arXiv subjects

Nina Otter

Publications and source records attributed to Nina Otter.

18 recordsLinked to original sources

arXiv2026

Coalgebraic analysis of social systems

The algebraic analysis of social systems, or algebraic social network analysis, refers to a collection of methods designed to extract information about the structure of a social system represented as a directed graph. Central among these are methods to determine the roles that exist within a given system, and the positions. The analysis of roles and positions is highly developed for social systems that involve only pairwise interactions among actors - however, in contemporary social network analysis it is increasingly common to use models that can take into account higher-order interactions as well. In this paper we take a category-theoretic approach to the question of how to lift role and positional analysis from graphs to hypergraphs, which can accommodate higher-order interactions. We use the framework of universal coalgebra - a 'theory of systems' with origins in computer science and logic - to formalize the main concepts of role and positional analysis and extend them to a large class of structures that includes both graphs and hypergraphs. As evidence for the validity of our definitions, we prove a very general functoriality theorem that specializes, in the case of graphs, to a folkloric observation about the compatibility of positional and role analysis.

cs.SI↗

arXiv2025

Geometric characterisation of structural and regular equivalences in undirected (hyper)graphs

Similarity notions between vertices in a graph, such as structural and regular equivalence, are one of the main ingredients in clustering tools in complex network science. We generalise structural and regular equivalences for undirected hypergraphs and provide a characterisation of structural and regular equivalences of undirected graphs and hypergraphs through neighbourhood graphs and Ollivier-Ricci curvature. Our characterisation sheds new light on these similarity notions opening a new avenue for their exploration. These characterisations also enable the construction of a possibly wide family of regular partitions, thereby offering a new route to a task that has so far been computationally challenging.

math.CO↗

arXiv2025

Fractal dimensions of complex networks: advocating for a topological approach

Topological Data Analysis (TDA) uses insights from topology to create representations of data able to capture global and local geometric and topological properties. Its methods have successfully been used to develop estimations of fractal dimensions for metric spaces that have been shown to outperform existing techniques. In a parallel line of work, networks are ubiquitously used to model a variety of complex systems. Higher-order interactions, i.e., simultaneous interactions between more than two nodes, are wide-spread in social and biological systems, and simplicial complexes, used in TDA, can capture important structural and topological properties of networks by modelling such higher-order interactions. In this position paper, we advocate for methods from TDA to be used to estimate fractal dimensions of complex networks, we discuss the possible advantages of such an approach and outline some of the challenges to be addressed.

math.AT↗

arXiv2024

Shoving tubes through shapes gives a sufficient and efficient shape statistic

The Persistent Homology Transform (PHT) was introduced in the field of Topological Data Analysis about 10 years ago, and has since been proven to be a very powerful descriptor of Euclidean shapes. The PHT consists of scanning a shape from all possible directions $v\in S^{n-1}$ and then computing the persistent homology of sublevel set filtrations of the respective height functions $h_v$; this results in a sufficient and continuous descriptor of Euclidean shapes. We introduce a generalisation of the PHT in which we consider arbitrary parameter spaces and sublevel sets with respect to any function. In particular, we study transforms, defined on the Grassmannian $\mathbb{A}\mathbb{G}(m,n)$ of affine subspaces of $\mathbb{R}^n$, that allow to scan a shape by probing it with all possible affine $m$-dimensional subspaces $P\subset \mathbb{R}^n$, for fixed dimension $m$, and by computing persistent homology of sublevel set filtrations of the function $\mathrm{dist}(\cdot, P)$ encoding the distance from the flat $P$. We call such transforms "distance-from-flat" PHTs. We show that these transforms are injective and continuous and that they provide computational advantages over the classical PHT. In particular, we show that it is enough to compute homology only in degrees up to $m-1$ to obtain injectivity; for $m=1$ this provides a very powerful and computationally advantageous tool for examining shapes, which in a previous work by a subset of the authors has proven to significantly outperform state-of-the-art neural networks for shape classification tasks.

math.AT↗

arXiv2024

Time-optimal persistent homology representatives for univariate time series

Persistent homology (PH) is one of the main methods used in Topological Data Analysis. An active area of research in the field is the study of appropriate notions of PH representatives, which allow to interpret the meaning of the information provided by PH, making it an important problem in the application of PH, and in the study of its interpretability. Computing optimal PH representatives is a problem that is known to be NP-hard, and one is therefore interested in developing context-specific optimality notions that are computable in practice. Here we introduce time-optimal PH representatives for time-varying data, allowing one to extract representatives that are close in time in an appropriate sense. We illustrate our methods on quasi-periodic synthetic time series, as well as time series arising from climate models, and we show that our methods provide optimal PH representatives that are better suited for these types of problems than existing optimality notions, such as length-optimal PH representatives.

