Topological Measures Research Articles

Characterization of entropy measures with connection number based indices of boric acid hydrogen-bonded 2D lattice sheets.

The cut method is a computational approach utilized to predict the fundamental activities of physicochemical properties of chemical networks, also called topological indices. The connection number is a new idea, that gives interesting and good results of the topological indices (TIs) and entropy measures (EMs) for structural representation of chemical compounds and networks. The physical density of chemical networks is characterized by these indices. In this paper, we determined the computational results for indices based on connection numbers for a two-dimensional lattice sheet of hydrogen-bonded boric acid. Boric acid, an inorganic compound, is not very harmful when applied to the skin or consumed. Finally, graphical and numerical comparisons of topological numbers including the number of borate hydrogen-bonded double lattice forms are also included in this study.

Symplectic topology and ideal-valued measures

We adapt Gromov’s notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections. Furthermore, it allows us to discuss three “big fiber theorems”—the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic topology—from a unified viewpoint. Our main technical tool is an enhancement of the symplectic cohomology theory recently developed by Varolgüneş.

Exploring topological indices and entropy measure via rational curve fitting models for calcium hydroxide network

This work uses rational curve fitting models to investigate the complex link between topological indices and entropy metrics in calcium hydroxide (Ca(OH)2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(Ca(OH)_{2})$$\\end{document} networks. Entropy measurements shed light on the complexity and disorder of these networks, whereas topological indices are essential instruments for comprehending the structural characteristics of chemical substances. Our goal is to find new patterns and connections by merging these two fields, improving materials science’s prediction capacity. We calculate many topological indices, such as the Randic, Balaban, and Zagreb indices, and examine the relationship between them and entropy measurements obtained from (Ca(OH)2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(Ca(OH)_{2})$$\\end{document} structural arrangement.

Reducing tree volume overestimation in quantitative structure models using modeled branch topology and direct twig measurements

Abstract Quantitative Structure Models (QSMs) are fit to tree point clouds to represent the topology of trees as a network of cylinders. QSMs allow for the calculation of metrics difficult to measure without destructive sampling, including total tree volume. Current limitations in terrestrial laser scanning technology make small branches difficult to accurately resolve, causing overestimation of small branch volume in QSMs, which can translate into overestimating tree biomass. We present a new method called Real Twig to correct overestimated small branch and twig cylinders in QSMs. Real Twig differs from current methods by using twig diameters measured directly from corresponding tree species to model a unique taper for every path in the QSM, using the QSM’s inherent branching topology, but without relying on predefined mathematical or allometric relationships. To test Real Twig, we generated QSMs for different sets of trees that had detailed dry mass and density measurements obtained via felling after scanning. QSM-based biomass estimates were obtained by multiplying the tree’s QSM-based volume estimate by the tree’s specific basic density value. We trained our method with high-quality data consisting of five northern red oak (Quercus rubra L.) and five red maple (Acer rubrum L.) trees, using two different versions of TreeQSM, a widely used algorithm for generating QSMs. We further tested our method on three publicly available datasets, including managed forests and large tropical trees, collected with both phase-shift or time-of-flight sensors. QSMs corrected with our Real Twig method showed a very large improvement in tree biomass estimation, with a relative mean error of −1.2%, a relative root mean square error of 10.5%, and a concordance correlation coefficient of 0.999, compared to a relative mean error 76.8%, a relative root mean square error of 48.7%, and a concordance correlation coefficient of 0.982, when using the standard outputs of TreeQSM.

Percolation Statistics in the MillenniumTNG Simulations

The statistical analysis of cosmic large-scale structure is most often based on simple two-point summary statistics, like the power spectrum or the two-point correlation function of a sample of galaxies or other types of tracers. In contrast, topological measures of clustering are also sensitive to higher-order correlations and thus offer the prospect to access additional information that may harbor important constraining power. We here revisit one such geometric measure of the cosmic web in the form of the so-called percolation analysis, using the recent MillenniumTNG simulation suite of the ΛCDM paradigm. We analyze continuum percolation statistics both for high-resolution dark matter particle distributions and for galaxy mock catalogs from a semianalytic galaxy formation model within a periodic simulation volume of 3000 Mpc on a side. For comparison, we also investigate the percolation statistics of random particle sets and neutrino distributions with two different summed particle masses. We find that the percolation statistics of the dark matter distribution evolves strongly with redshift and thus clustering strength, yielding a progressively lower percolation threshold toward later times. However, there is a sizable residual dependence on numerical resolution, which we interpret as a residual influence of different levels of shot noise. This is corroborated by our analysis of galaxy mock catalogs, whose results depend on sampling density more strongly than on galaxy selection criteria. While this limits the discriminative power of percolation statistics, our results suggest that it still remains useful as a complementary cosmological test when controlled for sampling density.

