This paper provides a comprehensive survey of the PageRank algorithm, focusing on its mathematical foundations, solution methods, storage challenges, and variations. It builds upon the earlier work by Bianchini et al. and explores various aspects of the PageRank model, including its convergence properties, sensitivity, and computational efficiency. The paper introduces new results and discusses potential future research directions. PageRank is a link analysis algorithm used by Google to rank web pages. It uses the hyperlink structure of the web to create a Markov chain, where the stationary distribution of the chain represents the PageRank scores of the web pages. The Markov chain is constructed using a transition probability matrix P, which is modified to ensure it is both stochastic and irreducible. This is achieved by adding a uniform matrix E to the original matrix P, ensuring that all nodes are connected and the chain is irreducible. The paper discusses various solution methods for computing the PageRank vector, including the power method, which is traditionally used due to its simplicity and effectiveness. However, the paper also explores alternative methods and optimizations to improve the efficiency of the PageRank computation. It addresses storage issues related to the large size of the web graph, discussing techniques such as compressed storage and adjacency lists to manage the data efficiently. The paper also examines the impact of dangling nodes (nodes with no outgoing links) on the PageRank computation. It suggests methods to handle these nodes, such as replacing zero rows in the transition matrix with uniform vectors, to ensure the matrix remains stochastic and irreducible. The paper further discusses the convergence properties of the PageRank algorithm, the sensitivity of the solution to changes in the input data, and the conditioning of the problem. In addition, the paper explores various acceleration techniques for the PageRank computation, including methods that reduce the number of iterations required and optimize the work per iteration. It also presents a linear system formulation of the PageRank problem, which provides a more efficient way to compute the PageRank vector by leveraging the properties of the matrix involved. Overall, the paper provides a detailed overview of the PageRank algorithm, its mathematical foundations, and its practical implementation, highlighting the importance of the algorithm in modern search engines and the ongoing research into improving its efficiency and effectiveness.This paper provides a comprehensive survey of the PageRank algorithm, focusing on its mathematical foundations, solution methods, storage challenges, and variations. It builds upon the earlier work by Bianchini et al. and explores various aspects of the PageRank model, including its convergence properties, sensitivity, and computational efficiency. The paper introduces new results and discusses potential future research directions. PageRank is a link analysis algorithm used by Google to rank web pages. It uses the hyperlink structure of the web to create a Markov chain, where the stationary distribution of the chain represents the PageRank scores of the web pages. The Markov chain is constructed using a transition probability matrix P, which is modified to ensure it is both stochastic and irreducible. This is achieved by adding a uniform matrix E to the original matrix P, ensuring that all nodes are connected and the chain is irreducible. The paper discusses various solution methods for computing the PageRank vector, including the power method, which is traditionally used due to its simplicity and effectiveness. However, the paper also explores alternative methods and optimizations to improve the efficiency of the PageRank computation. It addresses storage issues related to the large size of the web graph, discussing techniques such as compressed storage and adjacency lists to manage the data efficiently. The paper also examines the impact of dangling nodes (nodes with no outgoing links) on the PageRank computation. It suggests methods to handle these nodes, such as replacing zero rows in the transition matrix with uniform vectors, to ensure the matrix remains stochastic and irreducible. The paper further discusses the convergence properties of the PageRank algorithm, the sensitivity of the solution to changes in the input data, and the conditioning of the problem. In addition, the paper explores various acceleration techniques for the PageRank computation, including methods that reduce the number of iterations required and optimize the work per iteration. It also presents a linear system formulation of the PageRank problem, which provides a more efficient way to compute the PageRank vector by leveraging the properties of the matrix involved. Overall, the paper provides a detailed overview of the PageRank algorithm, its mathematical foundations, and its practical implementation, highlighting the importance of the algorithm in modern search engines and the ongoing research into improving its efficiency and effectiveness.