math.AT↗

arXiv2024

Amplitudes in persistence theory

The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates with data sets. Here we introduce a general framework to compare distances and invariants in multiparameter persistence, where there is no natural choice of invariants and distances between them. We define amplitudes, monotone, and subadditive invariants that arise from assigning a non-negative real number to objects of an abelian category. We then present different ways to associate distances to such invariants, and we provide a classification of classes of amplitudes relevant to topological data analysis. In addition, we study the the relationships as well as the discriminitative power of such amplitude distances arising in topological data analysis scenarios.

math.AT↗

arXiv2024

Pull-back Geometry of Persistent Homology Encodings

Persistent homology (PH) is a method for generating topology-inspired representations of data. Empirical studies that investigate the properties of PH, such as its sensitivity to perturbations or ability to detect a feature of interest, commonly rely on training and testing an additional model on the basis of the PH representation. To gain more intrinsic insights about PH, independently of the choice of such a model, we propose a novel methodology based on the pull-back geometry that a PH encoding induces on the data manifold. The spectrum and eigenvectors of the induced metric help to identify the most and least significant information captured by PH. Furthermore, the pull-back norm of tangent vectors provides insights about the sensitivity of PH to a given perturbation, or its potential to detect a given feature of interest, and in turn its ability to solve a given classification or regression problem. Experimentally, the insights gained through our methodology align well with the existing knowledge about PH. Moreover, we show that the pull-back norm correlates with the performance on downstream tasks, and can therefore guide the choice of a suitable PH encoding.

math.AT↗

arXiv2023

On the effectiveness of persistent homology

Persistent homology (PH) is one of the most popular methods in Topological Data Analysis. Even though PH has been used in many different types of applications, the reasons behind its success remain elusive; in particular, it is not known for which classes of problems it is most effective, or to what extent it can detect geometric or topological features. The goal of this work is to identify some types of problems where PH performs well or even better than other methods in data analysis. We consider three fundamental shape analysis tasks: the detection of the number of holes, curvature and convexity from 2D and 3D point clouds sampled from shapes. Experiments demonstrate that PH is successful in these tasks, outperforming several baselines, including PointNet, an architecture inspired precisely by the properties of point clouds. In addition, we observe that PH remains effective for limited computational resources and limited training data, as well as out-of-distribution test data, including various data transformations and noise. For convexity detection, we provide a theoretical guarantee that PH is effective for this task in $\mathbb{R}^d$, and demonstrate the detection of a convexity measure on the FLAVIA data set of plant leaf images. Due to the crucial role of shape classification in understanding mathematical and physical structures and objects, and in many applications, the findings of this work will provide some knowledge about the types of problems that are appropriate for PH, so that it can - to borrow the words from Wigner 1960 - ``remain valid in future research, and extend, to our pleasure", but to our lesser bafflement, to a variety of applications.

math.AT↗

arXiv2022

Alpha magnitude

Magnitude is an isometric invariant for metric spaces that was introduced by Leinster around 2010, and is currently the object of intense research, since it has been shown to encode many known invariants of metric spaces. In recent work, Govc and Hepworth introduced persistent magnitude, a numerical invariant of a filtered simplicial complex associated to a metric space. Inspired by Govc and Hepworth's definition, we introduce alpha magnitude and investigate some of its key properties. Heuristic observations lead us to conjecture a relationship with the Minkowski dimension of compact subspaces of Euclidean space. Finally, alpha magnitude presents computational advantages over both magnitude as well as Rips magnitude, and we thus propose it as a new measure for the estimation of fractal dimensions of real-world data sets that is easily computable.

math.AT↗

arXiv2022

Weisfeiler and Lehman Go Cellular: CW Networks

Graph Neural Networks (GNNs) are limited in their expressive power, struggle with long-range interactions and lack a principled way to model higher-order structures. These problems can be attributed to the strong coupling between the computational graph and the input graph structure. The recently proposed Message Passing Simplicial Networks naturally decouple these elements by performing message passing on the clique complex of the graph. Nevertheless, these models can be severely constrained by the rigid combinatorial structure of Simplicial Complexes (SCs). In this work, we extend recent theoretical results on SCs to regular Cell Complexes, topological objects that flexibly subsume SCs and graphs. We show that this generalisation provides a powerful set of graph "lifting" transformations, each leading to a unique hierarchical message passing procedure. The resulting methods, which we collectively call CW Networks (CWNs), are strictly more powerful than the WL test and not less powerful than the 3-WL test. In particular, we demonstrate the effectiveness of one such scheme, based on rings, when applied to molecular graph problems. The proposed architecture benefits from provably larger expressivity than commonly used GNNs, principled modelling of higher-order signals and from compressing the distances between nodes. We demonstrate that our model achieves state-of-the-art results on a variety of molecular datasets.

