A random walk on graphs provides a framework for ranking nodes based on their importance. The document discusses random walks, stationary distributions, and how the stationary distribution represents the probability that a random walker is at a node after a long time. It also covers properties like the Perron-Frobenius theorem and relationships to electrical networks, hitting times, and commute times. Applications include PageRank and recommendations.
Random walks are stochastic processes that can model many natural phenomena. A random walk is generated by successive random steps on a mathematical structure like integers or graphs. Random walks can simulate processes like molecular motion or animal foraging. They have applications in fields like recommender systems, investment theory, and generating fractal images. A random walk on a graph corresponds to a Markov chain, with transition probabilities defined by the graph structure. Random walks approach a unique stationary distribution if the graph is connected and aperiodic. The mixing time measures how fast this convergence occurs. Random walk algorithms are used for tasks like ranking genes by likelihood of having a property or learning vertex embeddings in networks.
Presented "Random Walk on Graphs" in the reading group for Knoesis. Specifically for Recommendation Context.
Referred: Purnamrita Sarkar, Random Walks on Graphs: An Overview
Linear regression with gradient descentSuraj Parmar
Intro to the very popular optimization Technique(Gradient descent) with linear regression . Linear regression with Gradient descent on www.landofai.com
How to validate a model?
What is a best model ?
Types of data
Types of errors
The problem of over fitting
The problem of under fitting
Bias Variance Tradeoff
Cross validation
K-Fold Cross validation
Boot strap Cross validation
This document provides an overview of stochastic processes and Markov chains. It defines stochastic processes as families of random variables indexed by time. Markov chains are a type of stochastic process where the future state depends only on the present state, not on the past. The document discusses examples of Markov chains, transition matrices, classification of states as transient or persistent, and properties like irreducibility. It aims to introduce key concepts in stochastic processes and Markov chains.
Mining Frequent Patterns, Association and CorrelationsJustin Cletus
This document summarizes Chapter 6 of the book "Data Mining: Concepts and Techniques" which discusses frequent pattern mining. It introduces basic concepts like frequent itemsets and association rules. It then describes several scalable algorithms for mining frequent itemsets, including Apriori, FP-Growth, and ECLAT. It also discusses optimizations to Apriori like partitioning the database and techniques to reduce the number of candidates and database scans.
Introduction to Maximum Likelihood EstimatorAmir Al-Ansary
This document provides an overview of maximum likelihood estimation (MLE). It discusses key concepts like probability models, parameters, and the likelihood function. MLE aims to find the parameter values that make the observed data most likely. This can be done analytically by taking derivatives or numerically using optimization algorithms. Practical considerations like removing constants and using the log-likelihood are also covered. The document concludes by introducing the likelihood ratio test for comparing nested models.
This document discusses linear regression with one variable using gradient descent. It introduces linear regression as a machine learning algorithm for predicting a real-valued output given a single input variable. Gradient descent is presented as an approach for minimizing the cost function to find the optimal parameters for the linear regression model. The gradient descent algorithm is explained as iteratively updating the parameters in the direction of the negative gradient to reduce the cost until reaching a minimum.
Random walks are stochastic processes that can model many natural phenomena. A random walk is generated by successive random steps on a mathematical structure like integers or graphs. Random walks can simulate processes like molecular motion or animal foraging. They have applications in fields like recommender systems, investment theory, and generating fractal images. A random walk on a graph corresponds to a Markov chain, with transition probabilities defined by the graph structure. Random walks approach a unique stationary distribution if the graph is connected and aperiodic. The mixing time measures how fast this convergence occurs. Random walk algorithms are used for tasks like ranking genes by likelihood of having a property or learning vertex embeddings in networks.
Presented "Random Walk on Graphs" in the reading group for Knoesis. Specifically for Recommendation Context.
Referred: Purnamrita Sarkar, Random Walks on Graphs: An Overview
Linear regression with gradient descentSuraj Parmar
Intro to the very popular optimization Technique(Gradient descent) with linear regression . Linear regression with Gradient descent on www.landofai.com
How to validate a model?
What is a best model ?
Types of data
Types of errors
The problem of over fitting
The problem of under fitting
Bias Variance Tradeoff
Cross validation
K-Fold Cross validation
Boot strap Cross validation
This document provides an overview of stochastic processes and Markov chains. It defines stochastic processes as families of random variables indexed by time. Markov chains are a type of stochastic process where the future state depends only on the present state, not on the past. The document discusses examples of Markov chains, transition matrices, classification of states as transient or persistent, and properties like irreducibility. It aims to introduce key concepts in stochastic processes and Markov chains.
Mining Frequent Patterns, Association and CorrelationsJustin Cletus
This document summarizes Chapter 6 of the book "Data Mining: Concepts and Techniques" which discusses frequent pattern mining. It introduces basic concepts like frequent itemsets and association rules. It then describes several scalable algorithms for mining frequent itemsets, including Apriori, FP-Growth, and ECLAT. It also discusses optimizations to Apriori like partitioning the database and techniques to reduce the number of candidates and database scans.
Introduction to Maximum Likelihood EstimatorAmir Al-Ansary
This document provides an overview of maximum likelihood estimation (MLE). It discusses key concepts like probability models, parameters, and the likelihood function. MLE aims to find the parameter values that make the observed data most likely. This can be done analytically by taking derivatives or numerically using optimization algorithms. Practical considerations like removing constants and using the log-likelihood are also covered. The document concludes by introducing the likelihood ratio test for comparing nested models.
This document discusses linear regression with one variable using gradient descent. It introduces linear regression as a machine learning algorithm for predicting a real-valued output given a single input variable. Gradient descent is presented as an approach for minimizing the cost function to find the optimal parameters for the linear regression model. The gradient descent algorithm is explained as iteratively updating the parameters in the direction of the negative gradient to reduce the cost until reaching a minimum.
