🍩 Database of Original & NonTheoretical Uses of Topology
(found 62 matches in 0.010114s)


A Simplified Algorithm for Identifying Abnormal Changes in Dynamic Networks (2022)
Bouchaib Azamir, Driss Bennis, Bertrand MichelAbstract
Topological data analysis has recently been applied to the study of dynamic networks. In this context, an algorithm was introduced and helps, among other things, to detect early warning signals of abnormal changes in the dynamic network under study. However, the complexity of this algorithm increases significantly once the database studied grows. In this paper, we propose a simplification of the algorithm without affecting its performance. We give various applications and simulations of the new algorithm on some weighted networks. The obtained results show clearly the efficiency of the introduced approach. Moreover, in some cases, the proposed algorithm makes it possible to highlight local information and sometimes early warning signals of local abnormal changes. 
Path Homologies of Motifs and Temporal Network Representations (2022)
Samir Chowdhury, Steve Huntsman, Matvey YutinAbstract
Path homology is a powerful method for attaching algebraic invariants to digraphs. While there have been growing theoretical developments on the algebrotopological framework surrounding path homology, bona fide applications to the study of complex networks have remained stagnant. We address this gap by presenting an algorithm for path homology that combines efficient pruning and indexing techniques and using it to topologically analyze a variety of realworld complex temporal networks. A crucial step in our analysis is the complete characterization of path homologies of certain families of small digraphs that appear as subgraphs in these complex networks. These families include all digraphs, directed acyclic graphs, and undirected graphs up to certain numbers of vertices, as well as some specially constructed cases. Using information from this analysis, we identify small digraphs contributing to path homology in dimension two for three temporal networks in an aggregated representation and relate these digraphs to network behavior. We then investigate alternative temporal network representations and identify complementary subgraphs as well as behavior that is preserved across representations. We conclude that path homology provides insight into temporal network structure, and in turn, emergent structures in temporal networks provide us with new subgraphs having interesting path homology. 
Toroidal Topology of Population Activity in Grid Cells (2022)
Richard J. Gardner, Erik Hermansen, Marius Pachitariu, Yoram Burak, Nils A. Baas, Benjamin A. Dunn, MayBritt Moser, Edvard I. MoserAbstract
The medial entorhinal cortex is part of a neural system for mapping the position of an individual within a physical environment1. Grid cells, a key component of this system, fire in a characteristic hexagonal pattern of locations2, and are organized in modules3 that collectively form a population code for the animal’s allocentric position1. The invariance of the correlation structure of this population code across environments4,5 and behavioural states6,7, independent of specific sensory inputs, has pointed to intrinsic, recurrently connected continuous attractor networks (CANs) as a possible substrate of the grid pattern1,8–11. However, whether grid cell networks show continuous attractor dynamics, and how they interface with inputs from the environment, has remained unclear owing to the small samples of cells obtained so far. Here, using simultaneous recordings from many hundreds of grid cells and subsequent topological data analysis, we show that the joint activity of grid cells from an individual module resides on a toroidal manifold, as expected in a twodimensional CAN. Positions on the torus correspond to positions of the moving animal in the environment. Individual cells are preferentially active at singular positions on the torus. Their positions are maintained between environments and from wakefulness to sleep, as predicted by CAN models for grid cells but not by alternative feedforward models12. This demonstration of network dynamics on a toroidal manifold provides a populationlevel visualization of CAN dynamics in grid cells. 
Topological Data Analysis of TaskBased fMRI Data From Experiments on Schizophrenia (2021)
Bernadette J. Stolz, Tegan Emerson, Satu Nahkuri, Mason A. Porter, Heather A. Harrington 
Filtration Curves for Graph Representation (2021)
Leslie O'Bray, Bastian Rieck, Karsten BorgwardtAbstract
The two predominant approaches to graph comparison in recent years are based on (i) enumerating matching subgraphs or (ii) comparing neighborhoods of nodes. In this work, we complement these two perspectives with a third way of representing graphs: using filtration curves from topological data analysis that capture both edge weight information and global graph structure. Filtration curves are highly efficient to compute and lead to expressive representations of graphs, which we demonstrate on graph classification benchmark datasets. Our work opens the door to a new form of graph representation in data mining. 
Homological Scaffold via Minimal Homology Bases (2021)
Marco Guerra, Alessandro De Gregorio, Ulderico Fugacci, Giovanni Petri, Francesco VaccarinoAbstract
The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global features onto individual network components, unless one provides a principled way to make such a choice. In this paper, we apply recent advances in the computation of minimal homology bases to introduce a quasicanonical version of the scaffold, called minimal, and employ it to analyze data both real and in silico. At the same time, we verify that, statistically, the standard scaffold is a good proxy of the minimal one for sufficiently complex networks. 
HiDeF: Identifying Persistent Structures in Multiscale ‘Omics Data (2021)
Fan Zheng, She Zhang, Christopher Churas, Dexter Pratt, Ivet Bahar, Trey IdekerAbstract
In any ‘omics study, the scale of analysis can dramatically affect the outcome. For instance, when clustering singlecell transcriptomes, is the analysis tuned to discover broad or specific cell types? Likewise, protein communities revealed from protein networks can vary widely in sizes depending on the method. Here, we use the concept of persistent homology, drawn from mathematical topology, to identify robust structures in data at all scales simultaneously. Application to mouse singlecell transcriptomes significantly expands the catalog of identified cell types, while analysis of SARSCOV2 protein interactions suggests hijacking of WNT. The method, HiDeF, is available via Python and Cytoscape. 
From Topological Analyses to Functional Modeling: The Case of Hippocampus (2021)
Yuri DabaghianAbstract
Topological data analyses are widely used for describing and conceptualizing large volumes of neurobiological data, e.g., for quantifying spiking outputs of large neuronal ensembles and thus understanding the functions of the corresponding networks. Below we discuss an approach in which convergent topological analyses produce insights into how information may be processed in mammalian hippocampus—a brain part that plays a key role in learning and memory. The resulting functional model provides a unifying framework for integrating spiking data at different timescales and following the course of spatial learning at different levels of spatiotemporal granularity. This approach allows accounting for contributions from various physiological phenomena into spatial cognition—the neuronal spiking statistics, the effects of spiking synchronization by different brain waves, the roles played by synaptic efficacies and so forth. In particular, it is possible to demonstrate that networks with plastic and transient synaptic architectures can encode stable cognitive maps, revealing the characteristic timescales of memory processing. 
