Past Applied Topology Seminar

19 June 2020
15:00
Nina Otter
Abstract

One of the main concerns in social network science is the study of positions and roles. By "position" social scientists usually mean a collection of actors who have similar ties to other actors, while a "role" is a specific pattern of ties among actors or positions. Since the 1970s a lot of research has been done to develop these concepts in a rigorous way. An open question in the field is whether it is possible to perform role and positional analysis simultaneously. In joint work in progress with Mason Porter we explore this question by proposing a framework that relies on the principle of functoriality in category theory. In this talk I will introduce role and positional analysis, present some well-studied examples from social network science, and what new insights this framework might give us.

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  • Applied Topology Seminar
12 June 2020
15:00
Abstract

Contagion maps are a family of maps that map nodes of a network to points in a high-dimensional space, based on the activations times in a threshold contagion on the network. A point cloud that is the image of such a map reflects both the structure underlying the network and the spreading behaviour of the contagion on it. Intuitively, such a point cloud exhibits features of the network's underlying structure if the contagion spreads along that structure, an observation which suggests contagion maps as a viable manifold-learning technique. We test contagion maps as a manifold-learning tool on several different data sets, and compare its performance to that of Isomap, one of the most well-known manifold-learning algorithms. We find that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, when Isomap is prone to noise-induced error. This consolidates contagion maps as a technique for manifold learning. 

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  • Applied Topology Seminar
5 June 2020
15:00
Ginestra Bianconi
Abstract

Synchronization is a collective phenomenon that pervades the natural systems from neurons to fireflies. In a network, synchronization of the dynamical variables associated to the nodes occurs when nodes are coupled to their neighbours as captured by the Kuramoto model. However many complex systems include also higher-order interactions among more than two nodes and sustain dynamical signals that might be related to higher-order simplices such as nodes of triangles. These dynamical topological signals include for instance fluxes which are dynamical variables associated to links.

In this talk I present a new topological approach [1] to synchronization on simplicial complexes. Here the theory of synchronization is combined with topology (specifically Hodge theory) for formulating the higher-order Kuramoto model that uses the higher-order Laplacians and provides the main synchronization route for topological signals. I will show that the dynamics defined on links can be projected to a dynamics defined on nodes and triangles that undergo a synchronization transition and I will discuss how this procedure can be immediately generalized for topological signals of higher dimension. Interestingly I will show that when the model includes an adaptive coupling of the two projected dynamics, the transition becomes explosive, i.e. synchronization emerges abruptly.

This model can be applied to study synchronization of topological signals in the brain and in biological transport networks as it proposes a new set of topological transformations that can reveal collective synchronization phenomena that could go unnoticed otherwise.

[1] Millán, A.P., Torres, J.J. and Bianconi, G., 2019. Explosive higher-order Kuramoto dynamics on simplicial complexes. Physical Review Letters (in press) arXiv preprint arXiv:1912.04405.

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  • Applied Topology Seminar
29 May 2020
15:00
Tadas Temcinas
Abstract


Eigenvalue-eigenvector pairs of combinatorial graph Laplacians are extensively used in graph theory and network analysis. It is well known that the spectrum of the Laplacian L of a given graph G encodes aspects of the geometry of G  - the multiplicity of the eigenvalue 0 counts the number of connected components while the second smallest eigenvalue (called the Fiedler eigenvalue) quantifies the well-connectedness of G . In network analysis, one uses Laplacian eigenvectors associated with small eigenvalues to perform spectral clustering. In graph signal processing, graph Fourier transforms are defined in terms of an orthonormal eigenbasis of L. Eigenvectors of L also play a central role in graph neural networks.

Motivated by this we study eigenvalue-eigenvector pairs of Laplacians of random graphs and their potential use in TDA. I will present simulation results on what persistent homology barcodes of Bernoulli random graphs G(n, p) look like when we use Laplacian eigenvectors as filter functions. Also, I will discuss the conjectures made from the simulations as well as the challenges that arise when trying to prove them. This is work in progress.
 

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  • Applied Topology Seminar
15 May 2020
15:00
Kelly Spendlove
Abstract

The last fifty years of dynamical systems theory have established that dynamical systems can exhibit extremely complex behavior with respect to both the system variables (chaos theory) and parameters (bifurcation theory). Such complex behavior found in theoretical work must be reconciled with the capabilities of the current technologies available for applications. For example, in the case of modelling biological phenomena, measurements may be of limited precision, parameters are rarely known exactly and nonlinearities often cannot be derived from first principles. 

The contrast between the richness of dynamical systems and the imprecise nature of available modeling tools suggests that we should not take models too seriously. Stating this a bit more formally, it suggests that extracting features which are robust over a range of parameter values is more important than an understanding of the fine structure at some particular parameter.

The goal of this talk is to present a high-level introduction/overview of computational Conley-Morse theory, a rigorous computational approach for understanding the global dynamics of complex systems.  This introduction will wander through dynamical systems theory, algebraic topology, combinatorics and end in game theory.

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  • Applied Topology Seminar
8 May 2020
15:00
Ambrose Yim
Abstract

We present a recipe for constructing filter functions on graphs with parameters that can optimised by gradient descent. This recipe, based on graph Laplacians and spectral wavelet signatures, do not require additional data to be defined on vertices. This allows any graph to be assigned a customised filter function for persistent homology computations and data science applications, such as graph classification. We show experimental evidence that this recipe has desirable properties for optimisation and machine learning pipelines that factors through persistent homology. 

  • Applied Topology Seminar
11 June 2018
14:00
Renaud Lamboitte
Abstract

In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research practice in mathematics. Holes die when a subset of their concepts appear in the same article, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the dimension of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas. We also show that authors' conceptual entropy is positively related with their contribution to homological holes, suggesting that polymaths tend to be on the frontier of research.

  • Applied Topology Seminar
1 June 2018
12:00
Maddie Weinstein
Abstract

We will discuss the algebraicity of two quantities central to the computation of persistent homology. We will also connect persistent homology and algebraic optimization. Namely, we will express the degree corresponding to the distance variable of the offset hypersurface in terms of the Euclidean distance degree of the starting variety, obtaining a new way to compute these degrees. Finally, we will describe the non-properness locus of the offset construction and use this to describe the set of points that are topologically interesting (the medial axis and center points of the bounded components of the complement of the variety) and relevant to the computation of persistent homology.

  • Applied Topology Seminar

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