arXiv subjects

Sarah Percival

Publications and source records attributed to Sarah Percival.

9 recordsLinked to original sources

arXiv2025

Effective Resistance in Simplicial Complexes as Bilinear Forms: Generalizations and Properties

The concept of effective resistance, originally introduced in electrical circuit theory, has been extended to the setting of graphs by interpreting each edge as a resistor. In this context, the effective resistance between two vertices quantifies the total opposition to current flow when a unit current is injected at one vertex and extracted at the other. Beyond its physical interpretation, the effective resistance encodes rich structural and geometric information about the underlying graph: it defines a metric on the vertex set, relates to the topology of the graph through Foster's theorem, and determines the probability of an edge appearing in a random spanning tree. Generalizations of effective resistance to simplicial complexes have been proposed in several forms, often formulated as matrix products of standard operators associated with the complex. In this paper, we present a twofold generalization of the effective resistance. First, we introduce a novel, basis-independent bilinear form, derived from an algebraic reinterpretation of circuit theory, that extends the classical effective resistance from graphs. Second, we extend this bilinear form to simplices, chains, and cochains within simplicial complexes. This framework subsumes and unifies all existing matrix-based formulations of effective resistance. Moreover, we establish higher-order analogues of several fundamental properties known in the graph case: (i) we prove that effective resistance induces a pseudometric on the space of chains and a metric on the space of cycles, and (ii) we provide a generalization of Foster's Theorem to simplicial complexes.

math.CO↗

arXiv2025

Finding the Cores of Higher Graphs Using Geometric and Topological Means: A Survey

In this survey, we explore recent literature on finding the cores of higher graphs using geometric and topological means. We study graphs, hypergraphs, and simplicial complexes, all of which are models of higher graphs. We study the notion of a core, which is a minimalist representation of a higher graph that retains its geometric or topological information. We focus on geometric and topological methods based on discrete curvatures, effective resistance, and persistent homology. We aim to connect tools from graph theory, discrete geometry, and computational topology to inspire new research on the simplification of higher graphs.

math.HO↗

arXiv2025

Towards an Optimal Bound for the Interleaving Distance on Mapper Graphs

Mapper graphs are widely used tools in topological data analysis and visualization. They can be understood as discrete approximations of Reeb graphs, providing insight into the shape and connectivity of complex data. Given a high-dimensional point cloud together with a real-valued function defined on it, a mapper graph summarizes the induced topological structure: each node represents a local neighborhood, and edges connect nodes whose corresponding neighborhoods overlap. Our focus is the interleaving distance for mapper graphs, arising as a discretized analogue of the interleaving distance for Reeb graphs-a quantity known to be NP-hard to compute. This distance measures how similar two mapper graphs are by quantifying how much they must be ``stretched'' to be made comparable. Recent work introduced a loss function that gives an upper bound on this distance. The loss evaluates how far a given collection of maps, called an assignment, is from being a true interleaving. Importantly, it is computationally tractable, offering a practical way to bound the distance, however the quality of the bound is dependent on the choice of assignment. In this paper, we develop the first framework for bounding the interleaving distance on mapper graphs. We present the bound in two ways: first, by formulating an integer linear program (ILP) that determines whether an $n$-interleaving exists for a given $n$; and second, by constructing an ILP that identifies an assignment with minimal loss for that $n$. We also evaluate the method on small examples where the interleaving distance is known, and on benchmark and simulated datasets, demonstrating the utility of the approach for classification tasks based on mapper graphs.

cs.CG↗

arXiv2024

Any Graph is a Mapper Graph

The Mapper algorithm is a popular tool for visualization and data exploration in topological data analysis. We investigate an inverse problem for the Mapper algorithm: Given a dataset $X$ and a graph $G$, does there exist a set of Mapper parameters such that the output Mapper graph of $X$ is isomorphic to $G$? We provide constructions that affirmatively answer this question. Our results demonstrate that it is possible to engineer Mapper parameters to generate a desired graph.

math.AT↗

arXiv2024

A Distance for Geometric Graphs via the Labeled Merge Tree Interleaving Distance

Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric graphs via merge trees. In order to preserve as much useful information as possible from the original data, we introduce a way of rotating the sublevel set to obtain the merge trees via the idea of the directional transform. We represent the merge trees using a surjective multi-labeling scheme and then compute the distance between two representative matrices. We show some theoretically desirable qualities and present two methods of computation: approximation via sampling and exact distance using a kinetic data structure, both in polynomial time. We illustrate its utility by implementing it on two data sets.

