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Showing posts with the label Topological Data Analysis and Geometric Graph Theory in Complex Networks

Topology-Driven Fault Detection in Smart Cities

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Topology-Driven Fault Detection in Smart Cities Topology-Driven Fault Detection in Smart Cities: Utilizing Dynamic Graph Filtrations and Persistent Homology for IoT Connectivity Patterns Shrishti Rastogi / Research Article Abstract: The rapid proliferation of IoT devices in smart cities necessitates robust, real-time monitoring systems to detect infrastructure faults. Traditional graph-theoretic methods often fail to capture multi-scale structural changes under fluctuating environmental conditions. This paper presents a topology-driven framework utilizing Topological Data Analysis (TDA), specifically persistent homology, to analyze time-varying graph filtrations of IoT connectivity patterns. We integrate Distance-to-Measure (DTM) filtrations to mitigate noise, coupled with machine learning classifiers applied to stable topological summaries. Empirical evaluation on simulated smart city datasets demonstrates superior F1-s...
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Topological Data Analysis and Geometric Graph Theory in Complex Networks

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Topological Data Analysis and Geometric Graph Theory in Complex Networks: From Social Dynamics to Spatial Omics. Abstract Complex networks have gone through a radical change from just a few graph-theoretic representations of networks that consider only pairs of nodes to advanced geometric and higher-order topological representations. This paper introduces a detailed methodological approach to build a bridge between the classical structural network analysis and contemporary spatial biological systems. We investigate the development of network modeling by the use of dynamic influence matrices, spectral geometry, and persistent homology, introducing a mathematical hierarchy. In conclusion, we show how a mathematical framework such as topological data analysis (TDA) and geometric graph theory are needed to unlock the architectural complexity of biological tissues, especially in the fast evolving field of spatial omics. Introduction Traditionally, complex networks have been thought ...
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Heuristic Computation and the Discovery of Mersenne Primes

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Heuristic Computation and the Discovery of Mersenne Primes Heuristic Computation and the Discovery of Mersenne Primes “Where Strategy Meets Infinity: The Quest for Mersenne Primes” Introduction: The Dance of Numbers and Heuristics Mersenne primes are not just numbers—they are milestones in the vast landscape of mathematics. Defined by the formula: \[ M_p = 2^p - 1 \] where \( p \) is itself prime, these giants challenge our computational limits and inspire new methods of discovery. But why are these primes so elusive? As \( p \) grows, the numbers become astronomically large, making brute-force testing impossible. This is where heuristic computation steps in—guiding us with smart, experience-driven strategies. “In the infinite sea of numbers, heuristics are our compass.” Let’s explore how heuristics and algorithms intertwine to unveil these mathematical treasures. 1. Mersenne Primes — Giants of Number Theory Definition: Numbers of the form \( M_p = 2^p - 1 \...
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Understanding the Laplacian of 1/r and the Dirac Delta Function Mathematical Foundations & SageMath Insights

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Unmasking the Laplacian: How Mathematics, Physics & AI Use This Powerful Operator Unveiling the Laplacian's Secrets: A Look at 1/r with SageMath Engage & Explore! Before we dive into the math, ask yourself: What does it mean when a function explodes to infinity at a point? Leave your thoughts in the comments below. We'll revisit this after exploring the Dirac delta function! Why Is the Laplacian of \( \frac{1}{r} \) Important? The function \( \frac{1}{r} \) frequently appears in physics to describe potentials like gravity and electrostatics, which depend inversely on distance. Understanding its Laplacian reveals the nature of sources concentrated at singular points. Mathematical Foundation: What Is the Laplacian and a Harmonic Function? The Laplacian operat...
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Neural Network Generalization in the Over-Parameterization Regime: Mechanisms, Benefits, and Limitations

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Neural Network Generalization in the Over-Parameterization Regime: Mechanisms, Benefits, and Limitations Neural Network Generalization in the Over-Parameterization Regime: Mechanisms, Benefits, and Limitations Introduction Over the past decade, deep neural networks (DNNs) have risen to prominence across a range of machine learning applications, achieving remarkable performance in domains such as computer vision, natural language processing, and reinforcement learning. A striking and counter-intuitive feature of modern DNNs is their propensity for over-parameterization: models often contain many more parameters than training samples, far exceeding the classical regime where statistical learning theory would predict rampant overfitting and poor generalization. Yet, these highly over-parameterized models not only fit the training data perfectly but also display outstanding generalization to unseen test data—often improving as the number of paramete...
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