🍩 Database of Original & NonTheoretical Uses of Topology
(found 11 matches in 0.001554s)


Complexes of Tournaments, Directionality Filtrations and Persistent Homology (2020)
Dejan Govc, Ran Levi, Jason P. SmithAbstract
Complete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semisimplicial complexes, that we refer to as "tournaplexes", whose simplices are tournaments. In particular, given a digraph \$\mathcal\G\\$, we associate with it a "flag tournaplex" which is a tournaplex containing the directed flag complex of \$\mathcal\G\\$, but also the geometric realisation of cliques that are not directed. We define several types of filtrations on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of these ideas by applying them to graph data arising from the Blue Brain Project's digital reconstruction of a rat's neocortex. 
Simplicial Neural Networks (2020)
Stefania Ebli, Michaël Defferrard, Gard SpreemannAbstract
We present simplicial neural networks (SNNs), a generalization of graph neural networks to data that live on a class of topological spaces called simplicial complexes. These are natural multidimensional extensions of graphs that encode not only pairwise relationships but also higherorder interactions between vertices  allowing us to consider richer data, including vector fields and \$n\$fold collaboration networks. We define an appropriate notion of convolution that we leverage to construct the desired convolutional neural networks. We test the SNNs on the task of imputing missing data on coauthorship complexes. 
Graph Filtration Learning (2020)
Christoph Hofer, Florian Graf, Bastian Rieck, Marc Niethammer, Roland KwittAbstract
We propose an approach to learning with graphstructured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graphlevel representation. To this end, we leverage persistent homology computed via a realvalued, learnable, filter function. We establish the theoretical foundation for differentiating through the persistent homology computation. Empirically, we show that this type of readout operation compares favorably to previous techniques, especially when the graph connectivity structure is informative for the learning problem. 
Graph Classification via Heat Diffusion on Simplicial Complexes (2020)
Mehmet Emin Aktas, Esra AkbasAbstract
In this paper, we study the graph classification problem in vertexlabeled graphs. Our main goal is to classify the graphs comparing their higherorder structures thanks to heat diffusion on their simplices. We first represent vertexlabeled graphs as simplexweighted supergraphs. We then define the diffusion Frechet function over their simplices to encode the higherorder network topology and finally reach our goal by combining the function values with machine learning algorithms. Our experiments on realworld bioinformatics networks show that using diffusion Fr\éḩet function on simplices is promising in graph classification and more effective than the baseline methods. To the best of our knowledge, this paper is the first paper in the literature using heat diffusion on higherdimensional simplices in a graph mining problem. We believe that our method can be extended to different graph mining domains, not only the graph classification problem. 
Path Homology as a Stronger Analogue of Cyclomatic Complexity (2020)
Steve HuntsmanAbstract
Cyclomatic complexity is an incompletely specified but mathematically principled software metric that can be usefully applied to both source and binary code. We consider the application of path homology as a stronger analogue of cyclomatic complexity. We have implemented an algorithm to compute path homology in arbitrary dimension and applied it to several classes of relevant flow graphs, including randomly generated flow graphs representing structured and unstructured control flow. We also compared path homology and cyclomatic complexity on a set of disassembled binaries obtained from the grep utility. There exist control flow graphs realizable at the assembly level with nontrivial path homology in arbitrary dimension. We exhibit several classes of examples in this vein while also experimentally demonstrating that path homology gives identicial results to cyclomatic complexity for at least one detailed notion of structured control flow. We also experimentally demonstrate that the two notions differ on disassembled binaries, and we highlight an example of extreme disagreement. Path homology empirically generalizes cyclomatic complexity for an elementary notion of structured code and appears to identify more structurally relevant features of control flow in general. Path homology therefore has the potential to substantially improve upon cyclomatic complexity. 
A Persistent WeisfeilerLehman Procedure for Graph Classification (2019)
Bastian Rieck, Christian Bock, Karsten BorgwardtAbstract
The Weisfeiler–Lehman graph kernel exhibits competitive performance in many graph classification tasks. However, its subtree features are not able to capture connected components and cycles, topological features known for characterising graphs. To extract such features, we leverage propagated node label information and transform unweighted graphs into metric ones. This permits us to augment the subtree features with topological information obtained using persistent homology, a concept from topological data analysis. Our method, which we formalise as a generalisation of Weisfeiler–Lehman subtree features, exhibits favourable classification accuracy and its improvements in predictive performance are mainly driven by including cycle information. 
