Topological Strata of Weighted Complex Networks

Higher-order connectomics of human brain function reveals local topological signatures of task decoding, individual identification, and behavior

Traditional models of human brain activity often represent it as a network of pairwise interactions between brain regions. Going beyond this limitation, recent approaches have been proposed to infer higher-order interactions from temporal brain signals involving three or more regions. However, to this day it remains unclear whether methods based on inferred higher-order interactions outperform traditional pairwise ones for the analysis of fMRI data. To address this question, we conducted a comprehensive analysis using fMRI time series of 100 unrelated subjects from the Human Connectome Project. We show that higher-order approaches greatly enhance our ability to decode dynamically between various tasks, to improve the individual identification of unimodal and transmodal functional subsystems, and to strengthen significantly the associations between brain activity and behavior. Overall, our approach sheds new light on the higher-order organization of fMRI time series, improving the characterization of dynamic group dependencies in rest and tasks, and revealing a vast space of unexplored structures within human functional brain data, which may remain hidden when using traditional pairwise approaches.

Persistent homology reveals robustness loss in inhaled substance abuse rs-fMRI networks

Analyzing functional brain activity through functional magnetic resonance imaging (fMRI) is commonly done using tools from graph theory for the analysis of the correlation matrices. A drawback of these methods is that the networks must be restricted to values of the weights of the edges within certain thresholds and there is no consensus about the best choice of such thresholds. Topological data analysis (TDA) is a recently-developed tool in algebraic topology which allows us to analyze networks through combinatorial spaces obtained from them, with the advantage that all the possible thresholds can be considered at once. In this paper we applied TDA, in particular persistent homology, to study correlation matrices from rs-fMRI, and through statistical analysis, we detected significant differences between the topological structures of adolescents with inhaled substance abuse disorder (ISAD) and healthy controls. We interpreted the topological differences as indicative of a loss of robustness in the functional brain networks of the ISAD population.

Network science: Ising states of matter

Network science provides very powerful tools for extracting information from interacting data. Although recently the unsupervised detection of phases of matter using machine learning has raised significant interest, the full prediction power of network science has not yet been systematically explored in this context. Here we fill this gap by providing an in-depth statistical, combinatorial, geometrical, and topological characterization of 2D Ising snapshot networks (IsingNets) extracted from Monte Carlo simulations of the 2D Ising model at different temperatures, going across the phase transition. Our analysis reveals the complex organization properties of IsingNets in both the ferromagnetic and paramagnetic phases and demonstrates the significant deviations of the IsingNets with respect to randomized null models. In particular percolation properties of the IsingNets reflect the existence of the symmetry between configurations with opposite magnetization below the critical temperature and the very compact nature of the two emerging giant clusters revealed by our persistent homology analysis of the IsingNets. Moreover, the IsingNets display a very broad degree distribution and significant degree-degree correlations and weight-degree correlations demonstrating that they encode relevant information present in the configuration space of the 2D Ising model. The geometrical organization of the critical IsingNets is reflected in their spectral properties deviating from the one of the null model. This work reveals the important insights that network science can bring to the characterization of phases of matter. The set of tools described hereby can be applied as well to numerical and experimental data.

Persistent Dirac for molecular representation

Molecular representations are of fundamental importance for the modeling and analysing molecular systems. The successes in drug design and materials discovery have been greatly contributed by molecular representation models. In this paper, we present a computational framework for molecular representation that is mathematically rigorous and based on the persistent Dirac operator. The properties of the discrete weighted and unweighted Dirac matrix are systematically discussed, and the biological meanings of both homological and non-homological eigenvectors are studied. We also evaluate the impact of various weighting schemes on the weighted Dirac matrix. Additionally, a set of physical persistent attributes that characterize the persistence and variation of spectrum properties of Dirac matrices during a filtration process is proposed to be molecular fingerprints. Our persistent attributes are used to classify molecular configurations of nine different types of organic-inorganic halide perovskites. The combination of persistent attributes with gradient boosting tree model has achieved great success in molecular solvation free energy prediction. The results show that our model is effective in characterizing the molecular structures, demonstrating the power of our molecular representation and featurization approach.

Hodge Laplacian of Brain Networks

The closed loops or cycles in a brain network embeds higher order signal transmission paths, which provide fundamental insights into the functioning of the brain. In this work, we propose an efficient algorithm for systematic identification and modeling of cycles using persistent homology and the Hodge Laplacian. Various statistical inference procedures on cycles are developed. We validate the our methods on simulations and apply to brain networks obtained through the resting state functional magnetic resonance imaging. The computer codes for the Hodge Laplacian are given in https://github.com/laplcebeltrami/hodge.

