-order interactions are described by the simplicial complexes, and the interlayer dependency is built via a one-to-one matching dependency link. Then, we propose a cascading failure model on MIHN and suggest a corresponding percolation-based theory to study the robustness of MIHN by investigating the giant connected components (GCC) and percolation threshold. We find that the density of the simplicial complexes and the number of layers of the network affect its penetration behavior. When the density of simplicial complexes exceeds a certain threshold, the network has a double transition, and the increase in network layers significantly enhances the vulnerability of MIHN. By comparing the simulation results of MIHNs with four types, we observe that under the same density of simplicial complexes, the size of the GCC is independent of the topological structures of MIHN after removing a certain number of nodes. Although the cascading failure process of MIHNs with different structures is different, the final results tend to be the same. We further analyze in detail the cascading failure process of MIHN with different structures and elucidate the factors influencing the speed of cascading failures. Among these four types of MIHNs, the chain-like MIHN has the slowest cascading failure rate and more stable robustness compared to the other three structures, followed by the tree-like MIHN and star-like MIHN. The ring-like MIHN has the fastest cascading failure rate and weakest robustness due to its ring structure. Finally, we give the time required for the MIHN with different structures to reach the stable state during the cascading failure process and find that the closer to the percolation threshold, the more time the network requires to reach the stable state.

Delaunay Triangulated Simplicial Complex Generation for EEG Signal Classification

Delaunay Triangulated Simplicial Complex Generation for EEG Signal Classification

Synchronization on fractional multiplex higher-order networks.

This paper explores the synchronization problem in fractional multiplex higher-order networks. Initially, a fractional multiplex higher-order network model is established, which seamlessly integrates multiplex structures with higher-order interactions. Subsequently, by leveraging a well-crafted Lyapunov function, the Lyapunov direct method, and fractional inequalities, it is demonstrated that the fractional multiplex higher-order network can achieve intra-layer synchronization, inter-layer synchronization, and complete synchronization. Finally, the theoretical findings are validated through two numerical examples featuring a simplicial complex or hypergraph structures within the intra-layer network.

Hopf Monoids of Ordered Simplicial Complexes

Abstract We study Hopf classes: families of pure ordered simplicial complexes that give rise to Hopf monoids under join and deletion/contraction. The prototypical Hopf class is the family of ordered matroids. The idea of a Hopf class leads to a systematic study of simplicial complexes related to matroids, including shifted complexes and broken-circuit complexes. We compute the Hopf antipodes in two cases: facet-initial complexes (which generalize shifted complexes) and unbounded ordered matroids. The latter calculation uses the topological method of Aguiar and Ardila, complicated by certain auxiliary simplicial complexes that we call Scrope complexes, whose Euler characteristics control the coefficients of the antipode. The resulting antipode formula is multiplicity-free and cancellation-free.

A general formula for the index of depth stability of edge ideals

By a classical result of Brodmann, the function d e p t h R / I t depthR/I^t is asymptotically a constant, i.e. there is a number s s such that d e p t h R / I t = d e p t h R / I s depthR/I^t = depthR/I^s for t > s t > s . One calls the smallest number s s with this property the index of depth stability of I I and denotes it by d s t a b ( I ) dstab(I) . This invariant remains mysterious till now. The main result of this paper gives an explicit formula for d s t a b ( I ) dstab(I) when I I is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize d s t a b ( I ) dstab(I) explicitly. The formula expresses d s t a b ( I ) dstab(I) in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.

The discrete flow category: structure and computation

In this article, we use concepts and methods from the theory of simplicial sets to study discrete Morse theory. We focus on the discrete flow category introduced by Vidit Nanda, and investigate its properties in the case where it is defined from a discrete Morse function on a regular CW complex. We design an algorithm to efficiently compute the Hom posets of the discrete flow category in this case. Furthermore, we show that in the special case where the discrete Morse function is defined on a simplicial complex, then each Hom poset has the structure of a face poset of a regular CW complex. Finally, we prove that the spectral sequence associated to the double nerve of the discrete flow category collapses on page 2.

Controllability of higher-order networks

Higher-order networks can comprehensively describe interactions among groups, thus emerging as a novel area of exploration in network science. This paper aims to delve into the controllability of higher-order networks, where the network topology is characterized by higher-order interactions and the nodes are higher-dimensional dynamical systems. The collective effects on the network controllability from the dynamics of higher-order interactions, node dynamics, inner interactions, and external control inputs are extensively explored. By applying matrix theory and control theory, some necessary and/or sufficient conditions are developed to determine the controllability of hypergraph networks and simplicial complex networks. Through simulated examples, it becomes evident that the controllability of higher-order networked system is far more complicated than that of traditional networked systems and the higher-order topological structures facilitate the controllability. Remarkably, the integrated network can achieve controllability even when the corresponding traditional network is uncontrollable by external inputs.

Pattern formation in reaction-diffusion information propagation model on multiplex simplicial complexes

Turing patterns for explaining spatial distribution in nature mostly focus on continuous media and existing networks, but only few attempts exist to study them on systems with high-order interactions. Considering that high-order interactions have a particularly significant impact on rumor propagation, this study establishes a generalized reaction-diffusion rumor propagation model based on a multiplex network, where simplicial complexes are employed to describe high-order structures. It aims to provide the spatial distribution patterns of the population participating in rumor propagation and identify the structural factors that affect such patterns. We theoretically provide the necessary conditions for Turing instability in single-layer and multiplex networks by considering high-order interactions. In the numerical simulation, we demonstrate that the Turing pattern could be controlled by adjusting the diffusion coefficient, high-order structure intensity, and average degree of the network. The results indicate that: (i) in a single-layer network, the Turing pattern only exists when high-order interactions appear, and the difference in diffusion rate plays a decisive role, (ii) in a multiplex network, the Turing pattern can still be observed under the same diffusion rates, which are affected by the difference in higher-order intensity between the two layers, and (iii) in existing networks, the average degree of the network has an important impact on Turing pattern. These findings contribute toward comprehending the impact of network structure on pattern formation, particularly the high-order interactions on the Turing pattern.