Logistic regression is a machine learning classification algorithm that predicts the probability of a categorical dependent variable. It models the probability of the dependent variable being in one of two possible categories, as a function of the independent variables. The model transforms the linear combination of the independent variables using the logistic sigmoid function to output a probability between 0 and 1. Logistic regression is optimized using maximum likelihood estimation to find the coefficients that maximize the probability of the observed outcomes in the training data. Like linear regression, it makes assumptions about the data being binary classified with no noise or highly correlated independent variables.
Stochastic modelling and its applicationsKartavya Jain
Stochastic processes and modelling have various applications in telecommunications. Token rings, continuous-time Markov chains, and fluid-flow models are used to model traffic flow and network performance. Aggregate dynamic stochastic models can model air traffic control by representing aircraft arrivals as Poisson processes. Disturbances like weather can be incorporated by altering flow rates. Wireless network models use search algorithms and location stochastic processes to track mobile users.
This presentation is aimed at fitting a Simple Linear Regression model in a Python program. IDE used is Spyder. Screenshots from a working example are used for demonstration.
This document provides a summary of Markov chains. It begins by defining stochastic processes and Markov chains. A Markov chain is a stochastic process where the probability of the next state depends only on the current state, not on the sequence of events that preceded it. The document discusses n-step transition probabilities, classification of states, and steady-state probabilities. It provides examples of Markov chains for cola purchases and camera store inventory to illustrate the concepts.
probabilistic, reasoning, artificial, computer, intelligence, IOE, Sushant, Pulchowk, AI,
Statistical techniques used in practical data analysis. e.g. t-tests, ANOVA, regression, correlation;
The use of probabilistic models in psychology and linguistics
Machine learning and computational linguistics/NLP
Measure theory (in fact, almost anything involving infinite sets)
Using logic and probability to handle uncertain situation
Probability based reasoning is same as understanding directly from the knowledge that a given probability rating based on uncertainty present
This document provides an introduction to logistic regression. It outlines key features such as using a logistic function to model a binary dependent variable that can take on values of 0 or 1. Logistic regression is a linear method that uses the logistic function to transform predictions. The document discusses applications in machine learning, medical science, social science, and industry. It also provides details on logistic regression models, including converting linear variables to logistic variables using a sigmoid function and examining the effects of varying the logistic growth and midpoint parameters on the logistic regression curve.
This presentation deals with the formal presentation of anomaly detection and outlier analysis and types of anomalies and outliers. Different approaches to tackel anomaly detection problems.
Truth maintenance systems (TMS) are used to represent beliefs and their justifications in a knowledge base and maintain consistency when new information is added or inferred facts are retracted. A TMS models inferences as a dependency network and uses an algorithm to track which inferences need to be retracted if assumptions change. It differs from traditional logic which treats all facts equally. Nonmonotonic logics also use TMS to represent default reasoning and maintain beliefs when assumptions are defeated. Fuzzy logic uses similar concepts to handle imprecise values and measurements in systems like control applications.
Mrbml004 : Introduction to Information Theory for Machine LearningJaouad Dabounou
La quatrième séance de lecture de livres en machine learning.
Vidéo : https://youtu.be/Ab5RvD7ieFg
Elle concernera une brève introduction à la théorie de l'information: Entropy, K-L divergence, mutual Information,... et son application dans la fonction de perte et notamment la cross-entropy.
Lecture de trois livres, dans le cadre de "Monday reading books on machine learning".
Le premier livre, qui constituera le fil conducteur de toute l'action :
Christopher Bishop; Pattern Recognition and Machine Learning, Springer-Verlag New York Inc, 2006
Seront utilisées des parties de deux livres, surtout du livre :
Ian Goodfellow, Yoshua Bengio, Aaron Courville; Deep Learning, The MIT Press, 2016
et du livre :
Ovidiu Calin; Deep Learning Architectures: A Mathematical Approach, Springer, 2020
This document discusses Markov chains, which are a type of stochastic process used to model randomly changing systems. It defines Markov chains and their key properties, like the Markov property and transition probabilities. It provides examples like modeling customer purchases over time and inventory management. It also covers concepts like steady state probabilities, transition matrices, and mean first passage times.
the fuzzy logic is very important for any automated or intelligent system.we are training the system with the help of fuzzy logic and fuzzy system so that it can behave and think like human beings. so, in this slide fuzzy inference system has been explained with some numerial problem.
Anomaly detection (or Outlier analysis) is the identification of items, events or observations which do not conform to an expected pattern or other items in a dataset. It is used is applications such as intrusion detection, fraud detection, fault detection and monitoring processes in various domains including energy, healthcare and finance. In this talk, we will introduce anomaly detection and discuss the various analytical and machine learning techniques used in in this field. Through a case study, we will discuss how anomaly detection techniques could be applied to energy data sets. We will also demonstrate, using R and Apache Spark, an application to help reinforce concepts in anomaly detection and best practices in analyzing and reviewing results.
This document discusses probabilistic models used for text mining. It introduces mixture models, Bayesian nonparametric models, and graphical models including Bayesian networks, hidden Markov models, Markov random fields, and conditional random fields. It provides details on the general framework of mixture models and examples like topic models PLSA and LDA. It also discusses learning algorithms for probabilistic models like EM algorithm and Gibbs sampling.
The document discusses representing knowledge using predicate logic. It provides examples of representing simple facts and relationships using predicates, variables, and quantification. It also discusses issues like representing classes and instances, exceptions to general rules, and using computable predicates to efficiently represent relationships that can be computed rather than explicitly stated.
Lect6 Association rule & Apriori algorithmhktripathy
The document discusses the Apriori algorithm for mining association rules from transactional data. The Apriori algorithm uses a level-wise search where frequent itemsets are used to explore longer itemsets. It determines frequent itemsets by identifying individual frequent items and extending them to larger sets as long as they meet a minimum support threshold. The algorithm takes advantage of the fact that subsets of frequent itemsets must also be frequent to prune the search space. It performs candidate generation and pruning to efficiently identify all frequent itemsets in the transactional data.
