Combinatorial Topology Research Articles

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.

Cell topology during coarsening of simulated three-dimensional dry liquid foams.

Cell topology provides a deep insight into the structure of dry liquid foams. This paper analyses the cell topology of simulated 3-dimensional monodisperse dry liquid foams through the process of coarsening, where gas slowly diffuses through the cell interfaces. The coarsened foams are polydisperse, yet show a difference in cell types to annealed, low energy foams of comparable polydispersity. These two types of foams are analysed using average face degree of the cells and a measure of combinatorial roundedness, a new concept from combinatorial topology. We see that the spectrum of cell types changes drastically through the evolution of the foam via coarsening, from cells with a face degree between 12 and 14, with many 4-, 5-, and 6-side faces, to a combination of very large cells with many faces alongside a high frequency of tetrahedra and other cells with low face degree. These results demonstrate the insight that topological methods can give into foams and other complex structures.

Geometric, Algebraic and Topological Combinatorics

The 2023 Oberwolfach meeting “Geometric, Algebraic, and Topological Combinatorics” was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference were (1) Federico Ardila and Tom Braden discussed recent exciting developments in the intersection theory of matroids; (2) Stavros Papadakis and Vasiliki Petrotou presented their proof of the Lefschetz property for spheres, and, more generally, for pseudomanifolds and cycles (this second part is joint with Karim Adiprasito); (3) Gaku Liu reported on his joint work with Spencer Backman that establishes the existence of a regular unimodular triangulation of an arbitrary matroid base polytope.

Pseudospheres: combinatorics, topology and distributed systems

Pseudospheres are simplicial complexes defined in the late 1990s to model some aspects of distributed systems. Since then, combinatorial properties of pseudospheres combined with topological properties have been very useful to derive distributed computability results. The goal of this paper is to study pseudospheres in more depth as mathematical objects and to give an overview of the properties that have been used in distributed computing. In this work we focus in combinatorial and topological aspects of pseudospheres. While doing so, the paper shows that these structures can be viewed from different perspectives, in addition to models of distributed computing. We show that the properties of pseudospheres that have been proved in distributed computing, as well as new ones, can be derived using combinatorial topology techniques and other combinatorial techniques taken from matroids and partial orders. A subclass of pseudospheres is related to universal bundles, and the Borsuk–Ulam theorem can be extended to apply to them.

Impure Simplicial Complexes: Complete Axiomatization

Combinatorial topology is used in distributed computing to model concurrency and asynchrony. The basic structure in combinatorial topology is the simplicial complex, a collection of subsets called simplices of a set of vertices, closed under containment. Pure simplicial complexes describe message passing in asynchronous systems where all processes (agents) are alive, whereas impure simplicial complexes describe message passing in synchronous systems where processes may be dead (have crashed). Properties of impure simplicial complexes can be described in a three-valued multi-agent epistemic logic where the third value represents formulae that are undefined, e.g., the knowledge and local propositions of dead agents. In this work we present an axiomatization for the logic of the class of impure complexes and show soundness and completeness. The completeness proof involves the novel construction of the canonical simplicial model and requires a careful manipulation of undefined formulae.

Homological eigenvalues of graph p-Laplacians

Inspired by persistent homology in topological data analysis, we introduce the homological eigenvalues of the graph [Formula: see text]-Laplacian [Formula: see text], which allows us to analyze and classify non-variational eigenvalues. We show the stability of homological eigenvalues, and we prove that for any homological eigenvalue [Formula: see text], the function [Formula: see text] is locally increasing, while the function [Formula: see text] is locally decreasing. As a special class of homological eigenvalues, the min–max eigenvalues [Formula: see text] are locally Lipschitz continuous with respect to [Formula: see text]. We also establish the monotonicity of [Formula: see text] and [Formula: see text] with respect to [Formula: see text]. These results systematically establish a refined analysis of [Formula: see text]-eigenvalues for varying [Formula: see text], which lead to several applications, including: (1) settle an open problem by Amghibech on the monotonicity of some function involving eigenvalues of [Formula: see text]-Laplacian with respect to [Formula: see text]; (2) resolve a question asking whether the third eigenvalue of graph [Formula: see text]-Laplacian is of min–max form; (3) refine the higher-order Cheeger inequalities for graph [Formula: see text]-Laplacians by Tudisco and Hein, and extend the multi-way Cheeger inequality by Lee, Oveis Gharan and Trevisan to the [Formula: see text]-Laplacian case. Furthermore, for the 1-Laplacian case, we characterize the homological eigenvalues and min–max eigenvalues from the perspective of topological combinatorics, where our idea is similar to the authors’ work on discrete Morse theory.

The graph minor theorem in topological combinatorics

We study a variety of natural constructions from topological combinatorics, including matching complexes as well as other graph complexes, from the perspective of the graph minor category of [13]. We prove that these complexes must have universally bounded torsion in their homology across all graphs of bounded genus. One may think of these results as arising from an algebraic version of the graph minor theorem of Robertson and Seymour [19,20].

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.