cs.LG↗

arXiv2021

A topological perspective on weather regimes

It has long been suggested that the mid-latitude atmospheric circulation possesses what has come to be known as `weather regimes', loosely categorised as regions of phase space with above-average density and/or extended persistence. Their existence and behaviour has been extensively studied in meteorology and climate science, due to their potential for drastically simplifying the complex and chaotic mid-latitude dynamics. Several well-known, simple non-linear dynamical systems have been used as toy-models of the atmosphere in order to understand and exemplify such regime behaviour. Nevertheless, no agreed-upon and clear-cut definition of a `regime' exists in the literature. We argue here for an approach which equates the existence of regimes in a dynamical system with the existence of non-trivial topological structure of the system's attractor. We show using persistent homology, an algorithmic tool in topological data analysis, that this approach is computationally tractable, practically informative, and identifies the relevant regime structure across a range of examples.

physics.ao-ph↗

arXiv2021

Magnitude meets persistence. Homology theories for filtered simplicial sets

The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that counts the "effective number of points" of the space and has been shown to encode many invariants of metric spaces from integral geometry and geometric measure theory. In 2015, Hepworth and Willerton introduced a homology theory for metric graphs, called magnitude homology, which categorifies the magnitude of a finite metric graph. This work was subsequently generalised to enriched categories by Leinster and Shulman, and the homology theory that they introduced categorifies magnitude for arbitrary finite metric spaces. When studying a metric space, one is often only interested in the metric space up to a rescaling of the distance of the points by a non-negative real number. The magnitude function describes how the effective number of points changes as one scales the distance, and is completely encoded by magnitude homology. When studying a finite metric space in topological data analysis using persistent homology, one approximates the space through a nested sequence of simplicial complexes so as to recover topological information about the space by studying the homology of this sequence. Here we relate magnitude homology and persistent homology as two different ways of computing homology of filtered simplicial sets.

math.AT↗

arXiv2021

Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks

The pairwise interaction paradigm of graph machine learning has predominantly governed the modelling of relational systems. However, graphs alone cannot capture the multi-level interactions present in many complex systems and the expressive power of such schemes was proven to be limited. To overcome these limitations, we propose Message Passing Simplicial Networks (MPSNs), a class of models that perform message passing on simplicial complexes (SCs). To theoretically analyse the expressivity of our model we introduce a Simplicial Weisfeiler-Lehman (SWL) colouring procedure for distinguishing non-isomorphic SCs. We relate the power of SWL to the problem of distinguishing non-isomorphic graphs and show that SWL and MPSNs are strictly more powerful than the WL test and not less powerful than the 3-WL test. We deepen the analysis by comparing our model with traditional graph neural networks (GNNs) with ReLU activations in terms of the number of linear regions of the functions they can represent. We empirically support our theoretical claims by showing that MPSNs can distinguish challenging strongly regular graphs for which GNNs fail and, when equipped with orientation equivariant layers, they can improve classification accuracy in oriented SCs compared to a GNN baseline.

cs.LG↗

arXiv2020

Can neural networks learn persistent homology features?

Topological data analysis uses tools from topology -- the mathematical area that studies shapes -- to create representations of data. In particular, in persistent homology, one studies one-parameter families of spaces associated with data, and persistence diagrams describe the lifetime of topological invariants, such as connected components or holes, across the one-parameter family. In many applications, one is interested in working with features associated with persistence diagrams rather than the diagrams themselves. In our work, we explore the possibility of learning several types of features extracted from persistence diagrams using neural networks.

cs.LG↗

arXiv2020

A unified framework for equivalences in social networks

A key concern in network analysis is the study of social positions and roles of actors in a network. The notion of "position" refers to an equivalence class of nodes that have similar ties to other nodes, whereas a "role" is an equivalence class of compound relations that connect the same pairs of nodes. An open question in network science is whether it is possible to simultaneously perform role and positional analysis. Motivated by the principle of functoriality in category theory we propose a new method that allows to tie role and positional analysis together. We illustrate our methods on two well-studied data sets in network science.

cs.SI↗

arXiv2019

Stratifying multiparameter persistent homology

A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.

math.AT↗

arXiv2017

A roadmap for the computation of persistent homology

Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.

math.AT↗

arXiv2017

Operads and Phylogenetic Trees

We construct an operad $\mathrm{Phyl}$ whose operations are the edge-labelled trees used in phylogenetics. This operad is the coproduct of $\mathrm{Com}$, the operad for commutative semigroups, and $[0,\infty)$, the operad with unary operations corresponding to nonnegative real numbers, where composition is addition. We show that there is a homeomorphism between the space of $n$-ary operations of $\mathrm{Phyl}$ and $\mathcal{T}_n\times [0,\infty)^{n+1}$, where $\mathcal{T}_n$ is the space of metric $n$-trees introduced by Billera, Holmes and Vogtmann. Furthermore, we show that the Markov models used to reconstruct phylogenetic trees from genome data give coalgebras of $\mathrm{Phyl}$. These always extend to coalgebras of the larger operad $\mathrm{Com} + [0,\infty]$, since Markov processes on finite sets converge to an equilibrium as time approaches infinity. We show that for any operad $O$, its coproduct with $[0,\infty]$ contains the operad $W(O)$ constucted by Boardman and Vogt. To prove these results, we explicitly describe the coproduct of operads in terms of labelled trees.

math.CT↗