Topological energy of networks

Energy is an important network indicator defined by the eigenvalues of an adjacency matrix that includes the neighbor information for each node. This article expands the definition of network energy to include higher-order information between nodes. We use resistance distances to characterize the distances between nodes and order complexes to extract higher-order information. Topological energy ( T E), defined by the resistance distance and order complex, reveals the characteristics of the network structure from multiple scales. In particular, calculations show that the topological energy can be used to distinguish graphs with the same spectrum well. In addition, topological energy is robust, and small random perturbations of edges do not significantly affect the T E values. Finally, we find that the energy curve of the real network is significantly different from that of the random graph, thus showing that T E can be used to distinguish the network structure well. This study shows that T E is an indicator that distinguishes the structure of a network and has some potential applications for real-world problems.

Topological data analysis of human brain networks through order statistics

Understanding the common topological characteristics of the human brain network across a population is central to understanding brain functions. The abstraction of human connectome as a graph has been pivotal in gaining insights on the topological properties of the brain network. The development of group-level statistical inference procedures in brain graphs while accounting for the heterogeneity and randomness still remains a difficult task. In this study, we develop a robust statistical framework based on persistent homology using the order statistics for analyzing brain networks. The use of order statistics greatly simplifies the computation of the persistent barcodes. We validate the proposed methods using comprehensive simulation studies and subsequently apply to the resting-state functional magnetic resonance images. We found a statistically significant topological difference between the male and female brain networks.

Persistent Homology for RNA Data Analysis

Molecular representations are of great importance for machine learning models in RNA data analysis. Essentially, efficient molecular descriptors or fingerprints that characterize the intrinsic structural and interactional information of RNAs can significantly boost the performance of all learning modeling. In this paper, we introduce two persistent models, including persistent homology and persistent spectral, for RNA structure and interaction representations and their applications in RNA data analysis. Different from traditional geometric and graph representations, persistent homology is built on simplicial complex, which is a generalization of graph models to higher-dimensional situations. Hypergraph is a further generalization of simplicial complexes and hypergraph-based embedded persistent homology has been proposed recently. Moreover, persistent spectral models, which combine filtration process with spectral models, including spectral graph, spectral simplicial complex, and spectral hypergraph, are proposed for molecular representation. The persistent attributes for RNAs can be obtained from these two persistent models and further combined with machine learning models for RNA structure, flexibility, dynamics, and function analysis.

Geometry of spiking patterns in early visual cortex: a topological data analytic approach

In the brain, spiking patterns live in a high-dimensional space of neurons and time. Thus, determining the intrinsic structure of this space presents a theoretical and experimental challenge. To address this challenge, we introduce a new framework for applying topological data analysis (TDA) to spike train data and use it to determine the geometry of spiking patterns in the visual cortex. Key to our approach is a parametrized family of distances based on the timing of spikes that quantifies the dissimilarity between neuronal responses. We applied TDA to visually driven single-unit and multiple single-unit spiking activity in macaque V1 and V2. TDA across timescales reveals a common geometry for spiking patterns in V1 and V2 which, among simple models, is most similar to that of a low-dimensional space endowed with Euclidean or hyperbolic geometry with modest curvature. Remarkably, the inferred geometry depends on timescale and is clearest for the timescales that are important for encoding contrast, orientation and spatial correlations.

Weighted simplicial complexes and their representation power of higher-order network data and topology

Hypergraphs and simplical complexes both capture the higher-order interactions of complex systems, ranging from higher-order collaboration networks to brain networks. One open problem in the field is what should drive the choice of the adopted mathematical framework to describe higher-order networks starting from data of higher-order interactions. Unweighted simplicial complexes typically involve a loss of information of the data, though having the benefit to capture the higher-order topology of the data. In this work we show that weighted simplicial complexes allow one to circumvent all the limitations of unweighted simplicial complexes to represent higher-order interactions. In particular, weighted simplicial complexes can represent higher-order networks without loss of information, allowing one at the same time to capture the weighted topology of the data. The higher-order topology is probed by studying the spectral properties of suitably defined weighted Hodge Laplacians displaying a normalized spectrum. The higher-order spectrum of (weighted) normalized Hodge Laplacians is studied combining cohomology theory with information theory. In the proposed framework we quantify and compare the information content of higher-order spectra of different dimension using higher-order spectral entropies and spectral relative entropies. The proposed methodology is tested on real higher-order collaboration networks and on the weighted version of the simplicial complex model "Network Geometry with Flavor."

Measuring node centrality when local and global measures overlap

Centrality metrics aim to identify the most relevant nodes in a network. In the literature, a broad set of metrics exists, measuring either local or global centrality characteristics. Nevertheless, when networks exhibit a high spectral gap, the usual global centrality measures typically do not add significant information with respect to the degree, i.e., the simplest local metric. To extract different information from this class of networks, we propose the use of the Generalized Economic Complexity index (GENEPY). Despite its original definition within the economic field, the GENEPY can be easily applied and interpreted on a wide range of networks, characterized by high spectral gap, including monopartite and bipartite network systems. Tests on synthetic and real-world networks show that the GENEPY can shed light about the node centrality, carrying information generally poorly correlated with the node number of direct connections (node degree).