Bayes' Theorem relates prior probabilities, conditional probabilities, and posterior probabilities. It provides a mathematical rule for updating estimates based on new evidence or observations. The theorem states that the posterior probability of an event is equal to the conditional probability of the event given the evidence multiplied by the prior probability, divided by the probability of the evidence. Bayes' Theorem can be used to calculate conditional probabilities, like the probability of a woman having breast cancer given a positive mammogram result, or the probability that a part came from a specific supplier given that it is non-defective. It is widely applicable in science, medicine, and other fields for revising hypotheses based on new data.
The document introduces basic probability concepts and provides examples to illustrate them. It discusses the key properties of probability, types of probability (objective and subjective), mutually exclusive and collectively exhaustive events, and probabilities of independent and dependent events. It also explains Bayes' theorem and how it can be used to update probabilities as new information becomes available.
The exponential probability distribution is useful for describing the time it takes to complete random tasks. It can model the time between events like vehicle arrivals at a toll booth, time to complete a survey, or distance between defects on a highway. The distribution is defined by a probability density function that uses the mean time or rate of the process. It can calculate the probability that an event will occur within a certain time threshold, like the chance a car will arrive at a gas pump within 2 minutes. The mean and standard deviation of the exponential distribution are equal, and it is an extremely skewed distribution without a defined mode.
Improving predictions: Lasso, Ridge and Stein's paradoxMaarten van Smeden
Slides of masterclass "Improving predictions: Lasso, Ridge and Stein's paradox" at the (Dutch) National Institute for Public Health and the Environment (RIVM)
Random walks on graphs - link prediction by Rouhollah Nabatinabati
This document provides an overview of random walks on graphs and their applications. It begins with basic definitions such as adjacency matrix, transition matrix, and Laplacian matrix. It then discusses properties like the stationary distribution and Perron-Frobenius theorem. Random walks are related to electrical networks, where hitting times correspond to voltages. Commute times and Laplacians are also connected. Applications include PageRank and recommender systems that use proximity measures from random walks. The document outlines key topics and provides examples to explain random walk concepts and their significance.
Logistic regression is a machine learning classification algorithm that predicts the probability of a categorical dependent variable. It models the probability of the dependent variable being in one of two possible categories, as a function of the independent variables. The model transforms the linear combination of the independent variables using the logistic sigmoid function to output a probability between 0 and 1. Logistic regression is optimized using maximum likelihood estimation to find the coefficients that maximize the probability of the observed outcomes in the training data. Like linear regression, it makes assumptions about the data being binary classified with no noise or highly correlated independent variables.
Stochastic modelling and its applicationsKartavya Jain
Stochastic processes and modelling have various applications in telecommunications. Token rings, continuous-time Markov chains, and fluid-flow models are used to model traffic flow and network performance. Aggregate dynamic stochastic models can model air traffic control by representing aircraft arrivals as Poisson processes. Disturbances like weather can be incorporated by altering flow rates. Wireless network models use search algorithms and location stochastic processes to track mobile users.
This presentation is aimed at fitting a Simple Linear Regression model in a Python program. IDE used is Spyder. Screenshots from a working example are used for demonstration.
This document provides a summary of Markov chains. It begins by defining stochastic processes and Markov chains. A Markov chain is a stochastic process where the probability of the next state depends only on the current state, not on the sequence of events that preceded it. The document discusses n-step transition probabilities, classification of states, and steady-state probabilities. It provides examples of Markov chains for cola purchases and camera store inventory to illustrate the concepts.
probabilistic, reasoning, artificial, computer, intelligence, IOE, Sushant, Pulchowk, AI,
Statistical techniques used in practical data analysis. e.g. t-tests, ANOVA, regression, correlation;
The use of probabilistic models in psychology and linguistics
Machine learning and computational linguistics/NLP
Measure theory (in fact, almost anything involving infinite sets)
Using logic and probability to handle uncertain situation
Probability based reasoning is same as understanding directly from the knowledge that a given probability rating based on uncertainty present
This document provides an introduction to logistic regression. It outlines key features such as using a logistic function to model a binary dependent variable that can take on values of 0 or 1. Logistic regression is a linear method that uses the logistic function to transform predictions. The document discusses applications in machine learning, medical science, social science, and industry. It also provides details on logistic regression models, including converting linear variables to logistic variables using a sigmoid function and examining the effects of varying the logistic growth and midpoint parameters on the logistic regression curve.
This presentation deals with the formal presentation of anomaly detection and outlier analysis and types of anomalies and outliers. Different approaches to tackel anomaly detection problems.
Truth maintenance systems (TMS) are used to represent beliefs and their justifications in a knowledge base and maintain consistency when new information is added or inferred facts are retracted. A TMS models inferences as a dependency network and uses an algorithm to track which inferences need to be retracted if assumptions change. It differs from traditional logic which treats all facts equally. Nonmonotonic logics also use TMS to represent default reasoning and maintain beliefs when assumptions are defeated. Fuzzy logic uses similar concepts to handle imprecise values and measurements in systems like control applications.
Mrbml004 : Introduction to Information Theory for Machine LearningJaouad Dabounou
La quatrième séance de lecture de livres en machine learning.
Vidéo : https://youtu.be/Ab5RvD7ieFg
Elle concernera une brève introduction à la théorie de l'information: Entropy, K-L divergence, mutual Information,... et son application dans la fonction de perte et notamment la cross-entropy.
Lecture de trois livres, dans le cadre de "Monday reading books on machine learning".
Le premier livre, qui constituera le fil conducteur de toute l'action :
Christopher Bishop; Pattern Recognition and Machine Learning, Springer-Verlag New York Inc, 2006
Seront utilisées des parties de deux livres, surtout du livre :
Ian Goodfellow, Yoshua Bengio, Aaron Courville; Deep Learning, The MIT Press, 2016
et du livre :
Ovidiu Calin; Deep Learning Architectures: A Mathematical Approach, Springer, 2020
This document discusses Markov chains, which are a type of stochastic process used to model randomly changing systems. It defines Markov chains and their key properties, like the Markov property and transition probabilities. It provides examples like modeling customer purchases over time and inventory management. It also covers concepts like steady state probabilities, transition matrices, and mean first passage times.
the fuzzy logic is very important for any automated or intelligent system.we are training the system with the help of fuzzy logic and fuzzy system so that it can behave and think like human beings. so, in this slide fuzzy inference system has been explained with some numerial problem.