S-Prime Graph of S-meet Semilattice

In this paper, the S-prime ideal in a lattice \(\mathfrak{L}\) is introduced where S is the meet subset of \(\mathfrak{L}\). Also, it isshown that the prime ideal of \(\mathfrak{L}\) is an S-prime ideal of \(\mathfrak{L}\) and studied with suitable examples. Further, theS-prime ideal \(\mathfrak{I}\)\(\mathfrak{s}\) of S-meet semilattice Ls is introduced. Finally, a new graph called S-prime graph of S-meet semilattice is defined and their topological measures are generalized.

A topological network connectivity design problem based on spectral analysis

How to improve network connectivity and which parts of the network are vulnerable are critical issues. We begin by defining an equal distribution problem, in which supplies are distributed equally to all nodes in the network. We then derive a topological network connectivity measure from the convergence speed, which is the second minimum eigenvalue of a Laplacian network matrix. Based on the equal distribution problem, we propose a method for identifying critical links for network connectivity using the derivative of the second minimum eigenvalue. Furthermore, we develop a network design problem that maximizes topological connectivity within a budget creating strengthening network links. The problem is convex programming, and the solution is global. Furthermore, it can be converted into an identical semidefinite programming problem, which requires less computational effort. Finally, we test the developed problems on road networks in the Japanese prefectures of Ishikawa and Toyama to determine their applicability and validity.

Complex network analysis of climate and landscape satellite data to explore spatio-temporal patterns in urban environment: the case of Athens

Motivated by the significance and complexity of exploring spatiotemporal patterns - regions within an urban environment, particularly in the context of extreme heat events- this research analyzes meteorological time series through complex network analysis. The data collected for the examination area is focused on Athens, Greece, and covers sections of the city’s urban landscape. The data was obtained from the Copernicus observation component of the European Union. Initially, the time series are transformed into networks using correlation network methodology, followed by examination of the discriminative capability of the topological measures of networks degree and modularity as community - region detection methods. Of particular interest is that our findings suggest that the proposed complex network analysis can lead to the extraction of spatial urban regions closely linked to land use and building heights in corresponding areas. These results may help investigate the spatial variability of heat in the urban environment and inform urban planning and management strategies in policy decision-making regarding the intensity of urban heat throughout the city and the planning of climate change adaptation strategies.

Reorganization of brain connectivity across the spectrum of clinical cognitive decline.

Clinical cognitive decline, leading to Alzheimer's Disease Dementia (ADD), has long been interpreted as a disconnection syndrome, hindering the information flow capacity of the brain, hence leading to the well-known symptoms of ADD. The structural and functional brain connectome analyses play a central role in studies of brain from this perspective. However, most current research implicitly assumes that the changes accompanying the progression of cognitive decline are monotonous in time, whether measured across the entire brain or in fixed cortical regions. We investigate the structural and functional connectivity-wise reorganization of the brain without such assumptions across the entire spectrum. We utilize nodal assortativity as a local topological measure of connectivity and follow a data-centric approach to identify and verify relevant local regions, as well as to understand the nature of underlying reorganization. The analysis of our preliminary experimental data points to statistically significant, hyper and hypo-assortativity regions that depend on the disease's stage, and differ for structural and functional connectomes. Our results suggest a new perspective into the dynamic, potentially a mix of degenerative and compensatory, topological alterations that occur in the brain as cognitive decline progresses.

Spatial food webs in the Barents Sea: atlantification and the reorganization of the trophic structure.

Climate change affects ecosystems at several levels: by altering the spatial distribution of individual species, by locally rewiring interspecific interactions, and by reorganizing trophic networks at larger scales. The dynamics of marine food webs are becoming more and more sensitive to spatial processes and connections in the seascape. As a case study, we study the atlantification of the Barents Sea: we compare spatio-temporal subsystems at three levels: the identity of key organisms, critically important interactions and the entire food web. Network analysis offers quantitative measurements, including centrality indices, trophic similarity indices, a topological measure of interaction asymmetry and network-level measures. We found that atlantification alters the identity of key species (boreal demersals becoming hubs), results in strongly asymmetric interactions (dominated by haddock), changes the dominant regulation regime (from bottom-up to wasp-waist control) and makes the food web less modular. Since the results of food web analysis may be quite sensitive to network construction, the aggregation of food web data was explicitly studied to increase the robustness of food web analysis. We found that an alternative, mathematical aggregation algorithm better preserves some network properties (e.g. density) of the original, unaggregated network than the biologically inspired aggregation into functional groups. This article is part of the theme issue 'Connected interactions: enriching food web research by spatial and social interactions'.

