Persistent Homology Guided Force-Directed Graph Layouts

Graphs are commonly used to encode relationships among entities, yet their abstractness makes them difficult to analyze. Node-link diagrams are popular for drawing graphs, and force-directed layouts provide a flexible method for node arrangements that use local relationships in an attempt to reveal the global shape of the graph. However, clutter and overlap of unrelated structures can lead to confusing graph visualizations. This paper leverages the persistent homology features of an undirected graph as derived information for interactive manipulation of force-directed layouts. We first discuss how to efficiently extract 0-dimensional persistent homology features from both weighted and unweighted undirected graphs. We then introduce the interactive persistence barcode used to manipulate the force-directed graph layout. In particular, the user adds and removes contracting and repulsing forces generated by the persistent homology features, eventually selecting the set of persistent homology features that most improve the layout. Finally, we demonstrate the utility of our approach across a variety of synthetic and real datasets

Mapper on Graphs for Network Visualization

Networks are an exceedingly popular type of data for representing relationships between individuals, businesses, proteins, brain regions, telecommunication endpoints, etc. Network or graph visualization provides an intuitive way to explore the node-link structures of network data for instant sense-making. However, naive node-link diagrams can fail to convey insights regarding network structures, even for moderately sized data of a few hundred nodes. We propose to apply the mapper construction--a popular tool in topological data analysis--to graph visualization, which provides a strong theoretical basis for summarizing network data while preserving their core structures. We develop a variation of the mapper construction targeting weighted, undirected graphs, called mapper on graphs, which generates property-preserving summaries of graphs. We provide a software tool that enables interactive explorations of such summaries and demonstrates the effectiveness of our method for synthetic and real-world data. The mapper on graphs approach we propose represents a new class of techniques that leverages tools from topological data analysis in addressing challenges in graph visualization

TDANetVis: Suggesting temporal resolutions for graph visualization using zigzag persistent homology

Temporal graphs are commonly used to represent complex systems and track the evolution of their constituents over time. Visualizing these graphs is crucial as it allows one to quickly identify anomalies, trends, patterns, and other properties leading to better decision-making. In this context, the to-be-adopted temporal resolution is crucial in constructing and analyzing the layout visually. The choice of a resolution is critical, e.g., when dealing with temporally sparse graphs. In such cases, changing the temporal resolution by grouping events (i.e., edges) from consecutive timestamps, a technique known as timeslicing, can aid in the analysis and reveal patterns that might not be discernible otherwise. However, choosing a suitable temporal resolution is not trivial. In this paper, we propose TDANetVis, a methodology that suggests temporal resolutions potentially relevant for analyzing a given graph, i.e., resolutions that lead to substantial topological changes in the graph structure. To achieve this goal, TDANetVis leverages zigzag persistent homology, a well-established technique from Topological Data Analysis (TDA). To enhance visual graph analysis, TDANetVis also incorporates the colored barcode, a novel timeline-based visualization built on the persistence barcodes commonly used in TDA. We demonstrate the usefulness and effectiveness of TDANetVis through a usage scenario and a user study involving 27 participants.Comment: This document contains the main article and supplementary material. For associated code and software, see https://github.com/raphaeltinarrage/TDANetVi

Efficient Planning of Multi-Robot Collective Transport using Graph Reinforcement Learning with Higher Order Topological Abstraction

Efficient multi-robot task allocation (MRTA) is fundamental to various time-sensitive applications such as disaster response, warehouse operations, and construction. This paper tackles a particular class of these problems that we call MRTA-collective transport or MRTA-CT -- here tasks present varying workloads and deadlines, and robots are subject to flight range, communication range, and payload constraints. For large instances of these problems involving 100s-1000's of tasks and 10s-100s of robots, traditional non-learning solvers are often time-inefficient, and emerging learning-based policies do not scale well to larger-sized problems without costly retraining. To address this gap, we use a recently proposed encoder-decoder graph neural network involving Capsule networks and multi-head attention mechanism, and innovatively add topological descriptors (TD) as new features to improve transferability to unseen problems of similar and larger size. Persistent homology is used to derive the TD, and proximal policy optimization is used to train our TD-augmented graph neural network. The resulting policy model compares favorably to state-of-the-art non-learning baselines while being much faster. The benefit of using TD is readily evident when scaling to test problems of size larger than those used in training.Comment: This paper has been accepted to be presented at the IEEE International Conference on Robotics and Automation, 202

Persistent Topological Laplacians -- a Survey

Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA), motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combines multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of persistent homology, while their nonharmonic spectra provide supplementary information, such as the homotopic shape evolution of data. Persistent topological Laplacians have demonstrated superior performance over persistent homology in addressing large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated on various mathematical objects, including simplicial complexes, path complexes, flag complexes, diraphs, hypergraphs, hyperdigraphs, cellular sheaves, as well as $N$-chain complexes. Alongside fundamental mathematical concepts, we emphasize the theoretical formulations associated with various persistent topological Laplacians and illustrate their applications through numerous simple geometric shapes

Topology combined machine learning for consonant recognition

In artificial-intelligence-aided signal processing, existing deep learning models often exhibit a black-box structure, and their validity and comprehensibility remain elusive. The integration of topological methods, despite its relatively nascent application, serves a dual purpose of making models more interpretable as well as extracting structural information from time-dependent data for smarter learning. Here, we provide a transparent and broadly applicable methodology, TopCap, to capture the most salient topological features inherent in time series for machine learning. Rooted in high-dimensional ambient spaces, TopCap is capable of capturing features rarely detected in datasets with low intrinsic dimensionality. Applying time-delay embedding and persistent homology, we obtain descriptors which encapsulate information such as the vibration of a time series, in terms of its variability of frequency, amplitude, and average line, demonstrated with simulated data. This information is then vectorised and fed into multiple machine learning algorithms such as k-nearest neighbours and support vector machine. Notably, in classifying voiced and voiceless consonants, TopCap achieves an accuracy exceeding 96% and is geared towards designing topological convolutional layers for deep learning of speech and audio signals