Learning Representations of Persistence Barcodes (2019)
Christoph D. Hofer, Roland Kwitt, Marc NiethammerAbstract
We consider the problem of supervised learning with summary representations of topological features in data. In particular, we focus on persistent homology, the prevalent tool used in topological data analysis. As the summary representations, referred to as barcodes or persistence diagrams, come in the unusual format of multi sets, equipped with computationally expensive metrics, they can not readily be processed with conventional learning techniques. While different approaches to address this problem have been proposed, either in the context of kernelbased learning, or via carefully designed vectorization techniques, it remains an open problem how to leverage advances in representation learning via deep neural networks. Appropriately handling topological summaries as input to neural networks would address the disadvantage of previous strategies which handle this type of data in a taskagnostic manner. In particular, we propose an approach that is designed to learn a taskspecific representation of barcodes. In other words, we aim to learn a representation that adapts to the learning problem while, at the same time, preserving theoretical properties (such as stability). This is done by projecting barcodes into a finite dimensional vector space using a collection of parametrized functionals, so called structure elements, for which we provide a generic construction scheme. A theoretical analysis of this approach reveals sufficient conditions to preserve stability, and also shows that different choices of structure elements lead to great differences with respect to their suitability for numerical optimization. When implemented as a neural network input layer, our approach demonstrates compelling performance on various types of problems, including graph classification and eigenvalue prediction, the classification of 2D/3D object shapes and recognizing activities from EEG signals. 
Stable Signatures for Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence (2018)
Woojin Kim, Facundo MemoliAbstract
When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. In a similar vein, studying the dynamics of social networks leads to the problem of characterizing groups/communities as they form and disperse throughout time. Motivated by this, we study the problem of obtaining persistent homology based summaries of timedependent data. Given a finite dynamic graph (DG), we first construct a zigzag persistence module arising from linearizing the dynamic transitive graph naturally induced from the input DG. Based on standard results, we then obtain a persistence diagram or barcode from this zigzag persistence module. We prove that these barcodes are stable under perturbations in the input DG under a suitable distance between DGs that we identify. More precisely, our stability theorem can be interpreted as providing a lower bound for the distance between DGs. Since it relies on barcodes, and their bottleneck distance, this lower bound can be computed in polynomial time from the DG inputs. Since DGs can be given rise by applying the Rips functor (with a fixed threshold) to dynamic metric spaces, we are also able to derive related stable invariants for these richer class of dynamic objects. Along the way, we propose a summarization of dynamic graphs that captures their timedependent clustering features which we call formigrams. These setvalued functions generalize the notion of dendrogram, a prevalent tool for hierarchical clustering. In order to elucidate the relationship between our distance between two DGs and the bottleneck distance between their associated barcodes, we exploit recent advances in the stability of zigzag persistence due to Botnan and Lesnick, and to Bjerkevik. 
Deep Learning With Topological Signatures (2017)
Christoph Hofer, Roland Kwitt, Marc Niethammer, Andreas Uhl 
Persistent Homology for Path Planning in Uncertain Environments (2015)
S. Bhattacharya, R. Ghrist, V. KumarAbstract
We address the fundamental problem of goaldirected path planning in an uncertain environment represented as a probability (of occupancy) map. Most methods generally use a threshold to reduce the grayscale map to a binary map before applying offtheshelf techniques to find the best path. This raises the somewhat illposed question, what is the right (optimal) value to threshold the map? We instead suggest a persistent homology approach to the problema topological approach in which we seek the homology class of trajectories that is most persistent for the given probability map. In other words, we want the class of trajectories that is free of obstacles over the largest range of threshold values. In order to make this problem tractable, we use homology in ℤ2 coefficients (instead of the standard ℤ coefficients), and describe how graph searchbased algorithms can be used to find trajectories in different homology classes. Our simulation results demonstrate the efficiency and practical applicability of the algorithm proposed in this paper.paper.