Geometric Feature Performance Under Downsampling for EEG Classification Tasks (2021)
Bryan Bischof, Eric BunchAbstract
We experimentally investigate a collection of feature engineering pipelines for use with a CNN for classifying eyesopen or eyesclosed from electroencephalogram (EEG) timeseries from the Bonn dataset. Using the Takens' embeddinga geometric representation of timeserieswe construct simplicial complexes from EEG data. We then compare \$\epsilon\$series of Bettinumbers and \$\epsilon\$series of graph spectra (a novel construction)two topological invariants of the latent geometry from these complexesto raw time series of the EEG to fill in a gap in the literature for benchmarking. These methods, inspired by Topological Data Analysis, are used for feature engineering to capture local geometry of the timeseries. Additionally, we test these feature pipelines' robustness to downsampling and data reduction. This paper seeks to establish clearer expectations for both timeseries classification via geometric features, and how CNNs for timeseries respond to data of degraded resolution. 
Inferring COVID19 Biological Pathways From Clinical Phenotypes via Topological Analysis (2021)
Negin Karisani, Daniel E. Platt, Saugata Basu, Laxmi ParidaAbstract
COVID19 has caused thousands of deaths around the world and also resulted in a large international economic disruption. Identifying the pathways associated with this illness can help medical researchers to better understand the properties of the condition. This process can be carried out by analyzing the medical records. It is crucial to develop tools and models that can aid researchers with this process in a timely manner. However, medical records are often unstructured clinical notes, and this poses significant challenges to developing the automated systems. In this article, we propose a pipeline to aid practitioners in analyzing clinical notes and revealing the pathways associated with this disease. Our pipeline relies on topological properties and consists of three steps: 1) preprocessing the clinical notes to extract the salient concepts, 2) constructing a feature space of the patients to characterize the extracted concepts, and finally, 3) leveraging the topological properties to distill the available knowledge and visualize the result. Our experiments on a publicly available dataset of COVID19 clinical notes testify that our pipeline can indeed extract meaningful pathways. 
Determining Structural Properties of Artificial Neural Networks Using Algebraic Topology (2021)
David Pérez Fernández, Asier GutiérrezFandiño, Jordi ArmengolEstapé, Marta VillegasAbstract
Artificial Neural Networks (ANNs) are widely used for approximating complex functions. The process that is usually followed to define the most appropriate architecture for an ANN given a specific function is mostly empirical. Once this architecture has been defined, weights are usually optimized according to the error function. On the other hand, we observe that ANNs can be represented as graphs and their topological 'fingerprints' can be obtained using Persistent Homology (PH). In this paper, we describe a proposal focused on designing more principled architecture search procedures. To do this, different architectures for solving problems related to a heterogeneous set of datasets have been analyzed. The results of the evaluation corroborate that PH effectively characterizes the ANN invariants: when ANN density (layers and neurons) or sample feeding order is the only difference, PH topological invariants appear; in the opposite direction in different subproblems (i.e. different labels), PH varies. This approach based on topological analysis helps towards the goal of designing more principled architecture search procedures and having a better understanding of ANNs. 
Persistent Homology Based Graph Convolution Network for FineGrained 3D Shape Segmentation (2021)
ChiChong Wong, ChiMan VongAbstract
Finegrained 3D segmentation is an important task in 3D object understanding, especially in applications such as intelligent manufacturing or parts analysis for 3D objects. However, many challenges involved in such problem are yet to be solved, such as i) interpreting the complex structures located in different regions for 3D objects; ii) capturing finegrained structures with sufficient topology correctness. Current deep learning and graph machine learning methods fail to tackle such challenges and thus provide inferior performance in finegrained 3D analysis. In this work, methods in topological data analysis are incorporated with geometric deep learning model for the task of finegrained segmentation for 3D objects. We propose a novel neural network model called Persistent Homology based Graph Convolution Network (PHGCN), which i) integrates persistent homology into graph convolution network to capture multiscale structural information that can accurately represent complex structures for 3D objects; ii) applies a novel Persistence Diagram Loss (ℒPD) that provides sufficient topology correctness for segmentation over the finegrained structures. Extensive experiments on finegrained 3D segmentation validate the effectiveness of the proposed PHGCN model and show significant improvements over current stateoftheart methods. 
Go With the Flow? A LargeScale Analysis of Health Care Delivery Networks in the United States Using Hodge Theory (2021)
Thomas Gebhart, Xiaojun Fu, Russell J. FunkAbstract
Health care delivery is a collaborative process, requiring close coordination among networks of providers with specialized expertise. Yet in the United States, care is often spread across multiple disconnected providers (e.g., primary care physicians, specialists), leading to fragmented care delivery networks, and contributing to higher costs and lower quality. While this problem is well known, there are relatively few quantitative tools available for characterizing the dynamics of care delivery networks at scale, thereby inhibiting deeper understanding of care fragmentation and efforts to address it. In this, study, we conduct a largescale analysis of care delivery networks across the United States using the discrete Hodge decomposition, an emerging method of topological data analysis. Using this technique, we decompose networks of patient flows among physicians into three orthogonal subspaces: gradient (acyclic flow), harmonic (global cyclic flow), and curl (local cyclic flow). We document substantial variation in the relative importance of each subspace, suggesting that there may be systematic differences in the organization of care delivery networks across health care markets. Moreover, we find that the relative importance of each subspace is predictive of local care cost and quality, with outcomes tending to be better with greater curl flow and worse with greater harmonic flow. 
The Emergence of HigherOrder Structure in Scientific and Technological Knowledge Networks (2020)
Thomas Gebhart, Russell J. FunkAbstract
The growth of science and technology is primarily a recombinative process, wherein new discoveries and inventions are generally built from prior knowledge. While the recent past has seen rapid growth in scientific and technological knowledge, relatively little is known about the manner in which science and technology develop and coalesce knowledge into larger structures that enable or constrain future breakthroughs. Network science has recently emerged as a framework for measuring the structure and dynamics of knowledge. While helpful, these existing approaches struggle to capture the global structural properties of the underlying networks, leading to conflicting observations about the nature of scientific and technological progress. We bridge this methodological gap using tools from algebraic topology to characterize the higherorder structure of knowledge networks in science and technology across scale. We observe rapid and varied growth in the highdimensional structure in many fields of science and technology, and find this highdimensional growth coincides with decline in lowerdimensional structure. This higherorder growth in knowledge networks has historically far outpaced the growth in scientific and technological collaboration networks. We also characterize the relationship between higherorder structure and the nature of the science and technology produced within these structural environments and find a positive relationship between the abstractness of language used within fields and increasing highdimensional structure. We also find a robust relationship between highdimensional structure and number of metrics for publication success, implying this highdimensional structure may be linked to discovery and invention. 
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. 