Topological Features of Electroencephalography are Robust to Re-referencing and Preprocessing

Electroencephalography (EEG) is among the most widely diffused, inexpensive, and adopted neuroimaging techniques. Nonetheless, EEG requires measurements against a reference site(s), which is typically chosen by the experimenter, and specific pre-processing steps precede analyses. It is therefore valuable to obtain quantities that are minimally affected by reference and pre-processing choices. Here, we show that the topological structure of embedding spaces, constructed either from multi-channel EEG timeseries or from their temporal structure, are subject-specific and robust to re-referencing and pre-processing pipelines. By contrast, the shape of correlation spaces, that is, discrete spaces where each point represents an electrode and the distance between them that is in turn related to the correlation between the respective timeseries, was neither significantly subject-specific nor robust to changes of reference. Our results suggest that the shape of spaces describing the observed configurations of EEG signals holds information about the individual specificity of the underlying individual's brain dynamics, and that temporal correlations constrain to a large degree the set of possible dynamics. In turn, these encode the differences between subjects' space of resting state EEG signals. Finally, our results and proposed methodology provide tools to explore the individual topographical landscapes and how they are explored dynamically. We propose therefore to augment conventional topographic analyses with an additional-topological-level of analysis, and to consider them jointly. More generally, these results provide a roadmap for the incorporation of topological analyses within EEG pipelines.

Topological analysis of the latent geometry of a complex network

Most real-world networks are embedded in latent geometries. If a node in a network is found in the vicinity of another node in the latent geometry, the two nodes have a disproportionately high probability of being connected by a link. The latent geometry of a complex network is a central topic of research in network science, which has an expansive range of practical applications, such as efficient navigation, missing link prediction, and brain mapping. Despite the important role of topology in the structures and functions of complex systems, little to no study has been conducted to develop a method to estimate the general unknown latent geometry of complex networks. Topological data analysis, which has attracted extensive attention in the research community owing to its convincing performance, can be directly implemented into complex networks; however, even a small fraction (0.1%) of long-range links can completely erase the topological signature of the latent geometry. Inspired by the fact that long-range links in a network have disproportionately high loads, we develop a set of methods that can analyze the latent geometry of a complex network: the modified persistent homology diagram and the map of the latent geometry. These methods successfully reveal the topological properties of the synthetic and empirical networks used to validate the proposed methods.

Minimal Cycle Representatives in Persistent Homology Using Linear Programming: An Empirical Study With User's Guide

Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, the non-uniqueness of these representatives creates ambiguity and can lead to many different interpretations of the same set of classes. One approach to solving this problem is to optimize the choice of representative against some measure that is meaningful in the context of the data. In this work, we provide a study of the effectiveness and computational cost of severalℓ 1 minimization optimization procedures for constructing homological cycle bases for persistent homology with rational coefficients in dimension one, including uniform-weighted and length-weighted edge-loss algorithms as well as uniform-weighted and area-weighted triangle-loss algorithms. We conduct these optimizations via standard linear programming methods, applying general-purpose solvers to optimize over column bases of simplicial boundary matrices. Our key findings are: 1) optimization is effective in reducing the size of cycle representatives, though the extent of the reduction varies according to the dimension and distribution of the underlying data, 2) the computational cost of optimizing a basis of cycle representatives exceeds the cost of computing such a basis, in most data sets we consider, 3) the choice of linear solvers matters a lot to the computation time of optimizing cycles, 4) the computation time of solving an integer program is not significantly longer than the computation time of solving a linear program for most of the cycle representatives, using the Gurobi linear solver, 5) strikingly, whether requiring integer solutions or not, we almost always obtain a solution with the same cost and almost all solutions found have entries in{ - 1,0,1 } and therefore, are also solutions to a restrictedℓ 0 optimization problem, and 6) we obtain qualitatively different results for generators in Erdős-Rényi random clique complexes than in real-world and synthetic point cloud data.

Risk prediction of clinical adverse outcomes with machine learning in a cohort of critically ill patients with atrial fibrillation

Critically ill patients affected by atrial fibrillation are at high risk of adverse events: however, the actual risk stratification models for haemorrhagic and thrombotic events are not validated in a critical care setting. With this paper we aimed to identify, adopting topological data analysis, the risk factors for therapeutic failure (in-hospital death or intensive care unit transfer), the in-hospital occurrence of stroke/TIA and major bleeding in a cohort of critically ill patients with pre-existing atrial fibrillation admitted to a stepdown unit; to engineer newer prediction models based on machine learning in the same cohort. We selected all medical patients admitted for critical illness and a history of pre-existing atrial fibrillation in the timeframe 01/01/2002–03/08/2007. All data regarding patients’ medical history, comorbidities, drugs adopted, vital parameters and outcomes (therapeutic failure, stroke/TIA and major bleeding) were acquired from electronic medical records. Risk factors for each outcome were analyzed adopting topological data analysis. Machine learning was used to generate three different predictive models. We were able to identify specific risk factors and to engineer dedicated clinical prediction models for therapeutic failure (AUC: 0.974, 95%CI: 0.934–0.975), stroke/TIA (AUC: 0.931, 95%CI: 0.896–0.940; Brier score: 0.13) and major bleeding (AUC: 0.930:0.911–0.939; Brier score: 0.09) in critically-ill patients, which were able to predict accurately their respective clinical outcomes. Topological data analysis and machine learning techniques represent a concrete viewpoint for the physician to predict the risk at the patients’ level, aiding the selection of the best therapeutic strategy in critically ill patients affected by pre-existing atrial fibrillation.

