Graph Homology Research Articles

On finite generation in magnitude (co)homology and its torsion

AbstractThe aim of this paper is to apply the framework developed by Sam and Snowden to study structural properties of graph homologies, in the spirit of Ramos, Miyata and Proudfoot. Our main results concern the magnitude homology of graphs introduced by Hepworth and Willerton, and we prove that it is a finitely generated functor (on graphs of bounded genus). More precisely, for graphs of bounded genus, we prove that magnitude cohomology, in each homological degree, has rank which grows at most polynomially in the number of vertices, and that its torsion is bounded. As a consequence, we obtain analogous results for path homology of (undirected) graphs.

Mapping Graph Homology to $$K$$-Theory of Roe Algebras

Mapping Graph Homology to $$K$$-Theory of Roe Algebras

Characterizing Flow Complexity in Transportation Networks Using Graph Homology

Characterizing Flow Complexity in Transportation Networks Using Graph Homology

Cyclic Path Homology of (Di)Graphs

Cyclic Path Homology of (Di)Graphs

Reflection Representations of Coxeter Groups and Homology of Coxeter Graphs

Reflection Representations of Coxeter Groups and Homology of Coxeter Graphs

Wheel graph homology classes via Lie graph homology

We give a new proof of the non-triviality of wheel graph homology classes using higher operations on Lie graph homology and a derived version of Koszul duality for modular operads.

Lie graph homology model for [formula omitted

This paper develops a new chain model for the commutative graph complex GC2 which takes Lie graph homology as an input. Our main technical result is the identification of a large contractible complex of (certain) tadpoles and higher genus vertices of the Feynman transform of Lie graph homology. Using this result we identify the anti-invariants of Lie graph homology in genus 2 with relations between bracketings of conjectural generators of grt1 in depth 2 modulo depth 3, unifying two a priori disparate appearances of the space of modular cusp forms in the study of graph homology.

Topological data analysis of Korean music in Jeongganbo: a cycle structure

Jeongganbo is a unique music representation invented by Sejong the Great. Contrary to the Western music notation, the pitch of each note is encrypted and the length is visualized directly in a matrix form. We use topological data analysis (TDA) to analyze the Korean music written in Jeongganbo for Suyeonjang, Songuyeo, and Taryong, those well-known pieces played among noble community. We define the nodes of each music with pitch and length and transform the music into a graph with the distance between the nodes defined as their adjacent occurrence rate. The graph homology is investigated by TDA. We identify cycles of each music and show how those cycles are interconnected. We found that the cycles of Suyeonjang and Songuyeo, categorized as a special type of cyclic music, frequently overlap each other in the music, while those of Taryong, which does not belong to the same class, appear only individually.

Girth, magnitude homology and phase transition of diagonality

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. The magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth–Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behaviour is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines the magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.

On Chromatic Symmetric Homology and Planarity of Graphs


 
 
 Sazdanovic and Yip (2018) defined a categorification of Stanley’s chromatic symmetric function called the chromatic symmetric homology, given by a suitable family of representations of the symmetric group. In this paper we prove that, as conjectured by Chandler, Sazdanovic, Stella and Yip (2019), if a graph $G$ is non-planar, then its chromatic symmetric homology in bidegree (1,0) contains $\mathbb{Z}_2$-torsion. Our proof follows a recursive argument based on Kuratowsky’s theorem.
 
 

Graph complexes and Feynman rules

We investigate Feynman graphs and their Feynman rules from the viewpoint of graph complexes. We focus on graph homology and on the appearance of cubical complexes when either reducing internal edges or when removing them by putting them on the massshell.

Magnitude homology of graphs and discrete Morse theory on Asao–Izumihara complexes

Magnitude homology of graphs and discrete Morse theory on Asao–Izumihara complexes

First Betti number of the path homology of random directed graphs

Path homology is a topological invariant for directed graphs, which is sensitive to their asymmetry and can discern between digraphs which are indistinguishable to the directed flag complex. In Erdős–Rényi directed random graphs, the first Betti number undergoes two distinct transitions, appearing at a low-density boundary and vanishing again at a high-density boundary. Through a novel, combinatorial condition for digraphs we describe both sparse and dense regimes under which the first Betti number of path homology is zero with high probability. We combine results of Grigor’yan et al., regarding generators for chain groups, with methods of Kahle and Meckes in order to determine regimes under which the first Betti number is positive with high probability. Together, these results describe the gradient of the lower boundary and yield bounds for the gradient of the upper boundary. With a view towards hypothesis testing, we obtain tighter bounds on the probability of observing a positive first Betti number in a high-density digraph of finite size. For comparison, we apply these techniques to the directed flag complex and derive analogous results