Anomaly detection (or Outlier analysis) is the identification of items, events or observations which do not conform to an expected pattern or other items in a dataset. It is used is applications such as intrusion detection, fraud detection, fault detection and monitoring processes in various domains including energy, healthcare and finance. In this talk, we will introduce anomaly detection and discuss the various analytical and machine learning techniques used in in this field. Through a case study, we will discuss how anomaly detection techniques could be applied to energy data sets. We will also demonstrate, using R and Apache Spark, an application to help reinforce concepts in anomaly detection and best practices in analyzing and reviewing results.
This document discusses probabilistic models used for text mining. It introduces mixture models, Bayesian nonparametric models, and graphical models including Bayesian networks, hidden Markov models, Markov random fields, and conditional random fields. It provides details on the general framework of mixture models and examples like topic models PLSA and LDA. It also discusses learning algorithms for probabilistic models like EM algorithm and Gibbs sampling.
The document discusses representing knowledge using predicate logic. It provides examples of representing simple facts and relationships using predicates, variables, and quantification. It also discusses issues like representing classes and instances, exceptions to general rules, and using computable predicates to efficiently represent relationships that can be computed rather than explicitly stated.
Lect6 Association rule & Apriori algorithmhktripathy
The document discusses the Apriori algorithm for mining association rules from transactional data. The Apriori algorithm uses a level-wise search where frequent itemsets are used to explore longer itemsets. It determines frequent itemsets by identifying individual frequent items and extending them to larger sets as long as they meet a minimum support threshold. The algorithm takes advantage of the fact that subsets of frequent itemsets must also be frequent to prune the search space. It performs candidate generation and pruning to efficiently identify all frequent itemsets in the transactional data.
Bayes' Theorem relates prior probabilities, conditional probabilities, and posterior probabilities. It provides a mathematical rule for updating estimates based on new evidence or observations. The theorem states that the posterior probability of an event is equal to the conditional probability of the event given the evidence multiplied by the prior probability, divided by the probability of the evidence. Bayes' Theorem can be used to calculate conditional probabilities, like the probability of a woman having breast cancer given a positive mammogram result, or the probability that a part came from a specific supplier given that it is non-defective. It is widely applicable in science, medicine, and other fields for revising hypotheses based on new data.
The document introduces basic probability concepts and provides examples to illustrate them. It discusses the key properties of probability, types of probability (objective and subjective), mutually exclusive and collectively exhaustive events, and probabilities of independent and dependent events. It also explains Bayes' theorem and how it can be used to update probabilities as new information becomes available.
The exponential probability distribution is useful for describing the time it takes to complete random tasks. It can model the time between events like vehicle arrivals at a toll booth, time to complete a survey, or distance between defects on a highway. The distribution is defined by a probability density function that uses the mean time or rate of the process. It can calculate the probability that an event will occur within a certain time threshold, like the chance a car will arrive at a gas pump within 2 minutes. The mean and standard deviation of the exponential distribution are equal, and it is an extremely skewed distribution without a defined mode.
Improving predictions: Lasso, Ridge and Stein's paradoxMaarten van Smeden
Slides of masterclass "Improving predictions: Lasso, Ridge and Stein's paradox" at the (Dutch) National Institute for Public Health and the Environment (RIVM)
Random walks on graphs - link prediction by Rouhollah Nabatinabati
This document provides an overview of random walks on graphs and their applications. It begins with basic definitions such as adjacency matrix, transition matrix, and Laplacian matrix. It then discusses properties like the stationary distribution and Perron-Frobenius theorem. Random walks are related to electrical networks, where hitting times correspond to voltages. Commute times and Laplacians are also connected. Applications include PageRank and recommender systems that use proximity measures from random walks. The document outlines key topics and provides examples to explain random walk concepts and their significance.
The document discusses Markov chains and their relationship to random walks on graphs and electrical networks. Some key points:
- A Markov chain is a process that transitions between a finite set of states based on transition probabilities that depend only on the current state.
- For a strongly connected Markov chain, there exists a unique stationary distribution that the long-term probabilities of the chain converge to, regardless of the starting state.
- Random walks on undirected graphs can be modeled as Markov chains, where the transition probabilities are proportional to edge conductances in an analogous electrical network. The stationary distribution of such a random walk is proportional to vertex degrees or conductances.
Tensor Spectral Clustering is an algorithm that generalizes graph partitioning and spectral clustering methods to account for higher-order network structures. It defines a new objective function called motif conductance that measures how partitions cut motifs like triangles in addition to edges. The algorithm represents a tensor of higher-order random walk transitions as a matrix and computes eigenvectors to find a partition that minimizes the number of motifs cut, allowing networks to be clustered based on higher-order connectivity patterns. Experiments on synthetic and real networks show it can discover meaningful partitions by accounting for motifs that capture important structural relationships.
The document discusses using neural networks to solve the traveling salesman problem (TSP). It describes Hopfield neural networks and how they can be used for optimization problems like TSP. It then discusses a concurrent neural network approach that requires fewer neurons (N(logN)) than Hopfield for TSP. The document compares the performance of concurrent neural networks to Hopfield networks and self-organizing maps on TSP test cases, finding concurrent networks converge faster but are less reliable than self-organizing maps.
The document summarizes research on spacey random walks, which are a type of stochastic process that can model higher-order Markov chains. Key points:
1. Spacey random walks generalize higher-order Markov chains by forgetting history but pretending to remember a random previous state, with the stationary distribution given by a tensor eigenvector of the transition tensor.
2. This connects higher-order Markov chains to tensor eigenvectors and provides a stochastic interpretation of tensor eigenvectors as stationary distributions.
3. The dynamics of spacey random walks can be modeled as an ordinary differential equation, allowing tensor eigenvectors to be computed by numerically integrating the dynamical system.