The Middle Science: Traversing Scale In Complex Many-Body Systems

A roadmap is developed that integrates simulation methodology and data science methods to target new theories that traverse the multiple length- and time-scale features of many-body phenomena.

Topological Data Analysis of C. elegans Locomotion and Behavior

We apply topological data analysis to the behavior of C. elegans, a widely studied model organism in biology. In particular, we use topology to produce a quantitative summary of complex behavior which may be applied to high-throughput data. Our methods allow us to distinguish and classify videos from various environmental conditions and we analyze the trade-off between accuracy and interpretability. Furthermore, we present a novel technique for visualizing the outputs of our analysis in terms of the input. Specifically, we use representative cycles of persistent homology to produce synthetic videos of stereotypical behaviors.

Promises and pitfalls of topological data analysis for brain connectivity analysis

Developing sensitive and reliable methods to distinguish normal and abnormal brain states is a key neuroscientific challenge. Topological Data Analysis, despite its relative novelty, already generated many promising applications, including in neuroscience. We conjecture its prominent tool of persistent homology may benefit from going beyond analysing structural and functional connectivity to effective connectivity graphs capturing the direct causal interactions or information flows. Therefore, we assess the potential of persistent homology to directed brain network analysis by testing its discriminatory power in two distinctive examples of disease-related brain connectivity alterations: epilepsy and schizophrenia. We estimate connectivity from functional magnetic resonance imaging and electrophysiology data, employ Persistent Homology and quantify its ability to distinguish healthy from diseased brain states by applying a support vector machine to features quantifying persistent homology structure. We show how this novel approach compares to classification using standard undirected approaches and original connectivity matrices. In the schizophrenia classification, topological data analysis generally performs close to random, while classifications from raw connectivity perform substantially better; potentially due to topographical, rather than topological, specificity of the differences. In the easier task of seizure discrimination from scalp electroencephalography data, classification based on persistent homology features generally reached comparable performance to using raw connectivity, albeit with typically smaller accuracies obtained for the directed (effective) connectivity compared to the undirected (functional) connectivity. Specific applications for topological data analysis may open when direct comparison of connectivity matrices is unsuitable - such as for intracranial electrophysiology with individual number and location of measurements. While standard homology performed overall better than directed homology, this could be due to notorious technical problems of accurate effective connectivity estimation.

Simplicial and topological descriptions of human brain dynamics

While brain imaging tools like functional magnetic resonance imaging (fMRI) afford measurements of whole-brain activity, it remains unclear how best to interpret patterns found amid the data’s apparent self-organization. To clarify how patterns of brain activity support brain function, one might identify metric spaces that optimally distinguish brain states across experimentally defined conditions. Therefore, the present study considers the relative capacities of several metric spaces to disambiguate experimentally defined brain states. One fundamental metric space interprets fMRI data topographically, that is, as the vector of amplitudes of a multivariate signal, changing with time. Another perspective compares the brain’s functional connectivity, that is, the similarity matrix computed between signals from different brain regions. More recently, metric spaces that consider the data’s topology have become available. Such methods treat data as a sample drawn from an abstract geometric object. To recover the structure of that object, topological data analysis detects features that are invariant under continuous deformations (such as coordinate rotation and nodal misalignment). Moreover, the methods explicitly consider features that persist across multiple geometric scales. While, certainly, there are strengths and weaknesses of each brain dynamics metric space, wefind that those that track topological features optimally distinguish experimentally defined brain states.

Time-varying functional connectivity interprets brain function as time-varying patterns of coordinated brain activity. While many questions remain regarding how brain function emerges from multiregional interactions, advances in the mathematics of topological data analysis (TDA) may provide new insights. One tool from TDA, “persistent homology,” observes the occurrence and persistence of n-dimensional holes in a sequence of simplicial complexes extracted from a weighted graph. In the present study, we compare the use of persistent homology versus more traditional metrics at the task of segmenting brain states that differ across experimental conditions. We find that the structures identified by persistent homology more accurately segment the stimuli, more accurately segment high versus low performance levels under common stimuli, and generalize better across volunteers.

Homological scaffold via minimal homology bases

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 quasi-canonical 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.

