Flag Complex Research Articles

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.

FlaG competes with FliS–flagellin complexes for access to FlhA in the flagellar T3SS to control Campylobacter jejuni filament length

Bacteria power rotation of an extracellular flagellar filament for swimming motility. Thousands of flagellin subunits compose the flagellar filament, which extends several microns from the bacterial surface. It is unclear whether bacteria actively control filament length. Many polarly flagellated bacteria produce shorter flagellar filaments than peritrichous bacteria, and FlaG has been reported to limit flagellar filament length in polar flagellates. However, a mechanism for how FlaG may function is unknown. We observed that deletion of flaG in the polarly flagellated pathogens Vibrio cholerae, Pseudomonas aeruginosa, and Campylobacter jejuni caused extension of flagellar filaments to lengths comparable to peritrichous bacteria. Using C. jejuni as a model to understand how FlaG controls flagellar filament length, we found that FlaG and FliS chaperone-flagellin complexes antagonize each other for interactions with FlhA in the flagellar type III secretion system (fT3SS) export gate. FlaG interacted with an understudied region of FlhA, and this interaction appeared to be enhanced in ΔfliS and FlhA FliS-binding mutants. Our data support that FlaG evolved in polarly flagellated bacteria as an antagonist to interfere with the ability of FliS to interact with and deliver flagellins to FlhA in the fT3SS export gate to control flagellar filament length so that these bacteria produce relatively shorter flagella than peritrichous counterparts. This mechanism is similar to how some gatekeepers in injectisome T3SSs prevent chaperones from delivering effector proteins until completion of the T3SS and host contact occurs. Thus, flagellar and injectisome T3SSs have convergently evolved protein antagonists to negatively impact respective T3SSs to secrete their major terminal substrates.

Structure of higher-order interactions in social-ecological networks through Q-analysis of their neighbourhood and clique complex.

This paper studies higher-order interactions in social-ecological networks, which formally represent interactions within the social and ecological units of an ecosystem. Many real-world social ecosystems exhibit not only pairwise interactions but also higher-order interactions among their units. Therefore, the conventional graph-theoretic description of networks falls short of capturing these higher-order interactions due to the inherent limitations of the graph definition. In this work, a mathematical framework for capturing the higher-order interactions of a social-ecological system has been given by incorporating notions from combinatorial algebraic topology. In order to achieve this, two different simplicial complexes, the clique and the neighbourhood complex, have been constructed from a pairwise social-ecological network. As a case study, the Q-analysis and a structural study of the interactions in the rural agricultural system of southern Madagascar have been done at various structural levels denoted by q. The results obtained by calculating all the structural vectors for both simplicial complexes, along with exciting results about the participation of facets of the clique complex at different q-levels, have been discussed. This work also establishes significant theorems concerning the dimension of the neighbourhood complex and clique complex obtained from the parent pairwise network.

Homotopy properties of the complex of frames of a unitary space

AbstractLet be a finite‐dimensional vector space equipped with a nondegenerate Hermitian form over a field . Let be the graph with vertex set the one‐dimensional nondegenerate subspaces of and adjacency relation given by orthogonality. We give a complete description of when is connected in terms of the dimension of and the size of the ground field . Furthermore, we prove that if , then the clique complex of is simply connected. For finite fields , we also compute the eigenvalues of the adjacency matrix of . Then, by Garland's method, we conclude that for all , where is a field of characteristic 0, provided that . Under these assumptions, we deduce that the barycentric subdivision of deformation retracts to the order complex of the certain rank selection of that is Cohen–Macaulay over . Finally, we apply our results to the Quillen poset of elementary abelian ‐subgroups of a finite group and to the study of geometric properties of the poset of nondegenerate subspaces of and the poset of orthogonal decompositions of .

MCMC sampling of directed flag complexes with fixed undirected graphs

Constructing null models to test the significance of extracted information is a crucial step in data analysis. In this work, we provide a uniformly sampleable null model of directed graphs with the same (or similar) number of simplices in the flag complex, with the restriction of retaining the underlying undirected graph. We describe an MCMC-based algorithm to sample from this null model and statistically investigate the mixing behaviour. This is paired with a high-performance, Rust-based, publicly available implementation. The motivation comes from topological data analysis of connectomes in neuroscience. In particular, we answer the fundamental question: are the high Betti numbers observed in the investigated graphs evidence of an interesting topology, or are they merely a byproduct of the high numbers of simplices? Indeed, by applying our new tool on the connectome of C. elegans and parts of the statistical reconstructions of the Blue Brain Project, we find that the Betti numbers observed are considerable statistical outliers with respect to this new null model. We thus, for the first time, statistically confirm that topological data analysis in microscale connectome research is extracting statistically meaningful information.

Loop Space Decompositions of Moment-Angle Complexes Associated to Flag Complexes

Abstract We prove that the loop space of the moment-angle complex associated with the k-skeleton of a flag complex belongs to the class $\mathcal{P}$ of spaces homotopy equivalent to a finite-type product of spheres and loops on simply connected spheres. To do this, a general result showing $\mathcal{P}$ is closed under retracts is proved.

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.

Persistent homology based Bottleneck distance in hypergraph products

In this paper, we extended the technique of measuring similarity between topological spaces using bottle neck distance between persistence diagrams to hypergraph networks. Finding a relationship between the bottleneck distance of the Cartesian product of topological spaces and the bottleneck distance of individual spaces, we are trying to ease the comparative study of the Cartesian product of topological spaces. The Cartesian product and the strong product of weighted hypergraphs are defined, and the relationship between the bottleneck distance between hypergraph products and the bottleneck distance between individual hypergraphs is determined. For this, clique complex filtration and the Vietoris–Rips filtration in unweighted and weighted hypergraphs are defined and used.

