Related papers: Towards Homological Methods in Graphic Statics
We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential…
We revisit Maxwell theory in 4d with a boundary, with particular attention to the global properties of the boundary conditions, both in the free (topological) and interacting (conformal) cases. We analyze the fate of Wilson-'t Hooft lines,…
We study abelian lattice gauge theory defined on a simplicial complex with arbitrary topology. The use of dual objects allows one to reformulate the theory in terms of new dynamical variables; however, we avoid the use of the dual lattice…
We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and…
Two global symmetries are holo-equivalent if their algebras of local symmetric operators are isomorphic. Holo-equivalent classes of global symmetries are classified by gappable-boundary topological orders (TO) in one higher dimension…
Deep learning methods for graphs have seen rapid progress in recent years with much focus awarded to generalising Convolutional Neural Networks (CNN) to graph data. CNNs are typically realised by alternating convolutional and pooling layers…
We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for {\em graphs with semi-edges}. The notion of graph covering is a discretization of coverings between surfaces or topological…
The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally…
For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating…
The study of the homology of diagram algebras has emerged as an interesting and important field. In many cases, the homology of a diagram algebra can be identified with the homology of a group. In this paper we have two main aims. Firstly,…
Graphs are structured data that models complex relations between real-world entities. Heterophilic graphs, where linked nodes are prone to be with different labels or dissimilar features, have recently attracted significant attention and…
Node2vec is a graph embedding method that learns a vector representation for each node of a weighted graph while seeking to preserve relative proximity and global structure. Numerical experiments suggest Node2vec struggles to recreate the…
The integration of Large Language Models (LLMs) and Graph Neural Networks (GNNs) promises to unify semantic understanding with structural reasoning, yet existing methods typically rely on static, unidirectional pipelines. These approaches…
Learning a graph topology to reveal the underlying relationship between data entities plays an important role in various machine learning and data analysis tasks. Under the assumption that structured data vary smoothly over a graph, the…
Persistent homology is a tool that can be employed to summarize the shape of data by quantifying homological features. When the data is an object in $\mathbb{R}^d$, the (augmented) persistent homology transform ((A)PHT) is a family of…
The approximate graph colouring problem, whose complexity is unresolved in most cases, concerns finding a $c$-colouring of a graph that is promised to be $k$-colourable, where $c\geq k$. This problem naturally generalises to promise graph…
We present new algorithm for growth of non-clustered planar graphs by aggregation of cells with given distribution of size and constraint of connectivity k=3 per node. The emergent graph structures are controlled by two parameters--chemical…
We introduce the \verb|Macaulay2| package \verb|RepHomology| for the computations of representation homology of certain spaces. The main methods implement computing the representation homology of surfaces (with group coefficients, and…
The effectiveness of Spatio-temporal Graph Neural Networks (STGNNs) in time-series applications is often limited by their dependence on fixed, hand-crafted input graph structures. Motivated by insights from the Topological Data Analysis…
The following article is an application of commutative algebra to the study of multiparameter persistent homology in topological data analysis. In particular, the theory of finite free resolutions of modules over polynomial rings is applied…