Stability of synchronization in simplicial complexes

Various systems in physics, biology, social sciences and engineering have been successfully modeled as networks of coupled dynamical systems, where the links describe pairwise interactions. This is, however, too strong a limitation, as recent studies have revealed that higher-order many-body interactions are present in social groups, ecosystems and in the human brain, and they actually affect the emergent dynamics of all these systems. Here, we introduce a general framework to study coupled dynamical systems accounting for the precise microscopic structure of their interactions at any possible order. We show that complete synchronization exists as an invariant solution, and give the necessary condition for it to be observed as a stable state. Moreover, in some relevant instances, such a necessary condition takes the form of a Master Stability Function. This generalizes the existing results valid for pairwise interactions to the case of complex systems with the most general possible architecture.

Weighted-persistent-homology-based machine learning for RNA flexibility analysis

With the great significance of biomolecular flexibility in biomolecular dynamics and functional analysis, various experimental and theoretical models are developed. Experimentally, Debye-Waller factor, also known as B-factor, measures atomic mean-square displacement and is usually considered as an important measurement for flexibility. Theoretically, elastic network models, Gaussian network model, flexibility-rigidity model, and other computational models have been proposed for flexibility analysis by shedding light on the biomolecular inner topological structures. Recently, a topology-based machine learning model has been proposed. By using the features from persistent homology, this model achieves a remarkable high Pearson correlation coefficient (PCC) in protein B-factor prediction. Motivated by its success, we propose weighted-persistent-homology (WPH)-based machine learning (WPHML) models for RNA flexibility analysis. Our WPH is a newly-proposed model, which incorporate physical, chemical and biological information into topological measurements using a weight function. In particular, we use local persistent homology (LPH) to focus on the topological information of local regions. Our WPHML model is validated on a well-established RNA dataset, and numerical experiments show that our model can achieve a PCC of up to 0.5822. The comparison with the previous sequence-information-based learning models shows that a consistent improvement in performance by at least 10% is achieved in our current model.

HONEM: Learning Embedding for Higher Order Networks

Representation learning on networks offers a powerful alternative to the oft painstaking process of manual feature engineering, and, as a result, has enjoyed considerable success in recent years. However, all the existing representation learning methods are based on the first-order network, that is, the network that only captures the pairwise interactions between the nodes. As a result, these methods may fail to incorporate non-Markovian higher order dependencies in the network. Thus, the embeddings that are generated may not accurately represent the underlying phenomena in a network, resulting in inferior performance in different inductive or transductive learning tasks. To address this challenge, this study presents higher order network embedding (HONEM), a higher order network (HON) embedding method that captures the non-Markovian higher order dependencies in a network. HONEM is specifically designed for the HON structure and outperforms other state-of-the-art methods in node classification, network reconstruction, link prediction, and visualization for networks that contain non-Markovian higher order dependencies.

Architecture and evolution of semantic networks in mathematics texts

Knowledge is a network of interconnected concepts. Yet, precisely how the topological structure of knowledge constrains its acquisition remains unknown, hampering the development of learning enhancement strategies. Here, we study the topological structure of semantic networks reflecting mathematical concepts and their relations in college-level linear algebra texts. We hypothesize that these networks will exhibit structural order, reflecting the logical sequence of topics that ensures accessibility. We find that the networks exhibit strong core–periphery architecture, where a dense core of concepts presented early is complemented with a sparse periphery presented evenly throughout the exposition; the latter is composed of many small modules each reflecting more narrow domains. Using tools from applied topology, we find that the expositional evolution of the semantic networks produces and subsequently fills knowledge gaps, and that the density of these gaps tracks negatively with community ratings of each textbook. Broadly, our study lays the groundwork for future efforts developing optimal design principles for textbook exposition and teaching in a classroom setting.

Weighted persistent homology for osmolyte molecular aggregation and hydrogen-bonding network analysis

It has long been observed that trimethylamine N-oxide (TMAO) and urea demonstrate dramatically different properties in a protein folding process. Even with the enormous theoretical and experimental research work on these two osmolytes, various aspects of their underlying mechanisms still remain largely elusive. In this paper, we propose to use the weighted persistent homology to systematically study the osmolytes molecular aggregation and their hydrogen-bonding network from a local topological perspective. We consider two weighted models, i.e., localized persistent homology (LPH) and interactive persistent homology (IPH). Boltzmann persistent entropy (BPE) is proposed to quantitatively characterize the topological features from LPH and IPH, together with persistent Betti number (PBN). More specifically, from the localized persistent homology models, we have found that TMAO and urea have very different local topology. TMAO is found to exhibit a local network structure. With the concentration increase, the circle elements in these networks show a clear increase in their total numbers and a decrease in their relative sizes. In contrast, urea shows two types of local topological patterns, i.e., local clusters around 6 Å and a few global circle elements at around 12 Å. From the interactive persistent homology models, it has been found that our persistent radial distribution function (PRDF) from the global-scale IPH has same physical properties as the traditional radial distribution function. Moreover, PRDFs from the local-scale IPH can also be generated and used to characterize the local interaction information. Other than the clear difference of the first peak value of PRDFs at filtration size 4 Å, TMAO and urea also shows very different behaviors at the second peak region from filtration size 5 Å to 10 Å. These differences are also reflected in the PBNs and BPEs of the local-scale IPH. These localized topological information has never been revealed before. Since graphs can be transferred into simplicial complexes by the clique complex, our weighted persistent homology models can be used in the analysis of various networks and graphs from any molecular structures and aggregation systems.