This document provides an introduction to quantum Monte Carlo methods. It discusses using Monte Carlo integration to evaluate multi-dimensional integrals that arise in quantum mechanical problems. Variational Monte Carlo is introduced as using a trial wavefunction to sample configuration space and estimate observables, like the energy. The Metropolis algorithm is described as a way to generate Markov chains that sample a given probability distribution. This allows using Monte Carlo methods to solve the electronic structure problem by approximating many-body wavefunctions and integrals over configuration space.
240401_JW_labseminar[LINE: Large-scale Information Network Embeddin].pptxthanhdowork
This document presents the LINE algorithm for embedding large-scale information networks. LINE aims to represent each network node in a low-dimensional space while preserving both first-order and second-order proximities. It defines objective functions to minimize the distance between the empirical and learned joint/conditional distributions. An edge sampling method is used to optimize the objective functions efficiently in large networks. Experiments on language, social, and citation networks demonstrate that LINE can scale to networks with millions of nodes and billions of edges while effectively preserving network structure.
Fourier-transform analysis of a unilateral fin line and its derivativesYong Heui Cho
This document presents a Fourier-transform analysis of a unilateral n-line and its derivatives. The key points are:
1) A unilateral n-line is transformed into an equivalent problem of multiple suspended substrate microstrip lines using the image theorem and Fourier transform.
2) New rigorous dispersion relations are derived in the form of rapidly convergent series solutions, providing an accurate yet efficient method for numerical computation.
3) The dispersion relations for derivatives of the unilateral n-line including suspended substrate microstrip lines, microstrip lines, slot lines and coplanar waveguides are also presented.
This document provides an overview of quantum computing, including:
- Quantum computers store and process information using quantum bits (qubits) that can exist in superpositions of states allowing exponential increases in processing power over classical computers.
- Key concepts include qubit representation and superpositions, entanglement, measurement and computational complexity classes like BQP.
- Quantum algorithms show exponential speedups over classical for factoring, discrete log, and some other problems.
- Implementation challenges include building reliable qubits, controlling operations, and error correction. Leading approaches use trapped ions, NMR, photonics, and solid state systems.
The document discusses ab initio molecular dynamics simulation methods. It begins by introducing molecular dynamics and Monte Carlo simulations using empirical potentials. It then describes limitations of empirical potentials and the need for ab initio molecular dynamics which calculates the potential from quantum mechanics. The document outlines several ab initio molecular dynamics methods including Ehrenfest molecular dynamics, Born-Oppenheimer molecular dynamics, and Car-Parrinello molecular dynamics. It provides details on how these methods treat the quantum mechanical potential and classical nuclear motion.
1. The document discusses entanglement generation and state transfer in a Heisenberg spin-1/2 chain under an external magnetic field.
2. It analyzes the fidelity and concurrence of the system over time and temperature using the density matrix and Hamiltonian equations for a 2-qubit system.
3. The results show that maximally entangled states are difficult to achieve but desirable for quantum computation applications like quantum teleportation.
A quantum computer uses quantum mechanics phenomena like superposition and entanglement to perform computations. In a quantum computer, a qubit can represent a 0 and 1 simultaneously using superposition. This allows quantum computers to evaluate functions on all possible inputs at once. Measurement causes the superposition to collapse to a single value. Quantum computers may be able to solve certain problems like factoring exponentially faster than classical computers due to these quantum effects. However, building large-scale, reliable quantum computers remains a significant technical challenge.
This document is Scott Shermer's master's thesis on instantons and perturbation theory in the 1-D quantum mechanical quartic oscillator. It begins by reviewing the harmonic oscillator and perturbation theory. It then discusses non-perturbative phenomena like instantons and Borel resummation. The focus is on obtaining the ground state energy of the quartic oscillator Hamiltonian using both perturbative and non-perturbative techniques, and addressing ambiguities that arise for negative coupling.
This document provides resources and an overview of topics for a course on crystallography and protein structure determination using X-ray crystallography. The course will involve crystallizing a protein, collecting data at the Advanced Light Source, and determining the protein's atomic structure. Key topics covered include X-ray scattering, the phase problem, structure refinement, and sources of errors. Online resources and contacts are provided for computing, tutorials, and beamline information.
1. The document discusses x-ray crystallography techniques for determining atomic structures, including crystallizing a protein, collecting diffraction data at an advanced light source, and determining structures.
2. Key aspects covered are x-ray diffraction, where x-rays scatter off electrons and the resolution is typically 1-3 Angstroms. Determining phases is also discussed as a challenge.
3. Resources provided include computing tools, online courses, and contacts for further information on crystallography techniques and resources.
Applications of artificial Intelligence in Mechanical Engineering.pdfAtif Razi
Historically, mechanical engineering has relied heavily on human expertise and empirical methods to solve complex problems. With the introduction of computer-aided design (CAD) and finite element analysis (FEA), the field took its first steps towards digitization. These tools allowed engineers to simulate and analyze mechanical systems with greater accuracy and efficiency. However, the sheer volume of data generated by modern engineering systems and the increasing complexity of these systems have necessitated more advanced analytical tools, paving the way for AI.
AI offers the capability to process vast amounts of data, identify patterns, and make predictions with a level of speed and accuracy unattainable by traditional methods. This has profound implications for mechanical engineering, enabling more efficient design processes, predictive maintenance strategies, and optimized manufacturing operations. AI-driven tools can learn from historical data, adapt to new information, and continuously improve their performance, making them invaluable in tackling the multifaceted challenges of modern mechanical engineering.
Rainfall intensity duration frequency curve statistical analysis and modeling...bijceesjournal
Using data from 41 years in Patna’ India’ the study’s goal is to analyze the trends of how often it rains on a weekly, seasonal, and annual basis (1981−2020). First, utilizing the intensity-duration-frequency (IDF) curve and the relationship by statistically analyzing rainfall’ the historical rainfall data set for Patna’ India’ during a 41 year period (1981−2020), was evaluated for its quality. Changes in the hydrologic cycle as a result of increased greenhouse gas emissions are expected to induce variations in the intensity, length, and frequency of precipitation events. One strategy to lessen vulnerability is to quantify probable changes and adapt to them. Techniques such as log-normal, normal, and Gumbel are used (EV-I). Distributions were created with durations of 1, 2, 3, 6, and 24 h and return times of 2, 5, 10, 25, and 100 years. There were also mathematical correlations discovered between rainfall and recurrence interval.