Persistent Homology based Wasserstein distance for graph networks

The technique of measuring similarity between topological spaces using Wasserstein distance
 between persistence diagrams is extended to graph networks in this paper. A relationship
 between the Wasserstein distance of the Cartesian product of topological spaces and the
 Wasserstein distance of individual spaces is found to ease the comparative study of the
 Cartesian product of topological spaces. The Cartesian product and the strong product of
 weighted graphs are defined, and the relationship between the Wasserstein distance between
 graph products and the Wasserstein distance between individual graphs is determined. For
 this, clique complex filtration and the Vietoris- Rips filtration are used.

Planar matrices and arrays of Feynman diagrams: poles for higher k

Planar arrays of tree diagrams were introduced as a generalization of Feynman diagrams that enable the computation of biadjoint amplitudes mn(k) for k > 2. In this follow-up work, we investigate the poles of mn(k) from the perspective of such arrays. For general k, we characterize the underlying polytope as a Flag Complex and propose a computation of the amplitude-based solely on the knowledge of the poles, whose number is drastically less than the number of the full arrays. As an example, we first provide all the poles for the cases (k, n) = (3, 7), (3, 8), (3, 9), (3, 10), (4, 8) and (4, 9) in terms of their planar arrays of degenerate Feynman diagrams. We then implement simple compatibility criteria together with an addition operation between arrays and recover the full collections/arrays for such cases. Along the way, we implement hard and soft kinematical limits, which provide a map between the poles in kinematic space and their combinatoric arrays. We use the operation to give a proof of a previously conjectured combinatorial duality for arrays in (k, n) and (n − k, n). We also outline the relation to boundary maps of the hypersimplex Δ k,n and rays in the tropical Grassmannian Tr(k,n) .

One‐sided sharp thresholds for homology of random flag complexes

AbstractWe prove that the random flag complex has a probability regime where the probability of nonvanishing homology is asymptotically bounded away from zero and away from one. Related to this main result, we also establish new bounds on a sharp threshold for the fundamental group of a random flag complex to be a free group. In doing so, we show that there is an intermediate probability regime in which the random flag complex has fundamental group that is neither free nor has Kazhdan's property (T).

On the frame complex of symplectic spaces

For a symplectic space V of dimension 2n over Fq, we compute the eigenvalues of its orthogonality graph. This is the simple graph with vertices the 2-dimensional non-degenerate subspaces of V and edges between orthogonal vertices. As a consequence of Garland's method, we obtain vanishing results on the homology groups of the frame complex of V, which is the clique complex of this graph. We conclude that if n<q+3 then the poset of frames of size ≠0,n−1, which is homotopy equivalent to the frame complex, is Cohen-Macaulay over a field of characteristic 0. However, we also show that this poset is not Cohen-Macaulay if the dimension is big enough.

Multivariate central limit theorems for random clique complexes

Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: the vector of critical simplex counts attained by a lexicographical Morse matching,the vector of simplex counts in the link of a fixed simplex, andthe vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U-statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein’s method.

Mixing is hard for triangle-free reflexive graphs

In the problem Mix(H) one is given a graph G and must decide if the Hom-graph Hom(G,H) is connected. We show that if H is a triangle-free reflexive graph with at least one cycle, Mix(H) is coNP-complete. The main part of this is a reduction to the problem NonFlat(H) for a simplicial complex H, in which one is given a simplicial complex G and must decide if there are any simplicial maps ϕ from G to H under which some 1-cycles of G maps to homologically nontrivial cycle of H. We show that for any reflexive graph H, if the clique complex H of H has a free, nontrivial homology group H1(H), then NonFlat(H) is NP-complete.

Double cohomology of moment-angle complexes

We put a cochain complex structure CH⁎(ZK) on the cohomology of a moment-angle complex ZK and call the resulting cohomology the double cohomology, HH⁎(ZK). We give three equivalent definitions for the differential, and compute HH⁎(ZK) for a family of simplicial complexes containing clique complexes of chordal graphs.

A note on virtual duality and automorphism groups of right-angled Artin groups

AbstractA theorem of Brady and Meier states that a right-angled Artin group is a duality group if and only if the flag complex of the defining graph is Cohen–Macaulay. We use this to give an example of a RAAG with the property that its outer automorphism group is not a virtual duality group. This gives a partial answer to a question of Vogtmann. In an appendix, Brück describes how he used a computer-assisted search to find further examples.

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 f-vectors of r-multichain subdivisions

For a poset P and an integer r≥1, let Pr be the collection of all r-multichains in P. Corresponding to each strictly increasing map ı:[r]→[2r], there is an order ⪯ı on Pr. Let Δ(Gı(Pr)) be the clique complex of the graph Gı associated to Pr and ı. In a recent paper [14], it is shown that Δ(Gı(Pr)) is a subdivision of P for a class of strictly increasing maps. In this paper, we show that all these subdivisions have the same f-vector. We give an explicit description of the transformation matrices from the f- and h-vectors of Δ to the f- and h-vectors of these subdivisions when P is a poset of faces of Δ. We study two important subdivisions, namely Cheeger-Müller-Schrader's subdivision and the r-colored barycentric subdivision which fall in our class of r-multichain subdivisions.

Systolic complexes and group presentations

We give conditions on a presentation of a group, which imply that its Cayley complex is simplicial, and the flag complex of the Cayley complex is systolic. We then apply this to Garside groups and Artin groups. We give a classification of the Garside groups whose presentation using the simple elements as generators satisfy our conditions. We then also give a dual presentation for Artin groups and identify in which cases the flag complex of the Cayley complex is systolic.

Line Graphs with a Cohen–Macaulay or Gorenstein Clique Complex

Line Graphs with a Cohen–Macaulay or Gorenstein Clique Complex