Fundamental Limits of Combinatorial Multi-Access Caching

This work identifies the fundamental limits of multi-access coded caching (MACC) where each user is connected to multiple caches in a manner that follows a generalized combinatorial topology. This topology stands out as it allows for unprecedented coding gains, even with very modest cache resources. First, we extend the setting and the scheme presented by Muralidhar et al. to a much more general topology that supports both a much denser range of users and the coexistence of users connected to different numbers of caches, all while maintaining the astounding coding gains — here proven to be exactly optimal — associated with the combinatorial topology. This is achieved, for this generalized topology, with a novel information-theoretic converse that we present here, which establishes, together with the scheme, the exact optimal performance under the assumption of uncoded placement. We subsequently consider different connectivity ensembles, including the very general scenario of the entire ensemble of all possible network connectivities/topologies, where any subset of caches can serve any arbitrary number of users. For these settings, we develop novel converse bounds on the optimal performance averaged over the ensemble’s different connectivities. This novel analysis of topological ensembles leaves open the possibility that currently-unknown topologies may yield even higher gains, a hypothesis that is part of the bigger question of which network topology yields the most caching gains.

Biomolecular Topology: Modelling and Analysis.

With the great advancement of experimental tools, a tremendous amount of biomolecular data has been generated and accumulated in various databases. The high dimensionality, structural complexity, the nonlinearity, and entanglements of biomolecular data, ranging from DNA knots, RNA secondary structures, protein folding configurations, chromosomes, DNA origami, molecular assembly, to others at the macromolecular level, pose a severe challenge in their analysis and characterization. In the past few decades, mathematical concepts, models, algorithms, and tools from algebraic topology, combinatorial topology, computational topology, and topological data analysis, have demonstrated great power and begun to play an essential role in tackling the biomolecular data challenge. In this work, we introduce biomolecular topology, which concerns the topological problems and models originated from the biomolecular systems. More specifically, the biomolecular topology encompasses topological structures, properties and relations that are emerged from biomolecular structures, dynamics, interactions, and functions. We discuss the various types of biomolecular topology from structures (of proteins, DNAs, and RNAs), protein folding, and protein assembly. A brief discussion of databanks (and databases), theoretical models, and computational algorithms, is presented. Further, we systematically review related topological models, including graphs, simplicial complexes, persistent homology, persistent Laplacians, de Rham-Hodge theory, Yau-Hausdorff distance, and the topology-based machine learning models.

Compression with wildcards: abstract simplicial complexes

Despite the more handy terminology of abstract simplicial complexes , in its core this article is about antitone Boolean functions. Given the maximal faces (=facets) of , our main algorithm, called Facets-To-Faces, outputs in a compressed format. The degree of compression of Facets-To-Faces, which is programmed using the high-level programming language Mathematica, compares favorably to both the hardwired Mathematica command BooleanConvert, and to the hardwired binary decision diagrams provided by Python. A novel way to calculate the face-numbers from the facets is also presented. Both algorithms can be parallelized and are applicable (e.g.) to reliability analysis, combinatorial topology, and frequent-set-mining.

Bisecting envelopes of convex polygons

We introduce a new subject – bisecting deltoids – by giving status to two curves, the area-bisecting deltoid and the perimeter-bisecting deltoid of a convex polygon, of which the second is a new curve. These are the envelopes of all lines that bisect either the area or the perimeter of the polygon. The subject is located between the Euclidean and convex geometries in the same way as the elliptic and hyperbolic geometries are located between the Euclidean and differential geometries. We show that these curves have numerous geometric, differential, analytic, topological and combinatorial properties.From the direction of the Euclidean, differential and analytic geometries, we describe the two curves both as geometric and analytic loci. Particular emphasis is given to deltoids of quadrilaterals, which are classified, and those of regular polygons. Our results here extend the known results of [5], [15] and [51] on bisecting deltoids of triangles and those of [58] and [66] on area-bisecting deltoids of convex polygons.From the direction of convex geometry and combinatorial topology, we describe the area- or perimeter-bisecting partitions, or the incidence partitions – two partitions of the plane that are common to these curves. The area (perimeter) bisecting partition is the partition of the plane given by labeling each point by the number lines through it that bisect the area (perimeter) of the polygon. The same label is achieved by counting the number of tangents to the area (perimeter) bisecting deltoid that pass through the point. The bisecting partition is different from the geometric partition, which is common to any closed curve in the plane. Our description of the bisecting partition extends the known results of [45] and [66]. This is made possible by a new counting technique, the total sweep count.

Combinatorial topology and geometry of fracture networks.

A map is proposed from the space of planar surface fracture networks to a four-parameter mathematical space, summarizing the average topological connectivity and geometrical properties of a network idealized as a convex polygonal mesh. The four parameters are identified as the average number of nodes and edges, the angular defect with respect to regular polygons, and the isoperimetric ratio. The map serves as a low-dimensional signature of the fracture network and is visually presented as a pair of three-dimensional graphs. A systematic study is made of a wide collection of real crack networks for various materials, collected from different sources. To identify the characteristics of the real materials, several well-known mathematical models of convex polygonal networks are presented and worked out. These geometric models may correspond to different physical fracturing processes. The proposed map is shown to be discriminative, and the points corresponding to materials of similar properties are found to form closely spaced groups in the parameter space. Results for the real and simulated systems are compared in an attempt to identify crack networks of unknown materials.