Findings: Based on findings, the Gumbel approach produced the highest intensity values, whereas the other approaches produced values that were close to each other. The data indicates that 461.9 mm of rain fell during the monsoon season’s 301st week. However, it was found that the 29th week had the greatest average rainfall, 92.6 mm. With 952.6 mm on average, the monsoon season saw the highest rainfall. Calculations revealed that the yearly rainfall averaged 1171.1 mm. Using Weibull’s method, the study was subsequently expanded to examine rainfall distribution at different recurrence intervals of 2, 5, 10, and 25 years. Rainfall and recurrence interval mathematical correlations were also developed. Further regression analysis revealed that short wave irrigation, wind direction, wind speed, pressure, relative humidity, and temperature all had a substantial influence on rainfall.
Originality and value: The results of the rainfall IDF curves can provide useful information to policymakers in making appropriate decisions in managing and minimizing floods in the study area.
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
The document further emphasizes on the importance of checking the connection between the Windows and WSL environments, providing instructions on how to ensure that the connection is optimal and ready for remote debugging.
It also offers an in-depth guide on how to configure the WSL interpreter and files within the PyCharm environment. This is essential for ensuring that the debugging process is set up correctly and that the program can be run effectively within the WSL terminal.
Additionally, the document provides guidance on how to set up breakpoints for debugging, a fundamental aspect of the debugging process which allows the developer to stop the execution of their code at certain points and inspect their program at those stages.
Finally, the document concludes by providing a link to a reference blog. This blog offers additional information and guidance on configuring the remote Python interpreter in PyCharm, providing the reader with a well-rounded understanding of the process.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
artificial intelligence and data science contents.pptxGauravCar
What is artificial intelligence? Artificial intelligence is the ability of a computer or computer-controlled robot to perform tasks that are commonly associated with the intellectual processes characteristic of humans, such as the ability to reason.
› ...
Artificial intelligence (AI) | Definitio
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
5. 5
Why graphs?
The underlying data is naturally a graph
Papers linked by citation
Authors linked by co-authorship
Bipartite graph of customers and products
Web-graph
Friendship networks: who knows whom
6. 6
What are we looking for
Rank nodes for a particular query
Top k matches for “Random Walks” from Citeseer
Who are the most likely co-authors of “Manuel
Blum”.
Top k book recommendations for Purna from
Amazon
Top k websites matching “Sound of Music”
Top k friend recommendations for Purna when she
joins “Facebook”
8. 8
Definitions
nxn Adjacency matrix A.
A(i,j) = weight on edge from i to j
If the graph is undirected A(i,j)=A(j,i), i.e. A is symmetric
nxn Transition matrix P.
P is row stochastic
P(i,j) = probability of stepping on node j from node i
= A(i,j)/∑iA(i,j)
nxn Laplacian Matrix L.
L(i,j)=∑iA(i,j)-A(i,j)
Symmetric positive semi-definite for undirected graphs
Singular
11. 11
What is a random walk
1
1/2
1/2
1
1
1/2
1/2
1
t=0 t=1
12. 12
What is a random walk
1
1/2
1/2
1
1
1/2
1/2
1
t=0 t=1
1
1/2
1/2
1
t=2
13. 13
What is a random walk
1
1/2
1/2
1
1
1/2
1/2
1
t=0 t=1
1
1/2
1/2
1
t=2
1
1/2
1/2
1
t=3
14. 14
Probability Distributions
xt(i) = probability that the surfer is at node i at time
t
xt+1(i) = ∑j(Probability of being at node j)*Pr(j->i)
=∑jxt(j)*P(j,i)
xt+1 = xtP = xt-1*P*P= xt-2*P*P*P = …=x0 Pt
What happens when the surfer keeps walking for a
long time?
15. 15
Stationary Distribution
When the surfer keeps walking for a long time
When the distribution does not change anymore
i.e. xT+1 = xT
For “well-behaved” graphs this does not depend on
the start distribution!!
16. 16
What is a stationary distribution?
Intuitively and Mathematically
17. 17
What is a stationary distribution?
Intuitively and Mathematically
The stationary distribution at a node is related to the
amount of time a random walker spends visiting that
node.
18. 18
What is a stationary distribution?
Intuitively and Mathematically
The stationary distribution at a node is related to the
amount of time a random walker spends visiting that
node.
Remember that we can write the probability
distribution at a node as
xt+1 = xtP
19. 19
What is a stationary distribution?
Intuitively and Mathematically
The stationary distribution at a node is related to the
amount of time a random walker spends visiting that
node.
Remember that we can write the probability
distribution at a node as
xt+1 = xtP
For the stationary distribution v0 we have
v0 = v0 P
20. 20
What is a stationary distribution?
Intuitively and Mathematically
The stationary distribution at a node is related to the
amount of time a random walker spends visiting that
node.
Remember that we can write the probability
distribution at a node as
xt+1 = xtP
For the stationary distribution v0 we have
v0 = v0 P
Whoa! that’s just the left eigenvector of the
transition matrix !
22. 22
Interesting questions
Does a stationary distribution always exist? Is it
unique?
Yes, if the graph is “well-behaved”.
What is “well-behaved”?
We shall talk about this soon.
How fast will the random surfer approach this
stationary distribution?
Mixing Time!
23. 23
Well behaved graphs
Irreducible: There is a path from every node to every
other node.
Irreducible Not irreducible
24. 24
Well behaved graphs
Aperiodic: The GCD of all cycle lengths is 1. The GCD
is also called period.
Aperiodic
Periodicity is 3
25. 25
Implications of the Perron Frobenius
Theorem
If a markov chain is irreducible and aperiodic then
the largest eigenvalue of the transition matrix will be
equal to 1 and all the other eigenvalues will be strictly
less than 1.
Let the eigenvalues of P be {σi| i=0:n-1} in non-increasing
order of σi .