Topological Data Analysis for Arrhythmia Detection Through Modular Neural Networks (2020)
Meryll Dindin, Yuhei Umeda, Frederic ChazalAbstract
This paper presents an innovative and generic deep learning approach to monitor heart conditions from ECG signals. We focus our attention on both the detection and classification of abnormal heartbeats, known as arrhythmia. We strongly insist on generalization throughout the construction of a shallow deeplearning model that turns out to be effective for new unseen patient. The novelty of our approach relies on the use of topological data analysis to deal with individual differences. We show that our structure reaches the performances of the stateoftheart methods for both arrhythmia detection and classification. 
A Topological Framework for Deep Learning (2020)
Mustafa Hajij, Kyle IstvanAbstract
We utilize classical facts from topology to show that the classification problem in machine learning is always solvable under very mild conditions. Furthermore, we show that a softmax classification network acts on an input topological space by a finite sequence of topological moves to achieve the classification task. Moreover, given a training dataset, we show how topological formalism can be used to suggest the appropriate architectural choices for neural networks designed to be trained as classifiers on the data. Finally, we show how the architecture of a neural network cannot be chosen independently from the shape of the underlying data. To demonstrate these results, we provide example datasets and show how they are acted upon by neural nets from this topological perspective. 
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. 
Spatial Applications of Topological Data Analysis: Cities, Snowflakes, Random Structures, and Spiders Spinning Under the Influence (2020)
Michelle Feng, Mason A. PorterAbstract
Spatial networks are ubiquitous in social, geographic, physical, and biological applications. To understand their largescale structure, it is important to develop methods that allow one to directly probe the effects of space on structure and dynamics. Historically, algebraic topology has provided one framework for rigorously and quantitatively describing the global structure of a space, and recent advances in topological data analysis (TDA) have given scholars a new lens for analyzing network data. In this paper, we study a variety of spatial networks  including both synthetic and natural ones  using novel topological methods that we recently developed specifically for analyzing spatial networks. We demonstrate that our methods are able to capture meaningful quantities, with specifics that depend on context, in spatial networks and thereby provide useful insights into the structure of those networks, including a novel approach for characterizing them based on their topological structures. We illustrate these ideas with examples of synthetic networks and dynamics on them, street networks in cities, snowflakes, and webs spun by spiders under the influence of various psychotropic substances. 
PersGNN: Applying Topological Data Analysis and Geometric Deep Learning to StructureBased Protein Function Prediction (2020)
Nicolas Swenson, Aditi S. Krishnapriyan, Aydin Buluc, Dmitriy Morozov, Katherine YelickAbstract
Understanding protein structurefunction relationships is a key challenge in computational biology, with applications across the biotechnology and pharmaceutical industries. While it is known that protein structure directly impacts protein function, many functional prediction tasks use only protein sequence. In this work, we isolate protein structure to make functional annotations for proteins in the Protein Data Bank in order to study the expressiveness of different structurebased prediction schemes. We present PersGNN  an endtoend trainable deep learning model that combines graph representation learning with topological data analysis to capture a complex set of both local and global structural features. While variations of these techniques have been successfully applied to proteins before, we demonstrate that our hybridized approach, PersGNN, outperforms either method on its own as well as a baseline neural network that learns from the same information. PersGNN achieves a 9.3% boost in area under the precision recall curve (AUPR) compared to the best individual model, as well as high F1 scores across different gene ontology categories, indicating the transferability of this approach. 
PINet: A Deep Learning Approach to Extract Topological Persistence Images (2020)
Anirudh Som, Hongjun Choi, Karthikeyan Natesan Ramamurthy, Matthew Buman, Pavan TuragaAbstract
Topological features such as persistence diagrams and their functional approximations like persistence images (PIs) have been showing substantial promise for machine learning and computer vision applications. This is greatly attributed to the robustness topological representations provide against different types of physical nuisance variables seen in realworld data, such as viewpoint, illumination, and more. However, key bottlenecks to their large scale adoption are computational expenditure and difﬁculty incorporating them in a differentiable architecture. We take an important step in this paper to mitigate these bottlenecks by proposing a novel onestep approach to generate PIs directly from the input data. We design two separate convolutional neural network architectures, one designed to take in multivariate time series signals as input and another that accepts multichannel images as input. We call these networks Signal PINet and Image PINet respectively. To the best of our knowledge, we are the ﬁrst to propose the use of deep learning for computing topological features directly from data. We explore the use of the proposed PINet architectures on two applications: human activity recognition using triaxial accelerometer sensor data and image classiﬁcation. We demonstrate the ease of fusion of PIs in supervised deep learning architectures and speed up of several orders of magnitude for extracting PIs from data. Our code is available at https://github.com/anirudhsom/PINet. 
The Growing Topology of the C. Elegans Connectome (2020)
Alec Helm, Ann S. Blevins, Danielle S. BassettAbstract
Probing the developing neural circuitry in Caenorhabditis elegans has enhanced our understanding of nervous systems. The C. elegans connectome, like those of other species, is characterized by a rich club of densely connected neurons embedded within a smallworld architecture. This organization of neuronal connections, captured by quantitative network statistics, provides insight into the system's capacity to perform integrative computations. Yet these network measures are limited in their ability to detect weakly connected motifs, such as topological cavities, that may support the systems capacity to perform segregated computations. We address this limitation by using persistent homology to track the evolution of topological cavities in the growing C. elegans connectome throughout neural development, and assess the degree to which the growing connectomes topology is resistant to biological noise. We show that the developing connectome topology is both relatively robust to changes in neuron birth times and not captured by similar growth models. Additionally, we quantify the consequence of a neurons specific birth time and ask if this metric tracks other biological properties of neurons. Our results suggest that the connectomes growing topology is a robust feature of the developing connectome that is distinct from other network properties, and that the growing topology is particularly sensitive to the exact birth times of a small set of predominantly motor neurons. By utilizing novel measurements that track biological features, we anticipate that our study will be helpful in the construction of more accurate models of neuronal development in C. elegans 
Community Structures in Simplicial Complexes: An Application to Wildlife Corridor Designing in Central India  Eastern Ghats Landscape Complex, India (2020)
Saurabh Shanu, Shashankaditya Upadhyay, Arijit Roy, Raghunandan Chundawat, Sudeepto BhattacharyaAbstract
The concept of simplicial complex from Algebraic Topology is applied to understand and model the flow of genetic information, processes and organisms between the areas of unimpaired habitats to design a network of wildlife corridors for Tigers (Panthera Tigris Tigris) in Central India Eastern Ghats landscape complex. The work extends and improves on a previous work that has made use of the concept of minimum spanning tree obtained from the weighted graph in the focal landscape, which suggested a viable corridor network for the tiger population of the Protected Areas (PAs) in the landscape complex. Centralities of the network identify the habitat patches and the critical parameters that are central to the process of tiger movement across the network. We extend the concept of vertex centrality to that of the simplicial centrality yielding intervertices adjacency and connection. As a result, the ecological information propagates expeditiously and even on a local scale in these networks representing a wellintegrated and selfexplanatory model as a community structure. A simplicial complex network based on the network centralities calculated in the landscape matrix presents a tiger corridor network in the landscape complex that is proposed to correspond better to reality than the previously proposed model. Because of the aforementioned functional and structural properties of the network, the work proposes an ecological network of corridors for the most tenable usage by the tiger populations both in the PAs and outside the PAs in the focal landscape. 