Towards Personalized Diagnosis of Glioblastoma in Fluid-Attenuated Inversion Recovery (FLAIR) by Topological Interpretable Machine Learning

Glioblastoma multiforme (GBM) is a fast-growing and highly invasive brain tumor, which tends to occur in adults between the ages of 45 and 70 and it accounts for 52 percent of all primary brain tumors. Usually, GBMs are detected by magnetic resonance images (MRI). Among MRI, a fluid-attenuated inversion recovery (FLAIR) sequence produces high quality digital tumor representation. Fast computer-aided detection and segmentation techniques are needed for overcoming subjective medical doctors (MDs) judgment. This study has three main novelties for demonstrating the role of topological features as new set of radiomics features which can be used as pillars of a personalized diagnostic systems of GBM analysis from FLAIR. For the first time topological data analysis is used for analyzing GBM from three complementary perspectives—tumor growth at cell level, temporal evolution of GBM in follow-up period and eventually GBM detection. The second novelty is represented by the definition of a new Shannon-like topological entropy, the so-called Generator Entropy. The third novelty is the combination of topological and textural features for training automatic interpretable machine learning. These novelties are demonstrated by three numerical experiments. Topological Data Analysis of a simplified 2D tumor growth mathematical model had allowed to understand the bio-chemical conditions that facilitate tumor growth—the higher the concentration of chemical nutrients the more virulent the process. Topological data analysis was used for evaluating GBM temporal progression on FLAIR recorded within 90 days following treatment completion and at progression. The experiment had confirmed that persistent entropy is a viable statistics for monitoring GBM evolution during the follow-up period. In the third experiment we developed a novel methodology based on topological and textural features and automatic interpretable machine learning for automatic GBM classification on FLAIR. The algorithm reached a classification accuracy up to 97%.

Reorderability of node-filtered order complexes

Growing graphs describe a multitude of developing processes from maturing brains to expanding vocabularies to burgeoning public transit systems. Each of these growing processes likely adheres to proliferation rules that establish an effective order of node and connection emergence. When followed, such proliferation rules allow the system to properly develop along a predetermined trajectory. But rules are rarely followed. Here we ask what topological changes in the growing graph trajectories might occur after the specific but basic perturbation of permuting the node emergence order. Specifically, we harness applied topological methods to determine which of six growing graph models exhibit topology that is robust to randomizing node order, termed global reorderability, and robust to temporally local node swaps, termed local reorderability. We find that the six graph models fall upon a spectrum of both local and global reorderability, and furthermore we provide theoretical connections between robustness to node pair ordering and robustness to arbitrary node orderings. Finally, we discuss real-world applications of reorderability analyses and suggest possibilities for designing reorderable networks.

weg2vec: Event embedding for temporal networks

Network embedding techniques are powerful to capture structural regularities in networks and to identify similarities between their local fabrics. However, conventional network embedding models are developed for static structures, commonly consider nodes only and they are seriously challenged when the network is varying in time. Temporal networks may provide an advantage in the description of real systems, but they code more complex information, which could be effectively represented only by a handful of methods so far. Here, we propose a new method of event embedding of temporal networks, called weg2vec, which builds on temporal and structural similarities of events to learn a low dimensional representation of a temporal network. This projection successfully captures latent structures and similarities between events involving different nodes at different times and provides ways to predict the final outcome of spreading processes unfolding on the temporal structure.

Visualising 2-simplex formation in metabolic reactions

Understanding in silico the dynamics of metabolic reactions made by a large number of molecules has led to the development of different tools for visualising molecular interactions. However, most of them are mainly focused on quantitative aspects. We investigate the potentiality of the topological interpretation of the interaction-as-perception at the basis of a multiagent system, to tackle the complexity of visualising the emerging behaviour of a complex system. We model and simulate the glycolysis process as a multiagent system, and we perform topological data analysis of the molecular perceptions graphs, gained during the formation of the enzymatic complexes, to visualise the set of emerging patterns. Identifying expected patterns in terms of simplicial structures allows us to characterise metabolic reactions from a qualitative point of view and conceivably reveal the simulation reactivity trend.