σ0 = 1 > σ1 > σ2 >= ……>= σn
26. 26
Implications of the Perron Frobenius
Theorem
If a markov chain is irreducible and aperiodic then
the largest eigenvalue of the transition matrix will be
equal to 1 and all the other eigenvalues will be strictly
less than 1.
Let the eigenvalues of P be {σi| i=0:n-1} in non-increasing
order of σi .
σ0 = 1 > σ1 > σ2 >= ……>= σn
These results imply that for a well behaved graph
there exists an unique stationary distribution.
More details when we discuss pagerank.
27. 27
Some fun stuff about undirected
graphs
A connected undirected graph is irreducible
A connected non-bipartite undirected graph has a
stationary distribution proportional to the degree
distribution!
Makes sense, since larger the degree of the node
more likely a random walk is to come back to it.
29. 29
Proximity measures from random walks
How long does it take to hit node b in a random walk
starting at node a ? Hitting time.
How long does it take to hit node b and come back to
node a ? Commute time.
a
b
30. 30
Hitting and Commute times
Hitting time from node i to node j
Expected number of hops to hit node j starting at node i.
Is not symmetric. h(a,b) > h(a,b)
h(i,j) = 1 + ΣkЄnbs(A) p(i,k)h(k,j)
a
b
31. 31
Hitting and Commute times
Commute time between node i and j
Is expected time to hit node j and come back to i
c(i,j) = h(i,j) + h(j,i)
Is symmetric. c(a,b) = c(b,a)
a
b
32. 32
Relationship with Electrical
networks1,2
Consider the graph as a n-node
resistive network.
Each edge is a resistor of 1 Ohm.
Degree of a node is number of
neighbors
Sum of degrees = 2*m
m being the number of edges
1. Random Walks and Electric Networks , Doyle and Snell, 1984
2. The Electrical Resistance Of A Graph Captures Its Commute And Cover Times, Ashok K. Chandra, Prabhakar Raghavan,
Walter L. Ruzzo, Roman Smolensky, Prasoon Tiwari, 1989
33. 33
Relationship with Electrical networks
Inject d(i) amp current in
each node
Extract 2m amp current from
node j.
Now what is the voltage
difference between i and j ?
i j
3
3
2
2
2
16
4
34. 34
Relationship with Electrical networks
Whoa!! Hitting time from i to
j is exactly the voltage drop
when you inject respective
degree amount of current in
every node and take out 2*m
from j!
i j
3
3
2
2
2
4
16
35. 35
Relationship with Electrical networks
Consider neighbors of i i.e. NBS(i)
Using Kirchhoff's law
d(i) = ΣkЄNBS(A) Φ(i,j) - Φ(k,j)
Oh wait, that’s also the definition of
hitting time from i to j!
)
(
)
,
(
)
(
1
1
)
,
(
i
NBS
k
j
k
i
d
j
i
)
(
)
,
(
)
,
(
1
)
,
(
i
NBS
k
j
k
h
k
i
P
j
i
h
16
i j
3
3
2
2
2
4
1Ω
1Ω
36. 36
Hitting times and Laplacians
1
0
.
.
.
.
.
n
j
i
=
1
0
.
2
.
.
.
.
n
j
i
d
m
d
d
d
h(i,j) = Φi- Φj
di
dj
-1 -1
-1
-1 -1
L
37. 37
Relationship with Electrical networks
i j
16
16
c(i,j) = h(i,j) + h(j,i) = 2m*Reff(i,j)
h(i,j) + h(j,i)
1. The Electrical Resistance Of i Graph Captures Its Commute And Cover Times, Ashok K. Chandra, Prabhakar Raghavan,
Walter L. Ruzzo, Roman Smolensky, Prasoon Tiwari, 1989
1
39. 39
Commute times and Laplacians
Why is this interesting ?
Because, this gives a very intuitive definition of
embedding the points in some Euclidian space, s.t. the
commute times is the squared Euclidian distances in
the transformed space.1
1. The Principal Components Analysis of a Graph, and its Relationships to Spectral Clustering . M. Saerens, et al, ECML ‘04
40. 40
L+ : some other interesting
measures of similarity1
L+
ij = xi
Txj = inner product of the position vectors
L+
ii = xi
Txi = square of length of position vector of i
Cosine similarity
jj
ii
ij
l
l
l
1. A random walks perspective on maximising satisfaction and profit. Matthew Brand, SIAM ‘05
42. 42
Recommender Networks1
1. A random walks perspective on maximising satisfaction and profit. Matthew Brand, SIAM ‘05
43. 43
Recommender Networks
For a customer node i define similarity as
H(i,j)
C(i,j)
Or the cosine similarity
Now the question is how to compute these quantities
quickly for very large graphs.
Fast iterative techniques (Brand 2005)
Fast Random Walk with Restart (Tong, Faloutsos 2006)
Finding nearest neighbors in graphs (Sarkar, Moore 2007)
jj
ii
ij
L
L
L
44. 44
Ranking algorithms on the web
HITS (Kleinberg, 1998) & Pagerank (Page & Brin,
1998)
We will focus on Pagerank for this talk.
An webpage is important if other important pages point to it.
Intuitively
v works out to be the stationary distribution of the markov
chain corresponding to the web.
i
j
out
j
j
v
i
v
)
(
deg
)
(
)
(
45. 45
Pagerank & Perron-frobenius
Perron Frobenius only holds if the graph is
irreducible and aperiodic.
But how can we guarantee that for the web graph?
Do it with a small restart probability c.
At any time-step the random surfer
jumps (teleport) to any other node with probability c
jumps to its direct neighbors with total probability 1-c.
j
i
n
c
c
ij ,
)
(
~
1
1
U
U
P
P
46. 46
Power iteration
Power Iteration is an algorithm for computing the
stationary distribution.
Start with any distribution x0
Keep computing xt+1 = xtP
Stop when xt+1 and xt are almost the same.
47. 47
Power iteration
Why should this work?
Write x0 as a linear combination of the left
eigenvectors {v0, v1, … , vn-1} of P
Remember that v0 is the stationary distribution.
x0 = c0v0 + c1v1 + c2v2 + … + cn-1vn-1
48. 48
Power iteration
Why should this work?