The Topological Basis of Function in Flow and Mechanical Networks (2019)
Jason Rocks, Andrea Liu, Eleni Katifori 
Persistent Homology to Quantify the Quality of SurfaceSupported Covalent Networks (2019)
Abraham Gutierrez, Mickaël Buchet, Sylvain ClairAbstract
Covalent networks formed by onsurface synthesis usually suffer from the presence of a large number of defects. We report on a methodology to characterize such twodimensional networks from their experimental images obtained by scanning probe microscopy. The computation is based on a persistent homology approach and provides a quantitative score indicative of the network homogeneity. We compare our scoring method with results previously obtained using minimal spanning tree analyses and we apply it to some molecular systems appearing in the existing literature. 
Exact Topological Inference of the RestingState Brain Networks in Twins (2019)
Moo K. Chung, Hyekyoung Lee, Hernando Ombao, Victor Solo 
A Survey of Topological Data Analysis Methods for Big Data in Healthcare Intelligence (2019)
Milan Joshi, Dhananjay Joshi 
Topological Data Analysis for Aviation Applications (2019)
Max Z. Li, Megan S. Ryerson, Hamsa BalakrishnanAbstract
Aviation data sets are increasingly highdimensional and sparse. Consequently, the underlying features and interactions are not easily uncovered by traditional data analysis methods. Recent advancements in applied mathematics introduce topological methods, offering a new approach to obtain these features. This paper applies the fundamental notions underlying topological data analysis and persistent homology (TDA/PH) to aviation data analytics. We review past aviation research that leverage topological methods, and present a new computational case study exploring the topology of airport surface connectivity. In each case, we connect abstract topological features with realworld processes in aviation, and highlight potential operational and managerial insights. 
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. 
Topological Machine Learning With Persistence Indicator Functions (2019)
Bastian Rieck, Filip Sadlo, Heike LeitteAbstract
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distancebased data analysis methods, such as multidimensional scaling, while providing a firm theoretical ground. Many modern machine learning algorithms, however, are based on kernels. This paper presents persistence indicator functions (PIFs), which summarize persistence diagrams, i.e., feature descriptors in topological data analysis. PIFs can be calculated and compared in linear time and have many beneficial properties, such as the availability of a kernelbased similarity measure. We demonstrate their usage in common data analysis scenarios, such as confidence set estimation and classification of complex structured data. 
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. 
The Importance of the Whole: Topological Data Analysis for the Network Neuroscientist (2019)
Ann E. Sizemore, Jennifer E. PhillipsCremins, Robert Ghrist, Danielle S. BassettAbstract
Data analysis techniques from network science have fundamentally improved our understanding of neural systems and the complex behaviors that they support. Yet the restriction of network techniques to the study of pairwise interactions prevents us from taking into account intrinsic topological features such as cavities that may be crucial for system function. To detect and quantify these topological features, we must turn to algebrotopological methods that encode data as a simplicial complex built from sets of interacting nodes called simplices. We then use the relations between simplices to expose cavities within the complex, thereby summarizing its topological features. Here we provide an introduction to persistent homology, a fundamental method from applied topology that builds a global descriptor of system structure by chronicling the evolution of cavities as we move through a combinatorial object such as a weighted network. We detail the mathematics and perform demonstrative calculations on the mouse structural connectome, synapses in C. elegans, and genomic interaction data. Finally, we suggest avenues for future work and highlight new advances in mathematics ready for use in neural systems., For the network neuroscientist, this exposition aims to communicate both the mathematics and the advantages of using tools from applied topology for the study of neural systems. Using data from the mouse connectome, electrical and chemical synapses in C. elegans, and chromatin interaction data, we offer example computations and applications to further demonstrate the power of topological data analysis in neuroscience. Finally, we expose the reader to novel developments in applied topology and relate these developments to current questions and methodological difficulties in network neuroscience. 
Persistent Homology Analysis of Osmolyte Molecular Aggregation and Their HydrogenBonding Networks (2019)
Kelin Xia, D. Vijay Anand, Saxena Shikhar, Yuguang MuAbstract
Dramatically different properties have been observed for two types of osmolytes, i.e., trimethylamine Noxide (TMAO) and urea, in a protein folding process. Great progress has been made in revealing the potential underlying mechanism of these two osmolyte systems. However, many problems still remain unsolved. In this paper, we propose to use the persistent homology to systematically study the osmolytes’ molecular aggregation and their hydrogenbonding network from a global topological perspective. It has been found that, for the first time, TMAO and urea show two extremely different topological behaviors, i.e., an extensive network and local clusters, respectively. In general, TMAO forms highly consistent large loop or circle structures in high concentrations. In contrast, urea is more tightly aggregated locally. Moreover, the resulting hydrogenbonding networks also demonstrate distinguishable features. With a concentration increase, TMAO hydrogenbonding networks vary greatly in their total number of loop structures and largesized loop structures consistently increase. In contrast, urea hydrogenbonding networks remain relatively stable with slight reduction of the total loop number. Moreover, the persistent entropy (PE) is, for the first time, used in characterization of the topological information of the aggregation and hydrogenbonding networks. The average PE systematically increases with the concentration for both TMAO and urea, and decreases in their hydrogenbonding networks. But their PE variances have totally different behaviors. Finally, topological features of the hydrogenbonding networks are found to be highly consistent with those from the ion aggregation systems, indicating that our topological invariants can characterize intrinsic features of the “structure making” and “structure breaking” systems. 
Knowledge Gaps in the Early Growth of Semantic Feature Networks (2018)
Ann E. Sizemore, Elisabeth A. Karuza, Chad Giusti, Danielle S. BassettAbstract
Understanding language learning and more general knowledge acquisition requires the characterization of inherently qualitative structures. Recent work has applied network science to this task by creating semantic feature networks, in which words correspond to nodes and connections correspond to shared features, and then by characterizing the structure of strongly interrelated groups of words. However, the importance of sparse portions of the semantic network—knowledge gaps—remains unexplored. Using applied topology, we query the prevalence of knowledge gaps, which we propose manifest as cavities in the growing semantic feature network of toddlers. We detect topological cavities of multiple dimensions and find that, despite word order variation, the global organization remains similar. We also show that nodal network measures correlate with filling cavities better than basic lexical properties. Finally, we discuss the importance of semantic feature network topology in language learning and speculate that the progression through knowledge gaps may be a robust feature of knowledge acquisition. 