cs.DS↗

arXiv2023

G-Mapper: Learning a Cover in the Mapper Construction

The Mapper algorithm is a visualization technique in topological data analysis (TDA) that outputs a graph reflecting the structure of a given dataset. However, the Mapper algorithm requires tuning several parameters in order to generate a ``nice" Mapper graph. This paper focuses on selecting the cover parameter. We present an algorithm that optimizes the cover of a Mapper graph by splitting a cover repeatedly according to a statistical test for normality. Our algorithm is based on G-means clustering which searches for the optimal number of clusters in $k$-means by iteratively applying the Anderson-Darling test. Our splitting procedure employs a Gaussian mixture model to carefully choose the cover according to the distribution of the given data. Experiments for synthetic and real-world datasets demonstrate that our algorithm generates covers so that the Mapper graphs retain the essence of the datasets, while also running significantly faster than a previous iterative method.

cs.LG↗

arXiv2023

Bounding the Interleaving Distance for Mapper Graphs with a Loss Function

Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances between them. In this work, we study the interleaving distance on discretization of these objects, called mapper graphs when $d=1$, where functor representations of the data can be compared by finding pairs of natural transformations between them. However, in many cases, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from recent work by Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation, called assignments. We then endow the functor images with the extra structure of a metric space and define a loss function which measures how far an assignment is from making the required diagrams of an interleaving commute. Finally we show that the computation of the loss function is polynomial with a given assignment. We believe this idea is both powerful and translatable, with the potential to provide approximations and bounds on interleavings in a broad array of contexts.

cs.CG↗

arXiv2021

Efficient computation of a semi-algebraic basis of the first homology group of a semi-algebraic set

Let $\mathrm{R}$ be a real closed field and $\mathrm{C}$ the algebraic closure of $\mathrm{R}$. We give an algorithm for computing a semi-algebraic basis for the first homology group, $\mathrm{H}_1(S,\mathbb{F})$, with coefficients in a field $\mathbb{F}$, of any given semi-algebraic set $S \subset \mathrm{R}^k$ defined by a closed formula. The complexity of the algorithm is bounded singly exponentially. It is not known how to compute such a basis for the higher homology groups with singly exponential complexity. As an intermediate step in our algorithm we construct a semi-algebraic subset $\Gamma$ of the given semi-algebraic set $S$, such that $\mathrm{H}_q(S,\Gamma) = 0$ for $q=0,1$. We relate this construction to a basic theorem in complex algebraic geometry stating that for any affine variety $X$ of dimension $n$, there exists Zariski closed subsets \[ Z^{(n-1)} \supset \cdots \supset Z^{(1)} \supset Z^{(0)} \] with $\dim_{\mathrm{C}} Z^{(i)} \leq i$, and $\mathrm{H}_q(X,Z^{(i)}) = 0$ for $0 \leq q \leq i$. We conjecture a quantitative version of this result in the semi-algebraic category, with $X$ and $Z^{(i)}$ replaced by closed semi-algebraic sets. We make initial progress on this conjecture by proving the existence of $Z^{(0)}$ and $Z^{(1)}$ with complexity bounded singly exponentially (previously, such an algorithm was known only for constructing $Z_0$).

math.AG↗

arXiv2018

On the Reeb spaces of definable maps

We prove that the Reeb space of a proper definable map $f:X \rightarrow Y$ in an arbitrary o-minimal expansion of a real closed field is realizable as a proper definable quotient. This result can be seen as an o-minimal analog of Stein factorization of proper morphisms in algebraic geometry. We also show that the Betti numbers of the Reeb space of $f$ can be arbitrarily large compared to those of $X$, unlike in the special case of Reeb graphs of manifolds. Nevertheless, in the special case when $f:X \rightarrow Y$ is a semi-algebraic map and $X$ is closed and bounded, we prove a singly exponential upper bound on the Betti numbers of the Reeb space of $f$ in terms of the number and degrees of the polynomials defining $X,Y$, and $f$.

math.AT↗