Extremal Khovanov homology and the girth of a knot

We show that Khovanov link homology is trivial in a range of gradings and utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and the chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper, we define the girth of a link, discuss relations to other knot invariants, and describe possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that the girth of the corresponding all-A graph is equal to [Formula: see text], then the girth of the link is equal to [Formula: see text]

Schwinger, ltd: loop-tree duality in the parametric representation

We derive a variant of the loop-tree duality for Feynman integrals in the Schwinger parametric representation. This is achieved by decomposing the integration domain into a disjoint union of cells, one for each spanning tree of the graph under consideration. Each of these cells is the total space of a fiber bundle with contractible fibers over a cube. Loop-tree duality emerges then as the result of first decomposing the integration domain, then integrating along the fibers of each fiber bundle.As a byproduct we obtain a new proof that the moduli space of graphs is homotopy equivalent to its spine. In addition, we outline a potential application to Kontsevich’s graph (co-)homology.

Simplex closing probabilities in directed graphs

Recent work in mathematical neuroscience has calculated the directed graph homology of the directed simplicial complex given by the brain's sparse adjacency graph, the so called connectome. These biological connectomes show an abundance of both high-dimensional directed simplices and Betti-numbers in all viable dimensions – in contrast to Erdős–Rényi-graphs of comparable size and density. An analysis of synthetically trained connectomes reveals similar findings, raising questions about the graphs comparability and the nature of origin of the simplices.We present a new method capable of delivering insight into the emergence of simplices and thus simplicial abundance. Our approach allows to easily distinguish simplex-rich connectomes of different origin. The method relies on the novel concept of an almost-d-simplex, that is, a simplex missing exactly one edge, and consequently the almost-d-simplex closing probability by dimension. We also describe a fast algorithm to identify almost-d-simplices in a given graph. Applying this method to biological and artificial data allows us to identify a mechanism responsible for simplex emergence, and suggests this mechanism is responsible for the simplex signature of the excitatory subnetwork of a statistical reconstruction of the mouse primary visual cortex. Our highly optimized code for this new method is publicly available.

Homology of étale groupoids a graded approach

We introduce a graded homology theory for graded étale groupoids. We prove that the graded homology of a strongly graded ample groupoid is isomorphic to the homology of its ε-th component with ε the identity of the grade group. For Z-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients Z is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras.

The Persistent Homology of Cyclic Graphs

We give an [Formula: see text] algorithm for computing the [Formula: see text]-dimensional persistent homology of a filtration of clique complexes of cyclic graphs on [Formula: see text] vertices. This is nearly quadratic in the number of vertices [Formula: see text], and therefore a large improvement upon the traditional persistent homology algorithm, which is cubic in the number of simplices of dimension at most [Formula: see text], and hence of running time [Formula: see text] in the number of vertices [Formula: see text]. Our algorithm applies, for example, to Vietoris–Rips complexes of points sampled from a curve in [Formula: see text] when the scale is bounded depending on the geometry of the curve, but still large enough so that the Vietoris–Rips complex may have non-trivial homology in arbitrarily high dimensions [Formula: see text]. In the case of the plane [Formula: see text], we prove that our algorithm applies for all scale parameters if the [Formula: see text] vertices are sampled from a convex closed differentiable curve whose convex hull contains its evolute. We ask if there are other geometric settings in which computing persistent homology is (say) quadratic or cubic in the number of vertices, instead of in the number of simplices.

Magnitude homology of enriched categories and metric spaces

Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories, which we call magnitude homology (in fact, it is a special sort of Hochschild homology), whose graded Euler characteristic is the magnitude. Magnitude homology of metric spaces generalizes the Hepworth--Willerton magnitude homology of graphs, and detects geometric information such as convexity.

Torsion in the magnitude homology of graphs

Magnitude homology is a bigraded homology theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced by Leinster. We analyze the structure and implications of torsion in magnitude homology. We show that any finitely generated abelian group may appear as a subgroup of the magnitude homology of a graph, and, in particular, that torsion of a given prime order can appear in the magnitude homology of a graph and that there are infinitely many such graphs. Finally, we provide complete computations of magnitude homology of a class of outerplanar graphs and focus on the ranks of the groups along the main diagonal of magnitude homology.