Topological Distance Between Nonplanar Transportation Networks (2018)
Ahmed Abdelkader, Geoff Boeing, Brittany Terese Fasy, David L. Millman 
Networked Data Analytics: Network Comparison and Applied Graph Signal Processing (2018)
Weiyu Huang 
Spatial Embedding Imposes Constraints on Neuronal Network Architectures (2018)
Jennifer Stiso, Danielle S. BassettAbstract
Recent progress towards understanding circuit function has capitalized on tools from network science to parsimoniously describe the spatiotemporal architecture of neural systems. Such tools often address systems topology divorced from its physical instantiation. Nevertheless, for embedded systems such as the brain, physical laws directly constrain the processes of network growth, development, and function. We review here the rules imposed by the space and volume of the brain on the development of neuronal networks, and show that these rules give rise to a specific set of complex topologies. These rules also affect the repertoire of neural dynamics that can emerge from the system, and thereby inform our understanding of network dysfunction in disease. We close by discussing new tools and models to delineate the effects of spatial embedding. 
Geometry and Topology of the Space of Sonar Target Echos (2018)
Michael Robinson, Sean Fennell, Brian DiZio, Jennifer DumiakAbstract
Successful synthetic aperture sonar target classification depends on the “shape” of the scatterers within a target signature. This article presents a workflow that computes a targettotarget distance from persistence diagrams, since the “shape” of a signature informs its persistence diagram in a structurepreserving way. The targettotarget distances derived from persistence diagrams compare favorably against those derived from spectral features and have the advantage of being substantially more compact. While spectral features produce clusters associated to each target type that are reasonably dense and well formed, the clusters are not wellseparated from one another. In rather dramatic contrast, a distance derived from persistence diagrams results in highly separated clusters at the expense of some misclassification of outliers. 
Visual Detection of Structural Changes in TimeVarying Graphs Using Persistent Homology (2018)
Mustafa Hajij, Bei Wang, Carlos Scheidegger, Paul RosenAbstract
Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, highdimensional data. In this paper, we propose a novel method using persistent homology to quantify structural changes in timevarying graphs. Specifically, we transform each instance of the timevarying graph into a metric space, extract topological features using persistent homology, and compare those features over time. We provide a visualization that assists in timevarying graph exploration and helps to identify patterns of behavior within the data. To validate our approach, we conduct several case studies on realworld datasets and show how our method can find cyclic patterns, deviations from those patterns, and onetime events in timevarying graphs. We also examine whether a persistencebased similarity measure satisfies a set of wellestablished, desirable properties for graph metrics. 
Connectivity in fMRI: Blind Spots and Breakthroughs (2018)
Victor Solo, JeanBaptiste Poline, Martin A. Lindquist, Sean L. Simpson, F. DuBois Bowman, Moo K. Chung, Ben CassidyAbstract
In recent years, driven by scientific and clinical concerns, there has been an increased interest in the analysis of functional brain networks. The goal of these analyses is to better understand how brain regions interact, how this depends upon experimental conditions and behavioral measures and how anomalies (disease) can be recognized. In this work we provide, firstly, a brief review of some of the main existing methods of functional brain network analysis. But rather than compare them, as a traditional review would do, instead, we draw attention to their significant limitations and blind spots. Then, secondly, relevant experts, sketch a number of emerging methods, which can break through these limitations. In particular we discuss five such methods. The first two, stochastic block models and exponential random graph models, provide an inferential basis for network analysis lacking in the exploratory graph analysis methods. The other three address: network comparison via persistent homology, timevarying connectivity that distinguishes sample fluctuations from neural fluctuations and, network system identification that draws inferential strength from temporal autocorrelation. 
Representability of Algebraic Topology for Biomolecules in Machine Learning Based Scoring and Virtual Screening (2018)
Zixuan Cang, Lin Mu, GuoWei WeiAbstract
This work introduces a number of algebraic topology approaches, including multicomponent persistent homology, multilevel persistent homology, and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes. In contrast to the conventional persistent homology, multicomponent persistent homology retains critical chemical and biological information during the topological simplification of biomolecular geometric complexity. Multilevel persistent homology enables a tailored topological description of inter and/or intramolecular interactions of interest. Electrostatic persistence incorporates partial charge information into topological invariants. These topological methods are paired with Wasserstein distance to characterize similarities between molecules and are further integrated with a variety of machine learning algorithms, including knearest neighbors, ensemble of trees, and deep convolutional neural networks, to manifest their descriptive and predictive powers for proteinligand binding analysis and virtual screening of small molecules. Extensive numerical experiments involving 4,414 proteinligand complexes from the PDBBind database and 128,374 ligandtarget and decoytarget pairs in the DUD database are performed to test respectively the scoring power and the discriminatory power of the proposed topological learning strategies. It is demonstrated that the present topological learning outperforms other existing methods in proteinligand binding affinity prediction and liganddecoy discrimination. 
Cliques and Cavities in the Human Connectome (2018)
Ann E. Sizemore, Chad Giusti, Ari Kahn, Jean M. Vettel, Richard F. Betzel, Danielle S. BassettAbstract
Encoding brain regions and their connections as a network of nodes and edges captures many of the possible paths along which information can be transmitted as humans process and perform complex behaviors. Because cognitive processes involve large, distributed networks of brain areas, principled examinations of multinode routes within larger connection patterns can offer fundamental insights into the complexities of brain function. Here, we investigate both densely connected groups of nodes that could perform local computations as well as larger patterns of interactions that would allow for parallel processing. Finding such structures necessitates that we move from considering exclusively pairwise interactions to capturing higher order relations, concepts naturally expressed in the language of algebraic topology. These tools can be used to study mesoscale network structures that arise from the arrangement of densely connected substructures called cliques in otherwise sparsely connected brain networks. We detect cliques (alltoall connected sets of brain regions) in the average structural connectomes of 8 healthy adults scanned in triplicate and discover the presence of more large cliques than expected in null networks constructed via wiring minimization, providing architecture through which brain network can perform rapid, local processing. We then locate topological cavities of different dimensions, around which information may flow in either diverging or converging patterns. These cavities exist consistently across subjects, differ from those observed in null model networks, and – importantly – link regions of early and late evolutionary origin in long loops, underscoring their unique role in controlling brain function. These results offer a first demonstration that techniques from algebraic topology offer a novel perspective on structural connectomics, highlighting looplike paths as crucial features in the human brain’s structural architecture. 