Disrupted Brain Network Measures in Parkinson's Disease Patients with Severe Hyposmia and Cognitively Normal Ability.

Neuroscience has revolved around brain structural changes, functional activity, and connectivity alteration in Parkinson's Disease (PD); however, how the network topology organization becomes altered is still unclear, specifically in Parkinson's patients with severe hyposmia. In this study, we have examined the functional network topological alteration in patients affected by Parkinson's Disease with normal cognitive ability (ODN), Parkinson's Disease with severe hyposmia (ODP), and healthy controls (HCs) using resting-state functional magnetic resonance imaging (rsfMRI) data. We have analyzed brain topological organization using popular graph measures such as network segregation (clustering coefficient, modularity), network integration (participation coefficient, path length), small-worldness, efficiency, centrality, and assortativity. Then, we used a feature ranking approach based on the diagonal adaptation of neighborhood component analysis, aiming to determine a graph measure that is sensitive enough to distinguish between these three different groups. We noted significantly lower segregation and local efficiency and small-worldness in ODP compared to ODN and HCs. On the contrary, we did not find differences in network integration in ODP compared to ODN and HCs, which indicates that the brain network becomes fragmented in ODP. At the brain network level, a progressive increase in the DMN (Default Mode Network) was observed from healthy controls to ODN to ODP, and a continuous decrease in the cingulo-opercular network was observed from healthy controls to ODN to ODP. Further, the feature ranking approach has shown that the whole-brain clustering coefficient and small-worldness are sensitive measures to classify ODP vs. ODN, as well as HCs. Looking at the brain regional network segregation, we have found that the cerebellum and limbic, fronto-parietal, and occipital lobes have higher ODP reductions than ODN and HCs. Our results suggest network topological measures, specifically whole-brain segregation and small-worldness decreases. At the network level, an increase in DMN and a decrease in the cingulo-opercular network could be used as biomarkers to characterize ODN and ODP.

Probabilistic Tractography-Based Tremor Network Connectivity in Tremor Dominant Parkinson's Disease and Essential Tremor plus.

Background: The basal ganglia-thalamocortical (BGTC) and cerebello-thalamocortical (CTC) networks are implicated in tremor genesis; however, exact contributions across disorders have not been studied. Objective: Evaluate the structural connectivity of BGTC and CTC in tremor-dominant Parkinson's disease (TDPD) and essential tremor plus (ETP) with the aid of probabilistic tractography and graph theory analysis. Methods: Structural connectomes of the BGTC and CTC were generated by probabilistic tractography for TDPD (n = 25), ETP (ET with rest tremor, n = 25), and healthy control (HC, n = 22). The Brain Connectivity Toolbox was used for computing standard topological graph measures of segregation, integration, and centrality. Tremor severity was ascertained using the Fahn-Tolosa-Marin tremor rating scale (FTMRS). Results: There was no difference in total FTMRS scores. Compared with HC, TDPD had a lower global efficiency and characteristic path length. Abnormality in segregation, integration, and centrality of bilateral putamen, globus pallidus externa (GPe), and GP interna (GPi), with reduction of centrality of right caudate and cerebellar lobule 8, was observed. ETP showed reduction in segregation and integration of right GPe and GPi, ventrolateral posterior nucleus, and centrality of right putamen, compared with HC. Differences between TDPD and ETP were a reduction of strength of the right putamen, and lower clustering coefficient, local efficiency, and strength of the left GPi in TDPD. Conclusions: Contrary to expectations, TDPD and ETP may not be significantly different with regard to tremor pathogenesis, with definite overlaps. There may be fundamental similarities in network disruption across different tremor disorders with the same tremor activation patterns, along with disease-specific changes.

Domination of nonlinear semigroups generated by regular, local Dirichlet forms

AbstractIn this article we study perturbations of local, nonlinear Dirichlet forms on arbitrary topological measure spaces. As a main result, we show that the semigroup generated by a local, regular, nonlinear Dirichlet form $${\mathcal {E}}$$ E dominates the semigroup generated by another local functional $${\mathcal {F}}$$ F if, and only if, $${\mathcal {F}}$$ F is a specific zero order perturbation of $${\mathcal {E}}$$ E . On the way, we prove a nonlinear version of the Riesz–Markov representation theorem, we define an abstract boundary of a topological measure space, and apply the notion of nonlinear capacity. The main result helps to classify the perturbations that lie between Neumann and Dirichlet boundary conditions.