Write x0 as a linear combination of the left
eigenvectors {v0, v1, … , vn-1} of P
Remember that v0 is the stationary distribution.
x0 = c0v0 + c1v1 + c2v2 + … + cn-1vn-1
c0 = 1 . WHY? (slide 71)
55. 55
Convergence Issues
Formally ||x0Pt – v0|| ≤ |λ|t
λ is the eigenvalue with second largest magnitude
The smaller the second largest eigenvalue (in
magnitude), the faster the mixing.
For λ<1 there exists an unique stationary distribution,
namely the first left eigenvector of the transition
matrix.
56. 56
Pagerank and convergence
The transition matrix pagerank uses really is
The second largest eigenvalue of can be proven1
to be ≤ (1-c)
Nice! This means pagerank computation will converge
fast.
1. The Second Eigenvalue of the Google Matrix, Taher H. Haveliwala and Sepandar D. Kamvar, Stanford University Technical
Report, 2003.
~
P
U
P
)
1
(
P
~
c
c
57. 57
Pagerank
We are looking for the vector v s.t.
r is a distribution over web-pages.
If r is the uniform distribution we get pagerank.
What happens if r is non-uniform?
cr
c
vP
)
1
(
v
58. 58
Pagerank
We are looking for the vector v s.t.
r is a distribution over web-pages.
If r is the uniform distribution we get pagerank.
What happens if r is non-uniform?
cr
c
vP
)
1
(
v
Personalization
59. 59
Personalized Pagerank1,2,3
The only difference is that we use a non-uniform
teleportation distribution, i.e. at any time step
teleport to a set of webpages.
In other words we are looking for the vector v s.t.
r is a non-uniform preference vector specific to an
user.
v gives “personalized views” of the web.
r
vP
)
1
(
v c
c
1. Scaling Personalized Web Search, Jeh, Widom. 2003
2. Topic-sensitive PageRank, Haveliwala, 2001
3. Towards scaling fully personalized pagerank, D. Fogaras and B. Racz, 2004
60. 60
Personalized Pagerank
Pre-computation: r is not known from before
Computing during query time takes too long
A crucial observation1 is that the personalized
pagerank vector is linear w.r.t r
Scaling Personalized Web Search, Jeh, Widom. 2003
1
0
0
r
,
0
0
1
r
)
(
)
1
(
)
(
)
(
1
0
r
2
0
2
0 r
v
r
v
r
v
61. 61
Topic-sensitive pagerank (Haveliwala’01)
Divide the webpages into 16 broad categories
For each category compute the biased personalized
pagerank vector by uniformly teleporting to websites
under that category.
At query time the probability of the query being from
any of the above classes is computed, and the final
page-rank vector is computed by a linear combination
of the biased pagerank vectors computed offline.
62. 62
Personalized Pagerank: Other
Approaches
Scaling Personalized Web Search (Jeh & Widom ’03)
Towards scaling fully personalized pagerank:
algorithms, lower bounds and experiments (Fogaras et
al, 2004)
Dynamic personalized pagerank in entity-relation
graphs. (Soumen Chakrabarti, 2007)
63. 63
Personalized Pagerank (Purna’s Take)
But, whats the guarantee that the new transition matrix will still
be irreducible?
Check out
The Second Eigenvalue of the Google Matrix, Taher H. Haveliwala
and Sepandar D. Kamvar, Stanford University Technical Report,
2003.
Deeper Inside PageRank, Amy N. Langville. and Carl D. Meyer.
Internet Mathematics, 2004.
As long as you are adding any rank one (where the matrix is a
repetition of one distinct row) matrix of form (1Tr) to your
transition matrix as shown before,
λ ≤ 1-c
65. 65
Rank stability
How does the ranking change when the link structure
changes?
The web-graph is changing continuously.
How does that affect page-rank?
66. 66
Rank stability1 (On the Machine Learning papers
from the CORA2 database)
1. Link analysis, eigenvectors, and stability, Andrew Y. Ng, Alice X. Zheng and Michael Jordan, IJCAI-01
2. Automating the contruction of Internet portals with machine learning, A. Mc Callum, K. Nigam, J. Rennie, K. Seymore, In
Information Retrieval Journel, 2000
Rank on 5 perturbed
datasets by deleting
30% of the papers
Rank on the
entire database.
67. 67
Rank stability
Ng et al 2001:
Theorem: if v is the left eigenvector of . Let the
pages i1, i2,…, ik be changed in any way, and let v’ be
the new pagerank. Then
So if c is not too close to 0, the system would be rank
stable and also converge fast!
U
P
P c
c
)
(
~
1
~
P
c
i
k
j j )
(
||
'
||
1
1
v
v
v
70. 70
Acknowledgements
Andrew Moore
Gary Miller
Check out Gary’s Fall 2007 class on “Spectral Graph Theory,
Scientific Computing, and Biomedical Applications”
http://www.cs.cmu.edu/afs/cs/user/glmiller/public/Scientific-
Computing/F-07/index.html
Fan Chung Graham’s course on
Random Walks on Directed and Undirected Graphs
http://www.math.ucsd.edu/~phorn/math261/
Random Walks on Graphs: A Survey, Laszlo Lov'asz
Reversible Markov Chains and Random Walks on Graphs, D
Aldous, J Fill
Random Walks and Electric Networks, Doyle & Snell
71. 71
Convergence Issues1
Lets look at the vectors x for t=1,2,…
Write x0 as a linear combination of the eigenvectors of
P
x0 = c0v0 + c1v1 + c2v2 + … + cn-1vn-1
c0 = 1 . WHY?
Remember that 1is the right eigenvector of P with
eigenvalue 1, since P is stochastic. i.e. P*1T = 1T. Hence
vi1T = 0 if i≠0.
1 = x*1T = c0v0*1T = c0 . Since v0 and x0 are both
distributions
1. We are assuming that P is diagonalizable. The non-diagonalizable case is trickier, you can take a
look at Fan Chung Graham’s class notes (the link is in the acknowledgements section).