Persistent Homology Analysis of Ion Aggregations and HydrogenBonding Networks (2018)
Kelin XiaAbstract
Despite the great advancement of experimental tools and theoretical models, a quantitative characterization of the microscopic structures of ion aggregates and their associated water hydrogenbonding networks still remains a challenging problem. In this paper, a newlyinvented mathematical method called persistent homology is introduced, for the first time, to quantitatively analyze the intrinsic topological properties of ion aggregation systems and hydrogenbonding networks. The two most distinguishable properties of persistent homology analysis of assembly systems are as follows. First, it does not require a predefined bond length to construct the ion or hydrogenbonding network. Persistent homology results are determined by the morphological structure of the data only. Second, it can directly measure the size of circles or holes in ion aggregates and hydrogenbonding networks. To validate our model, we consider two wellstudied systems, i.e., NaCl and KSCN solutions, generated from molecular dynamics simulations. They are believed to represent two morphological types of aggregation, i.e., local clusters and extended ion networks. It has been found that the two aggregation types have distinguishable topological features and can be characterized by our topological model very well. Further, we construct two types of networks, i.e., Onetworks and H2Onetworks, for analyzing the topological properties of hydrogenbonding networks. It is found that for both models, KSCN systems demonstrate much more dramatic variations in their local circle structures with a concentration increase. A consistent increase of largesized local circle structures is observed and the sizes of these circles become more and more diverse. In contrast, NaCl systems show no obvious increase of largesized circles. Instead a consistent decline of the average size of the circle structures is observed and the sizes of these circles become more and more uniform with a concentration increase. As far as we know, these unique intrinsic topological features in ion aggregation systems have never been pointed out before. More importantly, our models can be directly used to quantitatively analyze the intrinsic topological invariants, including circles, loops, holes, and cavities, of any networklike structures, such as nanomaterials, colloidal systems, biomolecular assemblies, among others. These topological invariants cannot be described by traditional graph and network models. 
Cliques of Neurons Bound Into Cavities Provide a Missing Link Between Structure and Function (2017)
Michael W. Reimann, Max Nolte, Martina Scolamiero, Katharine Turner, Rodrigo Perin, Giuseppe Chindemi, Paweł Dłotko, Ran Levi, Kathryn Hess, Henry MarkramAbstract
The lack of a formal link between neural network structure and its emergent function has hampered our understanding of how the brain processes information. We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reflect the direction of information flow, and analyzing these directed graphs using algebraic topology. Applying this approach to a local network of neurons in the neocortex revealed a remarkably intricate and previously unseen topology of synaptic connectivity. The synaptic network contains an abundance of cliques of neurons bound into cavities that guide the emergence of correlated activity. In response to stimuli, correlated activity binds synaptically connected neurons into functional cliques and cavities that evolve in a stereotypical sequence towards peak complexity. We propose that the brain processes stimuli by forming increasingly complex functional cliques and cavities. 
Deep Learning With Topological Signatures (2017)
Christoph Hofer, Roland Kwitt, Marc Niethammer, Andreas Uhl 
The Topology of the Cosmic Web in Terms of Persistent Betti Numbers (2017)
Pratyush Pranav, Herbert Edelsbrunner, Rien van de Weygaert, Gert Vegter, Michael Kerber, Bernard J. T. Jones, Mathijs WintraeckenAbstract
Abstract. We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence of 
Congestion Barcodes: Exploring the Topology of Urban Congestion Using Persistent Homology (2017)
Yu Wu, Gabriel Shindnes, Vaibhav Karve, Derrek Yager, Daniel B. Work, Arnab Chakraborty, Richard B. SowersAbstract
This work presents a new method to quantify connectivity in transportation networks. Inspired by the field of topological data analysis, we propose a novel approach to explore the robustness of road network connectivity in the presence of congestion on the roadway. The robustness of the pattern is summarized in a congestion barcode, which can be constructed directly from traffic datasets commonly used for navigation. As an initial demonstration, we illustrate the main technique on a publicly available traffic dataset in a neighborhood in New York City. 
Identification of Topological Network Modules in Perturbed Protein Interaction Networks (2017)
Mihaela E. Sardiu, Joshua M. Gilmore, Brad Groppe, Laurence Florens, Michael P. WashburnAbstract
Biological networks consist of functional modules, however detecting and characterizing such modules in networks remains challenging. Perturbing networks is one strategy for identifying modules. Here we used an advanced mathematical approach named topological data analysis (TDA) to interrogate two perturbed networks. In one, we disrupted the S. cerevisiae INO80 protein interaction network by isolating complexes after protein complex components were deleted from the genome. In the second, we reanalyzed previously published data demonstrating the disruption of the human Sin3 network with a histone deacetylase inhibitor. Here we show that disrupted networks contained topological network modules (TNMs) with shared properties that mapped onto distinct locations in networks. We define TMNs as proteins that occupy close network positions depending on their coordinates in a topological space. TNMs provide new insight into networks by capturing proteins from different categories including proteins within a complex, proteins with shared biological functions, and proteins disrupted across networks. 
Evasion Paths in Mobile Sensor Networks (2015)
Henry Adams, Gunnar CarlssonAbstract
Suppose that ballshaped sensors wander in a bounded domain. A sensor does not know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. In ‘Coordinatefree coverage in sensor networks with controlled boundaries via homology', Vin de Silva and Robert Ghrist give a necessary condition, depending only on the timevarying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with timevarying connectivity data as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends not only on the fibrewise homotopy type of the region covered by sensors but also on its embedding in spacetime. For planar sensors that also measure weak rotation and distance information, we provide necessary and sufficient conditions for the existence of an evasion path. 
Conserved Abundance and Topological Features in ChromatinRemodeling Protein Interaction Networks (2015)
Mihaela E Sardiu, Joshua M Gilmore, Brad D Groppe, Damir Herman, Sreenivasa R Ramisetty, Yong Cai, Jingji Jin, Ronald C Conaway, Joan W Conaway, Laurence Florens, Michael P WashburnAbstract
Abstract The study of conserved protein interaction networks seeks to better understand the evolution and regulation of protein interactions. Here, we present a quantitative proteomic analysis of 18 orthologous baits from three distinct chromatinremodeling complexes in Saccharomyces cerevisiae and Homo sapiens. We demonstrate that abundance levels of orthologous proteins correlate strongly between the two organisms and both networks have highly similar topologies. We therefore used the protein abundances in one species to crosspredict missing protein abundance levels in the other species. Lastly, we identified a novel conserved lowabundance subnetwork further demonstrating the value of quantitative analysis of networks. 