The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses

The paper presents the Mixed-Integer Non-linear Programming (MINLP) approach to the synthesis of trusses. The solution of continuous/discrete non-convex and non-linear optimization problems is discussed with respect to the simultaneous topology, shape and discrete sizing optimization of trusses. A truss MINLP superstructure of different topology and design alternatives has been generated, and a special MINLP model formulation for trusses has been developed. In the optimization model, a mass objective function of the structure has been defined and subjected to design, load and dimensioning constraints. The MINLP problems are solved using the Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm. Multi-level MINLP strategies are introduced to accelerate the convergence of the algorithm. The Modified Two-Phase and the Sequential Two-Phase MINLP strategies are proposed in order to solve highly combinatorial topology, shape and discrete sizing optimization problems. The importance of local buckling constraints on topology optimization is also discussed. Some simple numerical examples are shown at the end of the paper to demonstrate the suitability and efficiency of the proposed method.

Calculating Complete Lists of Belyi Pairs

Belyi pairs constitute an important element of the program developed by Alexander Grothendieck in 1972–1984. This program related seemingly distant domains of mathematics; in the case of Belyi pairs, such domains are two-dimensional combinatorial topology and one-dimensional arithmetic geometry. The paper contains an account of some computer-assisted calculations of Belyi pairs with fixed discrete invariants. We present three complete lists of polynomial-like Belyi pairs: (1) of genus 2 and (minimal possible) degree 5; (2) clean ones of genus 1 and degree 8; and (3) clean ones of genus 2 and degree 8. The explanation of some phenomena we encounter in these calculations will hopefully stimulate further development of the dessins d’enfants theory.

The totally nonnegative Grassmannian is a ball

We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.

Subdivisions of shellable complexes

In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of interlacing polynomials. Many of the open questions on stability and unimodality of polynomials pertain to the enumeration of faces of cell complexes. In this paper, we relate the theory of interlacing polynomials to the shellability of cell complexes. We first derive a sufficient condition for stability of the h-polynomial of a subdivision of a shellable complex. To apply it, we generalize the notion of reciprocal domains for convex embeddings of polytopes to abstract polytopes and use this generalization to define the family of stable shellings of a polytopal complex. We characterize the stable shellings of cubical and simplicial complexes, and apply this theory to answer a question of Brenti and Welker on barycentric subdivisions for the well-known cubical polytopes. We also give a positive solution to a problem of Mohammadi and Welker on edgewise subdivisions of cell complexes. We end by relating the family of stable line shellings to the combinatorics of hyperplane arrangements. We pose related questions, answers to which would resolve some long-standing problems while strengthening ties between the theory of interlacing polynomials and the combinatorics of hyperplane arrangements.

Monsky’s Theorem / O Teorema de Monsky

The main objective of this paper is to prove Monsky’s Theorem, that provides a beautiful application of the 2-adic valuation in order to solve a plane geometry problem. This theorem states that given any dissection of a square into finitely many nonoverlapping triangles of equal area the number of triangles must be even. In order to prove this statement, we will need some previous results from Combinatorial Topology and Algebra.

Hyperharmonic analysis for the study of high-order information-theoretic signals

Network representations often cannot fully account for the structural richness of complex systems spanning multiple levels of organisation. Recently proposed high-order information-theoretic signals are well-suited to capture synergistic phenomena that transcend pairwise interactions; however, the exponential-growth of their cardinality severely hinders their applicability. In this work, we combine methods from harmonic analysis and combinatorial topology to construct efficient representations of high-order information-theoretic signals. The core of our method is the diagonalisation of a discrete version of the Laplace–de Rham operator, that geometrically encodes structural properties of the system. We capitalise on these ideas by developing a complete workflow for the construction of hyperharmonic representations of high-order signals, which is applicable to a wide range of scenarios.

Wolof verbal particles and their syntactic and topological combinatorics

Dans cet article, nous abordons l'etude des particules verbales du wolof en nous placant resolument dans une approche distributionnelle basee sur l'etude exhaustive d'un corpus annote en syntaxe de dependance par Dione (2019). Ceci nous permet de reviser les travaux anterieurs en distinguant nettement, parmi les particules verbales, un groupe de 6 particules (a, la, na, da,-u, ngi), que nous appellerons les particules assertives. Les constructions dans lesquelles entrent ces particules se caracterisent par une distribution commune (impossibilite d'etre dans les subordonnees relatives et obligation d'etre dans certaines completives) et par une structure topologique commune caracterisee par la position de la particule assertive, des clitiques complements et de l'auxiliaire imperfectif di. Nous proposons une description complete de la combinatoire de l'ensemble des particules verbales, ainsi que l'ensemble des schemas topologiques des constructions predicatives du wolof.