Examining cognition and brain networks using magnetoencephalography in paediatric autoimmune encephalitis and acute disseminated encephalomyelitis: a preliminary study.

Paediatric autoimmune encephalitis, including acute disseminated encephalomyelitis, are inflammatory brain diseases presenting with cognitive deficits, psychiatric symptoms, seizures, MRI and EEG abnormalities. Despite improvements in disease recognition and early immunotherapy, long-term outcomes in paediatric autoimmune encephalitis remain poor. Our aim was to understand functional connectivity changes that could be associated with negative developmental outcomes across different types of paediatric autoimmune encephalitis using magnetoencephalography. Participants were children diagnosed with paediatric autoimmune encephalitis at least 18 months before testing and typically developing children. All completed magnetoencephalography recording at rest, T1 MRI scans and neuropsychology testing. Brain connectivity (specifically in delta and theta) was estimated with amplitude envelope correlation, and network efficiency was measured using graph measures (global efficiency, local efficiency and modularity). Twelve children with paediatric autoimmune encephalitis (11.2 ± 3.5 years, interquartile range 9 years; 5M:7F) and 12 typically developing controls (10.6 ± 3.2 years, interquartile range 7 years; 8M:4F) participated. Children with paediatric autoimmune encephalitis did not differ from controls in working memory (t(21) = 1.449; P = 0.162; d = 0.605) but had significantly lower processing speed (t(21) = 2.463; P = 0.023; Cohen's d = 1.028). Groups did not differ in theta network topology measures. The paediatric autoimmune encephalitis group had a significantly lower delta local efficiency across all thresholds tested (d = -1.60 at network threshold 14%). Theta modularity was associated with lower working memory (β = -0.781; t(8) = -2.588, P = 0.032); this effect did not survive correction for multiple comparisons (P(corr) = 0.224). Magnetoencephalography was able to capture specific network alterations in paediatric autoimmune encephalitis patients. This preliminary study demonstrates that magnetoencephalography is an appropriate tool for assessing children with paediatric autoimmune encephalitis and could be associated with cognitive outcomes.

A topological description of loss surfaces based on Betti Numbers

In the context of deep learning models, attention has recently been paid to studying the surface of the loss function in order to better understand training with methods based on gradient descent. This search for an appropriate description, both analytical and topological, has led to numerous efforts in identifying spurious minima and characterize gradient dynamics. Our work aims to contribute to this field by providing a topological measure for evaluating loss complexity in the case of multilayer neural networks. We compare deep and shallow architectures with common sigmoidal activation functions by deriving upper and lower bounds for the complexity of their respective loss functions and revealing how that complexity is influenced by the number of hidden units, training models, and the activation function used. Additionally, we found that certain variations in the loss function or model architecture, such as adding an ℓ2 regularization term or implementing skip connections in a feedforward network, do not affect loss topology in specific cases.

EEG connectivity in functional brain networks supporting visuomotor integration processes in dominant and non-dominant hand movements

Objective. This study explores the changes in the organization of functional brain networks induced by performing a visuomotor integration task, as revealed by noninvasive spontaneous electroencephalographic traces (EEG). Approach. EEG data were acquired during the execution of the Nine Hole Peg Test (NHPT) with the dominant and non-dominant hands in a group of 44 right-handed volunteers. Both spectral analysis and phase-based connectivity analysis were performed in the theta (ϑ), mu ( μ ) and beta (ß) bands. Graph Theoretical Analysis (GTA) was also performed to investigate the topological reorganization induced by motor task execution. Main results. Spectral analysis revealed an increase of frontoparietal ϑ power and a spatially diffused reduction of µ and ß contribution, regardless of the hand used. GTA showed a significant increase in network integration induced by movement performed with the dominant limb compared to baseline in the ϑ band. The µ and ß bands were associated with a reduction in network integration during the NHPT. In the µ rhythm, this result was more evident for the right-hand movement, while in the ß band, results did not show dependence on the laterality. Finally, correlation analysis highlighted an association between frequency-specific topology measures and task performance for both hands. Significance. Our results show that functional brain networks reorganize during visually guided movements in a frequency-dependent manner, differently depending on the hand used (dominant/non dominant).