Topological Data Analysis of Contagion Maps for Examining Spreading Processes on Networks (2015)
Dane Taylor, Florian Klimm, Heather A. Harrington, Miroslav Kramár, Konstantin Mischaikow, Mason A. Porter, Peter J. MuchaAbstract
Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earth’s surface; however, in modern contagions longrange edges—for example, due to airline transportation or communication media—allow clusters of a contagion to appear in distant locations. Here we study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct ‘contagion maps’ that use multiple contagions on a network to map the nodes as a point cloud. By analysing the topology, geometry and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modelling, forecast and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring lowdimensional structure in networks. 
Coverage Criterion in Sensor Networks Stable Under Perturbation (2014)
Yasuaki Hiraoka, Genki KusanoAbstract
To the coverage problem of sensor networks, V. de Silva and R. Ghrist (2007) developed several approaches based on (persistent) homology theory. Their criteria for the coverage are formulated on the Rips complexes constructed by the sensors, in which their locations are supposed to be fixed. However, the sensors are in general affected by perturbations (e.g., natural phenomena), and hence the stability of the coverage criteria should be also discussed. In this paper, we present a coverage theorem stable under perturbation. Furthermore, we also introduce a method of eliminating redundant cover after perturbation. The coverage theorem is derived by extending the Rips interleaving theorem studied by F. Chazal, V. de Silva, and S. Oudot (2013) into an appropriate relative version. 
Homological Scaffolds of Brain Functional Networks (2014)
G. Petri, P. Expert, F. Turkheimer, R. CarhartHarris, D. Nutt, P. J. Hellyer, F. VaccarinoAbstract
Networks, as efficient representations of complex systems, have appealed to scientists for a long time and now permeate many areas of science, including neuroimaging (Bullmore and Sporns 2009 Nat. Rev. Neurosci.10, 186–198. (doi:10.1038/nrn2618)). Traditionally, the structure of complex networks has been studied through their statistical properties and metrics concerned with node and link properties, e.g. degreedistribution, node centrality and modularity. Here, we study the characteristics of functional brain networks at the mesoscopic level from a novel perspective that highlights the role of inhomogeneities in the fabric of functional connections. This can be done by focusing on the features of a set of topological objects—homological cycles—associated with the weighted functional network. We leverage the detected topological information to define the homological scaffolds, a new set of objects designed to represent compactly the homological features of the correlation network and simultaneously make their homological properties amenable to networks theoretical methods. As a proof of principle, we apply these tools to compare restingstate functional brain activity in 15 healthy volunteers after intravenous infusion of placebo and psilocybin—the main psychoactive component of magic mushrooms. The results show that the homological structure of the brain's functional patterns undergoes a dramatic change postpsilocybin, characterized by the appearance of many transient structures of low stability and of a small number of persistent ones that are not observed in the case of placebo. 
Topological Data Analysis of Escherichia Coli O157:H7 and NonO157 Survival in Soils (2014)
Abasiofiok M. Ibekwe, Jincai Ma, David E. Crowley, ChingHong Yang, Alexis M. Johnson, Tanya C. Petrossian, Pek Y. LumAbstract
Shiga toxinproducing E. coli O157:H7 and nonO157 have been implicated in many foodborne illnesses caused by the consumption of contaminated fresh produce. However, data on their persistence in soils are limited due to the complexity in datasets generated from different environmental variables and bacterial taxa. There is a continuing need to distinguish the various environmental variables and different bacterial groups to understand the relationships among these factors and the pathogen survival. Using an approach called Topological Data Analysis (TDA); we reconstructed the relationship structure of E. coli O157 and nonO157 survival in 32 soils (16 organic and 16 conventionally managed soils) from California (CA) and Arizona (AZ) with a multiresolution output. In our study, we took a community approach based on total soil microbiome to study community level survival and examining the network of the community as a whole and the relationship between its topology and biological processes. TDA produces a geometric representation of complex data sets. Network analysis showed that Shiga toxin negative strain E. coli O157:H7 4554 survived significantly longer in comparison to E. coli O157:H7 EDL933, while the survival time of E. coli O157:NM was comparable to that of E. coli O157:H7 strain 933 in all of the tested soils. Two nonO157 strains, E. coli O26:H11 and E. coli O103:H2 survived much longer than E. coli O91:H21 and the three strains of E. coli O157. We show that there are complex interactions between E. coli strain survival, microbial community structures, and soil parameters. 
Extracting Insights From the Shape of Complex Data Using Topology (2013)
P. Y. Lum, G. Singh, A. Lehman, T. Ishkanov, M. VejdemoJohansson, M. Alagappan, J. Carlsson, G. CarlssonAbstract
This paper applies topological methods to study complex high dimensional data sets by extracting shapes (patterns) and obtaining insights about them. Our method combines the best features of existing standard methodologies such as principal component and cluster analyses to provide a geometric representation of complex data sets. Through this hybrid method, we often find subgroups in data sets that traditional methodologies fail to find. Our method also permits the analysis of individual data sets as well as the analysis of relationships between related data sets. We illustrate the use of our method by applying it to three very different kinds of data, namely gene expression from breast tumors, voting data from the United States House of Representatives and player performance data from the NBA, in each case finding stratifications of the data which are more refined than those produced by standard methods. 