Weighted persistent homology for biomolecular data analysis

In this paper, we systematically review weighted persistent homology (WPH) models and their applications in biomolecular data analysis. Essentially, the weight value, which reflects physical, chemical and biological properties, can be assigned to vertices (atom centers), edges (bonds), or higher order simplexes (cluster of atoms), depending on the biomolecular structure, function, and dynamics properties. Further, we propose the first localized weighted persistent homology (LWPH). Inspired by the great success of element specific persistent homology (ESPH), we do not treat biomolecules as an inseparable system like all previous weighted models, instead we decompose them into a series of local domains, which may be overlapped with each other. The general persistent homology or weighted persistent homology analysis is then applied on each of these local domains. In this way, functional properties, that are embedded in local structures, can be revealed. Our model has been applied to systematically study DNA structures. It has been found that our LWPH based features can be used to successfully discriminate the A-, B-, and Z-types of DNA. More importantly, our LWPH based principal component analysis (PCA) model can identify two configurational states of DNA structures in ion liquid environment, which can be revealed only by the complicated helical coordinate system. The great consistence with the helical-coordinate model demonstrates that our model captures local structure variations so well that it is comparable with geometric models. Moreover, geometric measurements are usually defined in local regions. For instance, the helical-coordinate system is limited to one or two basepairs. However, our LWPH can quantitatively characterize structure information in regions or domains with arbitrary sizes and shapes, where traditional geometrical measurements fail.

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.

Persistent homology of unweighted complex networks via discrete Morse theory

Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and undirected networks using persistent homology. Leveraging on the features of discrete Morse theory, our method not only captures the topology of the clique complex of such graphs via the concept of critical simplices, but also achieves close to the theoretical minimum number of critical simplices in several analyzed model and real networks. This leads to a reduced filtration scheme based on the subsequence of the corresponding critical weights, thereby leading to a significant increase in computational efficiency. We have employed our filtration scheme to explore the persistent homology of several model and real-world networks. In particular, we show that our method can detect differences in the higher-order structure of networks, and the corresponding persistence diagrams can be used to distinguish between different model networks. In summary, our method based on discrete Morse theory further increases the applicability of persistent homology to investigate the global topology of complex networks.

Topological phase transitions in functional brain networks

Functional brain networks are often constructed by quantifying correlations between time series of activity of brain regions. Their topological structure includes nodes, edges, triangles, and even higher-dimensional objects. Topological data analysis (TDA) is the emerging framework to process data sets under this perspective. In parallel, topology has proven essential for understanding fundamental questions in physics. Here we report the discovery of topological phase transitions in functional brain networks by merging concepts from TDA, topology, geometry, physics, and network theory. We show that topological phase transitions occur when the Euler entropy has a singularity, which remarkably coincides with the emergence of multidimensional topological holes in the brain network. The geometric nature of the transitions can be interpreted, under certain hypotheses, as an extension of percolation to high-dimensional objects. Due to the universal character of phase transitions and noise robustness of TDA, our findings open perspectives toward establishing reliable topological and geometrical markers for group and possibly individual differences in functional brain network organization.

Exact topological inference of the resting-state brain networks in twins

A cycle in a brain network is a subset of a connected component with redundant additional connections. If there are many cycles in a connected component, the connected component is more densely connected. Whereas the number of connected components represents the integration of the brain network, the number of cycles represents how strong the integration is. However, it is unclear how to perform statistical inference on the number of cycles in the brain network. In this study, we present a new statistical inference framework for determining the significance of the number of cycles through the Kolmogorov-Smirnov (KS) distance, which was recently introduced to measure the similarity between networks across different filtration values by using the zeroth Betti number. In this paper, we show how to extend the method to the first Betti number, which measures the number of cycles. The performance analysis was conducted using the random network simulations with ground truths. By using a twin imaging study, which provides biological ground truth, the methods are applied in determining if the number of cycles is a statistically significant heritable network feature in the resting-state functional connectivity in 217 twins obtained from the Human Connectome Project. The MATLAB codes as well as the connectivity matrices used in generating results are provided at http://www.stat.wisc.edu/∼mchung/TDA.

In this paper, we propose a new topological distance based on the Kolmogorov-Smirnov (KS) distance that is adapted for brain networks, and compare them against other topological network distances including the Gromov-Hausdorff (GH) distances. KS-distance is recently introduced to measure the similarity between networks across different filtration values by using the zeroth Betti number, which measures the number of connected components. In this paper, we show how to extend the method to the first Betti number, which measures the number of cycles. The performance analysis was conducted using random network simulations with ground truths. Using a twin imaging study, which provides biological ground truth (of network differences), we demonstrate that the KS distances on the zeroth and first Betti numbers have the ability to determine heritability.

The importance of the whole: Topological data analysis for the network neuroscientist

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 algebro-topological 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.

Simplicial models of social contagion

Complex networks have been successfully used to describe the spread of diseases in populations of interacting individuals. Conversely, pairwise interactions are often not enough to characterize social contagion processes such as opinion formation or the adoption of novelties, where complex mechanisms of influence and reinforcement are at work. Here we introduce a higher-order model of social contagion in which a social system is represented by a simplicial complex and contagion can occur through interactions in groups of different sizes. Numerical simulations of the model on both empirical and synthetic simplicial complexes highlight the emergence of novel phenomena such as a discontinuous transition induced by higher-order interactions. We show analytically that the transition is discontinuous and that a bistable region appears where healthy and endemic states co-exist. Our results help explain why critical masses are required to initiate social changes and contribute to the understanding of higher-order interactions in complex systems.

