arXiv subjects

Jānis Lazovskis

Publications and source records attributed to Jānis Lazovskis.

8 recordsLinked to original sources

arXiv2026

Higher-order rich clubs and configuration models on general directed hypergraphs

Detecting structure in complex networks, especially those arising from physical systems, is a central problem across the sciences. One approach is via rich club analysis, which identifies important vertices using a centrality metric and measures whether those vertices are more tightly interconnected than expected by chance. While informative, this approach captures only pairwise interactions, missing out on higher-order ones known to shape the structure and function of many complex systems. We propose a hyper-rich club pipeline that asks whether central vertices are more tightly interconnected than expected by chance through hyperedges encoding higher-order interactions, which also enables the inclusion of important, often omitted, directional information. We work in a broad class of hypergraphs, which we call general directed hypergraphs, that includes as special cases undirected hypergraphs, head-and-tail directed hypergraphs, and totally ordered hypergraphs (a hypergraph related to directed simplicial complexes from topological data analysis). This unifies several non-equivalent notions of directed hypergraph under one definition. On these hypergraphs we define a hyper-rich club framework whose concrete construction depends on explicit choices the domain scientist fixes according to their research goals. Particular choices recover the existing rich club notions for graphs and undirected hypergraphs, and yield the first such notion for each version of directed hypergraphs. We demonstrate that the pipeline recovers meaningful structure in data by studying networks of very different origins: connectomes, temporal networks of infectious spread, networks of poems, and the XGI hypergraph database, in each case detecting structure the standard graph rich club misses.

cs.SI↗

arXiv2026

A constructive proof for the simple connectedness of finite subset spaces

The space of all finite non-empty subsets of a topological space $X$, also known as the Ran space of $X$, is weakly contractible for $X$ path connected. We consider subspaces $\mathrm{Ran}_{\leqslant n}(X)$ of the Ran space given by all subsets of $X$ of size at most $n$, and their first homotopy groups. These groups are known to be trivial for $n\geqslant 3$ when $X$ is a path connected CW-complex, though the proofs are not constructive. We show that the induced map $π_1(\mathrm{Ran}_{\leqslant n}(X)) \to π_1(\mathrm{Ran}_{\leqslant n+2}(X))$ is trivial for all positive integers $n$, by explicitly drawing the path homotopies that contract any loop in $X$ to a point. From this we get a constructive proof for the triviality of $π_1(\mathrm{Ran}_{\leqslant n}(X))$, for all $n\geqslant 4$.

math.AT↗

arXiv2026

Pruning vineyards: updating barcodes and representative cycles by removing simplices

The barcode of a filtration and its representative cycles encode rich information often useful in data analysis. However, obtaining them can be computationally expensive. Therefore, it is useful to have methods that update them if the associated filtration undergoes small changes. There are already efficient algorithms updating a barcode if simplices exchange entrance order or are added, but not if simplices are removed. We provide an implementation to update a reduced boundary matrix when simplices in the filtration are removed. Our algorithm, the Simplicial Removal Update Procedure (SiRUP), intrinsically updates also the representative cycles, and is compatible with the clearing optimizations. We show that the complexity of our algorithm is lower than recomputing the barcode from scratch and that the number of executed matrix column additions is minimal, with both theoretical and experimental methods.

math.AT↗

arXiv2025

Stability of 0-dimensional persistent homology in enriched and sparsified point clouds

We give bounds for dimension 0 persistent homology and codimension 1 homology of Vietoris--Rips, alpha, and cubical complex filtrations from finite sets related by enrichment (adding new elements), sparsification (removing elements), and aligning to a grid (uniformly discretizing elements). For enrichment we use barycentric subdivision, for sparsification we use a minimum separating distance, and for aligning to a grid we take the quotient when dividing each coordinate value by a fixed step size. We are motivated by applications presenting large and irregular datasets, and the development of persistent homology to better work with them. In particular, we consider an application to ecology, in which the state of an observed species is inferred through a high-dimensional space with environmental variables as dimensions. This ``hypervolume'' has geometry (volume, convexity) and topology (connectedness, homology), which are known to be related to the current and potentially future status of the species. We offer an approach for the analysis of hypervolumes with topological guarantees, complementary to current statistical methods, giving precise bounds between persistence diagrams of Vietoris--Rips and alpha complexes, and a duality identity for cubical complexes. Implementation of our methods, called TopoAware, is made available in C++, Python, and R, building upon the GUDHI library.

math.AT↗

arXiv2023

DONUT -- Creation, Development, and Opportunities of a Database

DONUT is a database of papers about practical, real-world uses of Topological Data Analysis (TDA). Its original seed was planted in a group chat formed during the HIM Spring School on Applied and Computational Algebraic Topology in April 2017. This document describes the creation, curation, and maintenance process of the database.

cs.DL↗

arXiv2021

An application of neighbourhoods in digraphs to the classification of binary dynamics

A binary state on a graph means an assignment of binary values to its vertices. For example, if one encodes a network of spiking neurons as a directed graph, then the spikes produced by the neurons at an instant of time is a binary state on the encoding graph. Allowing time to vary and recording the spiking patterns of the neurons in the network produces an example of binary dynamics on the encoding graph, namely a one-parameter family of binary states on it. The central object of study in this article is the closed neighbourhood of a vertex $v$ in a graph $\mathcal{G}$, namely the subgraph of $\mathcal{G}$ that is induced by $v$ and all its neighbours in $\mathcal{G}$. We present a topological/graph theoretic method for extracting information out of binary dynamics on a graph, based on a selection of a relatively small number of vertices and their neighbourhoods. As a test case we demonstrate an application of the method to binary dynamics that arises from sample activity on the Blue Brain Project reconstruction of cortical tissue of a rat.

math.AT↗

arXiv2021

Stratifications on the Ran space

We describe a partial order on finite simplicial complexes. This partial order provides a poset stratification of the product of the Ran space of a metric space and the nonnegative real numbers, through the \v Cech simplicial complex. We show that paths in this product space respecting its stratification induce simplicial maps between the endpoints of the path.

math.AT↗

arXiv2020

Moduli Spaces of Morse Functions for Persistence

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate any two Morse--Smale vector fields on the sphere by a sequence of fundamental moves by considering graph-equivalent Morse functions. We also explore the combinatorially rich world of height-equivalent Morse functions, considered as height functions of embedded spheres in $\mathbf R^3$. Their level-set invariant, a poset generated by nested disks and annuli from levels sets, gives insight into the moduli space of Morse functions sharing the same persistence barcode.

math.AT↗