Applications of Persistent Homology to Time Varying Systems (2013)
Elizabeth MunchAbstract
\textlessp\textgreaterThis dissertation extends the theory of persistent homology to time varying systems. Most of the previous work has been dedicated to using this powerful tool in topological data analysis to study static point clouds. In particular, given a point cloud, we can construct its persistence diagram. Since the diagram varies continuously as the point cloud varies continuously, we study the space of time varying persistence diagrams, called vineyards when they were introduced by CohenSteiner, Edelsbrunner, and Morozov.\textless/p\textgreater\textlessp\textgreaterWe will first show that with a good choice of metric, these vineyards are stable for small perturbations of their associated point clouds. We will also define a new mean for a set of persistence diagrams based on the work of Mileyko et al. which, unlike the previously defined mean, is continuous for geodesic vineyards. \textless/p\textgreater\textlessp\textgreaterNext, we study the sensor network problem posed by Ghrist and de Silva, and their application of persistent homology to understand when a set of sensors covers a given region. Giving each of these sensors a probability of failure over time, we show that an exact computation of the probability of failure of the whole system is NPhard, but give an algorithm which can predict failure in the case of a monitored system.\textless/p\textgreater\textlessp\textgreaterFinally, we apply these methods to an automated system which can cluster agents moving in aerial images by their behaviors. We build a data structure for storing and querying the information in realtime, and define behavior vectors which quantify behaviors of interest. This clustering by behavior can be used to find groups of interest, for which we can also quantify behaviors in order to determine whether the group is working together to achieve a common goal, and we speculate that this work can be extended to improving tracking algorithms as well as behavioral predictors.\textless/p\textgreater 
Persistent Brain Network Homology From the Perspective of Dendrogram (2012)
Hyekyoung Lee, Hyejin Kang, Moo K. Chung, BungNyun Kim, Dong Soo LeeAbstract
The brain network is usually constructed by estimating the connectivity matrix and thresholding it at an arbitrary level. The problem with this standard method is that we do not have any generally accepted criteria for determining a proper threshold. Thus, we propose a novel multiscale framework that models all brain networks generated over every possible threshold. Our approach is based on persistent homology and its various representations such as the Rips filtration, barcodes, and dendrograms. This new persistent homological framework enables us to quantify various persistent topological features at different scales in a coherent manner. The barcode is used to quantify and visualize the evolutionary changes of topological features such as the Betti numbers over different scales. By incorporating additional geometric information to the barcode, we obtain a single linkage dendrogram that shows the overall evolution of the network. The difference between the two networks is then measured by the GromovHausdorff distance over the dendrograms. As an illustration, we modeled and differentiated the FDGPET based functional brain networks of 24 attentiondeficit hyperactivity disorder children, 26 autism spectrum disorder children, and 11 pediatric control subjects. 
Topological Extraction and Tracking of Defects in Crystal Structures (2011)
Sebastian Grottel, Carlos A. Dietrich, João L. D. Comba, Thomas ErtlAbstract
Interfaces between materials with different mechanical properties play an important role in technical applications. Nowadays molecular dynamics simulations are used to observe the behavior of such compound materials at the atomic level. Due to different atom crystal sizes, dislocations in the atom crystal structure occur once external forces are applied, and it has been observed that studying the change of thesedislocations can provide further understanding of macroscopic attributes like elasticity and plasticity. Standard visualization techniques such as the rendering of individual atoms work for 2D data or sectional views; however, visualizingdislocations in 3D using such methods usually fail due to occlusion and clutter. In this work we propose to extract and visualize the structure ofdislocations, which summarizes the commonly employed filtered atomistic renderings into a concise representation. The benefits of our approach are clearer images while retaining relevant data and easier visual tracking of topological changes over time. 
Coverage in Sensor Networks via Persistent Homology (2007)
Vin de Silva, Robert GhristAbstract
We introduce a topological approach to a problem of covering a region in Euclidean space by balls of fixed radius at unknown locations (this problem being motivated by sensor networks with minimal sensing capabilities). In particular, we give a homological criterion to rigorously guarantee that a collection of balls covers a bounded domain based on the homology of a certain simplicial pair. This pair of (Vietoris–Rips) complexes is derived from graphs representing a coarse form of distance estimation between nodes and a proximity sensor for the boundary of the domain. The methods we introduce come from persistent homology theory and are applicable to nonlocalized sensor networks with ad hoc wireless communications. 
CoordinateFree Coverage in Sensor Networks With Controlled Boundaries via Homology (2006)
V. de Silva, R. GhristAbstract
Tools from computational homology are introduced to verify coverage in an idealized sensor network. These methods are unique in that, while they are coordinatefree and assume no localization or orientation capabilities for the nodes, there are also no probabilistic assumptions. The key ingredient is the theory of homology from algebraic topology. The robustness of these tools is demonstrated by adapting them to a variety of settings, including static planar coverage, 3D barrier coverage, and timedependent sweeping coverage. Results are also given on hole repair, error tolerance, optimal coverage, and variable radii. An overview of implementation is given. 
Blind Swarms for Coverage in 2D (2005)
V. D. Silva, R. Ghrist, A. MuhammadAbstract
We consider coverage problems in robot sensor networks with minimal sensing capabilities. In particular, we demonstrate that a “blind” swarm of robots with no localization and only a weak form of distance estimation can rigorously determine coverage in a bounded planar domain of unknown size and shape. The methods we introduce come from algebraic topology. I. COVERAGE PROBLEMS Many of the potential applications of robot swarms require information about coverage in a given domain. For example, using a swarm of robot sensors for surveillance and security applications carries with it the charge to maximize, or, preferably, guarantee coverage. Such applications include networks of security cameras, mine field sweeping via networked robots [18], and oceanographic sampling [4]. In these contexts, each robot has some coverage domain, and one wishes to know about the union of these coverage domains. Such problems are also crucial in applications not involving robots directly, e.g., communication networks. As a preliminary analysis, we consider the static “field” coverage problem, in which robots are assumed stationary and the goal is to verify blanket coverage of a given domain. There is a large literature on this subject; see, e.g., [7], [1], [16]. In addition, there are variants on these problems involving “barrier” coverage to separate regions. Dynamic or “sweeping” coverage [3] is a common and challenging task with applications ranging from security to vacuuming. Although a sensor network composed of robots will have dynamic capabilities, we restrict attention in this brief paper to the static case in order to lay the groundwork for future inquiry. There are two primary approaches to static coverage problems in the literature. The first uses computational geometry tools applied to exact node coordinates. This typically involves ‘rulerandcompass’ style geometry [10] or Delaunay triangulations of the domain [16], [14], [20]. Such approaches are very rigid with regards to inputs: one must know exact node coordinates and one must know the geometry of the domain precisely to determine the Delaunay complex. To alleviate the former requirement, many authors have turned to probabilistic tools. For example, in [13], the author assumes a randomly and uniformly distributed collection of nodes in a domain with a fixed geometry and proves expected area coverage. Other approaches [15], [19] give percolationtype results about coverage and network integrity for randomly distributed nodes. The drawback of these methods is the need for strong assumptions about the exact shape of the domain, as well as the need for a uniform distribution of nodes. In the sensor networks community, there is a compelling interest (and corresponding burgeoning literature) in determining properties of a network in which the nodes do not possess coordinate data. One example of a coordinatefree approach is in [17], which gives a heuristic method for geographic routing without coordinate data: among the large literature arising from this paper, we note in particular the mathematical analysis of this approach in [11]. To our knowledge, noone has treated the coverage problem in a coordinatefree setting. In this note, we introduce a new set of tools for answering coverage problems in robotics and sensor networks with minimal assumptions about domain geometry and node localization. We provide a sufficiency criterion for coverage. We do not answer the problem of how the nodes should be placed in order to maximize coverage, nor the minimum number of such nodes necessary; neither do we address how to reallocate nodes to fill coverage holes.