From networks to optimal higher-order models of complex systems

Rich data is revealing that complex dependencies between the nodes of a network may escape models based on pairwise interactions. Higher-order network models go beyond these limitations, offering new perspectives for understanding complex systems.

Big Data: From Forecasting to Mesoscopic Understanding. Meta-Profiling as Complex Systems

We consider Big Data as a phenomenon with acquired properties, similar to collective behaviours, that establishes virtual collective beings. We consider the occurrence of ongoing non-equivalent multiple properties in the conceptual framework of structural dynamics given by sequences of structures and not only by different values assumed by the same structure. We consider the difference between modelling and profiling in a constructivist way, as De Finetti intended probability to exist, depending on the configuration taken into consideration. The past has little or no influence, while events and their configurations are not memorised. Any configuration of events is new, and the probabilistic values to be considered are reset. As for collective behaviours, we introduce methodological and conceptual proposals using mesoscopic variables and their property profiles and meta-profile Big Data and non-computable profiles which were inspired by the use of natural computing to deal with cyber-ecosystems. The focus is on ongoing profiles, in which the arising properties trace trajectories, rather than assuming that we can foresee them based on the past.

Spatial Embedding Imposes Constraints on Neuronal Network Architectures

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.

Topological Data Analysis of Single-Trial Electroencephalographic Signals

Epilepsy is a neurological disorder that can negatively affect the visual, audial and motor functions of the human brain. Statistical analysis of neurophysiological recordings, such as electroencephalogram (EEG), facilitates the understanding and diagnosis of epileptic seizures. Standard statistical methods, however, do not account for topological features embedded in EEG signals. In the current study, we propose a persistent homology (PH) procedure to analyze single-trial EEG signals. The procedure denoises signals with a weighted Fourier series (WFS), and tests for topological difference between the denoised signals with a permutation test based on their PH features persistence landscapes (PL). Simulation studies show that the test effectively identifies topological difference and invariance between two signals. In an application to a single-trial multichannel seizure EEG dataset, our proposed PH procedure was able to identify the left temporal region to consistently show topological invariance, suggesting that the PH features of the Fourier decomposition during seizure is similar to the process before seizure. This finding is important because it could not be identified from a mere visual inspection of the EEG data and was in fact missed by earlier analyses of the same dataset.

Knowledge gaps in the early growth of semantic feature networks

Understanding language learning, and more general knowledge acquisition, requires 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 to shared features, then characterizing the structure of strongly inter-related 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 within the growing semantic feature network of toddlers. We detect topological cavities of multiple dimensions and find that despite word order variation, 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.

Co-occurrence simplicial complexes in mathematics: identifying the holes of knowledge

In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research practice in mathematics. k-dimensional holes die when every concept in the hole appears in an article together with other k+1 concepts in the hole, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the size of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas. We provide further description of the conceptual space by looking for the simplicial analogs of stars and explore the likelihood of edges in a star to be also part of a homological cycle. We also show that authors' conceptual entropy is positively related with their contribution to homological holes, suggesting that polymaths tend to be on the frontier of research.

Using persistent homology to reveal hidden covariates in systems governed by the kinetic Ising model

We propose a method, based on persistent homology, to uncover topological properties of a priori unknown covariates in a system governed by the kinetic Ising model with time-varying external fields. As its starting point the method takes observations of the system under study, a list of suspected or known covariates, and observations of those covariates. We infer away the contributions of the suspected or known covariates, after which persistent homology reveals topological information about unknown remaining covariates. Our motivating example system is the activity of neurons tuned to the covariates physical position and head direction, but the method is far more general.

Quantifying the Effects of Topology and Weight for Link Prediction in Weighted Complex Networks

In weighted networks, both link weight and topological structure are significant characteristics for link prediction. In this study, a general framework combining null models is proposed to quantify the impact of the topology, weight correlation and statistics on link prediction in weighted networks. Three null models for topology and weight distribution of weighted networks are presented. All the links of the original network can be divided into strong and weak ties. We can use null models to verify the strong effect of weak or strong ties. For two important statistics, we construct two null models to measure their impacts on link prediction. In our experiments, the proposed method is applied to seven empirical networks, which demonstrates that this model is universal and the impact of the topology and weight distribution of these networks in link prediction can be quantified by it. We find that in the USAir, the Celegans, the Gemo, the Lesmis and the CatCortex, the strong ties are easier to predict, but there are a few networks whose weak edges can be predicted more easily, such as the Netscience and the CScientists. It is also found that the weak ties contribute more to link prediction in the USAir, the NetScience and the CScientists, that is, the strong effect of weak ties exists in these networks. The framework we proposed is versatile, which is not only used to link prediction but also applicable to other directions in complex networks.