The role of elasticity on adhesion and clustering of neurons on soft surfaces

The question of whether material stiffness enhances cell adhesion and clustering is still open to debate. Results from the literature are seemingly contradictory, with some reports illustrating that adhesion increases with surface stiffness and others suggesting that the performance of a system of cells is curbed by high values of elasticity. To address the role of elasticity as a regulator in neuronal cell adhesion and clustering, we investigated the topological characteristics of networks of neurons on polydimethylsiloxane (PDMS) surfaces - with values of elasticity (E) varying in the 0.55–2.65 MPa range. Results illustrate that, as elasticity increases, the number of neurons adhering on the surface decreases. Notably, the small-world coefficient – a topological measure of networks – also decreases. Numerical simulations and functional multi-calcium imaging experiments further indicated that the activity of neuronal cells on soft surfaces improves for decreasing E. Experimental findings are supported by a mathematical model, that explains adhesion and clustering of cells on soft materials as a function of few parameters - including the Young’s modulus and roughness of the material. Overall, results indicate that – in the considered elasticity interval – increasing the compliance of a material improves adhesion, improves clustering, and enhances communication of neurons.

Learning a complex network representation for shape classification

Shape contour is a key low-level characteristic, making shape description an important aspect in many computer vision problems, with several challenges such as variations in scale, rotation, and noise. In this paper, we introduce an approach for shape analysis and classification from binary images based on representations learned by applying Randomized Neural Networks (RNNs) on feature maps derived from a Complex Network (CN) framework. Our approach models the contour in a complex network and computes their topological measures using a dynamic evolution strategy. This evolution of the CN provides significant information into the physical aspects of the shape’s contour. Therefore, we propose embedding the topological measures computed from the dynamics of the CN evolution into a matrix representation, which we have named the Topological Feature Map (TFM). Then, we employ the RNN to learn representations from the TFM through a sliding window strategy. The proposed representation is formed by the learned weights between the hidden and output layers of the RNN. Our experimental results show performance improvements in shape classification using the proposed method across two generic shape datasets. We also applied our approach to the recognition of plant leaves, achieving high performance in this challenging task. Furthermore, the proposed approach has demonstrated robustness to noise and invariance to transformations in scale and orientation of the shapes.

Investigation of Indian stock markets using topological data analysis and geometry-inspired network measures

Financial markets are a quintessential example of complex systems wherein strategic connections arising through the trading (buying and selling) of different stocks gives rise to dynamic correlation patterns in stock prices. Network science based approaches have been employed for more than two decades to investigate the stock market as an inferential network, with nodes representing stocks and edges as pairwise cross-correlations between stock prices. However, a recent focus has been the characterization of the higher-order structure, beyond pairwise correlation, of such networks. To this end, geometry-inspired measures (such as discrete Ricci curvatures) and topological data analysis (TDA) based methods (such as persistent homology) have become attractive tools for characterizing the higher-order structure of networks representing the financial systems. In this study, our goal is to perform a comparative analysis of both these approaches, especially by assessing the fragility and systemic risk in the Indian stock markets, which is known for its high volatility and risk. To achieve this goal, we analyze the time series of daily log-returns of stocks comprising the National Stock Exchange (NSE) and the Bombay Stock Exchange (BSE). Specifically, our aim is to monitor the changes in standard network measures, edge-centric discrete Ricci curvatures, and persistent homology based topological measures computed from cross-correlation matrices of stocks. In this study, the edge-centric discrete Ricci curvatures have been employed for the first time in the analysis of the Indian stock markets. The Indian stock markets are known to be less diverse in comparison to the US market, and hence provides us an interesting example. Our results point that, among the persistent homology based topological measures, persistent entropy is simple and more robust than L1-norm and L2-norm of persistence landscape. In a broader comparison between network analysis and TDA, we highlight that the network analysis is sensitive to the way of constructing the networks (threshold or minimum spanning tree), as well as the threshold values used to construct the correlation-based threshold networks. On the other hand, the persistent homology is a more robust approach and is able to capture the higher-order interactions and eliminate noisy data in financial systems, since it does not take into account a single value of threshold but rather a range of values.

Pre-disaster resource allocation based on network topology and flow features

Protecting transportation systems from severe damage is crucial for their post-disaster functionality. We adopt network topological metrics and generalised topological metrics involving demand and traffic flow characteristics to identify critical links for enhancing post-event transportation network performance. A single-parameter, non-monotonic function is proposed to allocate limited resources based on these metrics. Monte Carlo simulation generates failure scenarios to evaluate residual network performance through network topological measures and network level of service indicators. This work introduces ‘Trip Efficiency’, a novel level of service indicator that considers both demand connectivity and travel efficiency for a network with multiple links interrupted. In Anaheim case study network, betweenness-based resource allocation leads to better performance on network topological measures; while generalised topological metrics result in more connected OD pairs and higher travel efficiency. The experimental results demonstrate the effectiveness of the proposed method in enhancing network resilience against multisite disruptive events through proactive resource allocations.