دانلود کتاب تجزیه و تحلیل و احتمال بر روی گراف‌ها

Analysis and Probability on Graphs, abbreviated as APG, forms a foundational framework across various advanced mathematical topics. It serves as a critical tool in disciplines such as computer science, physics, engineering, bioinformatics, economics, and social sciences, particularly in analyzing the dynamics of large-scale events on graphs. Many of these dynamic phenomena can be modeled through appropriate probabilistic processes applied to large graphs. One real-world application of combining graph theory with analysis and probability is the social network scenario: Consider a social network where users are connected based on their interactions or shared interests. The analysis of these connections through graph theory, combined with probability, can produce personalized recommendations. From graph theory perspective, represent users as nodes in a graph and connect users with edges if they have interacted or shared common interests. From probability perspective, assign probabilities to edges based on the strength or frequency of interactions between connected users. Analyze the graph to identify communities or groups of users who exhibit similar behavior or preferences. When a user expresses interest in a particular item (such as a movie, book, or product), the system may predict the likelihood of interest from their connected peers. Calculate the probability that the user will like the item based on the preferences of their connected neighbors in the graph. Recommend items with the highest probability of being liked by the user, considering the preferences of their network connections. Adjust recommendations dynamically as the user’s interactions and the graph evolve. Our goals. In our discussions, we prioritize understanding the motivation of each statement over getting immersed in details. Our approach is to aim for rigor where possible, but in other cases, we focus on conveying the concept of each statement. The main goal is to explore and explain various topics in this direction and provide a background for advanced undergraduate and graduate students in mathematics, physics, and computer science. The book is designed to be accessible to individuals with a basic understanding of graph theory and probability theory, making it suitable for upper-level undergraduates and those involved in independent study, using it as a reference, or exploring research with a basic background in probability and graph theory. The book may also be used as a textbook. Covering all these topics in a one-semester graduate course could be challenging. Nevertheless, since many sections of the book are independent, instructors have the flexibility to select and incorporate relevant sections based on their specific needs. Organization. From historical perspective, this textbook originates from years of teaching courses in probability theory, graph theory, combinatorics, and matrix theory by the first author at the University of Illinois, Chicago. The book has seven chapters. The first four chapters may be used as a textbook for advanced undergraduate courses related to probability theory, random walks, and probability on graphs in mathematics and mathematical computer science departments. The materials covered in these chapters may be used to investigate real-world problems. The last three chapters are intended for graduate students as a reference book or for individual study or research. We provide a brief overview of its contents. Chapter 1 focuses on fundamental concepts in probability theory, covering aspects such as probability space, random variables, expected values, and moments. Theorems related to assessing the probability of a random variable, including Markov’s inequality and Chebyshev’s inequality, are presented. Certain important random variables such as Bernoulli, binomial, and Poisson random variables and their properties are discussed. Chapter 2 is devoted to undirected and directed graphs. We first review the basic notions of undirected graphs. We explore walks, trails, and paths in undirected graphs, as well as examine these concepts in trees and bipartite graphs, along with associated results. Subsequently, we classify Eulerian circuits. We introduce the basic notions of digraphs (directed graphs) and shift our focus to connectedness in digraphs. Finally, we classify strongly connected digraphs by leveraging tools from number theory, such as results concerning Frobenius numbers. In Chapter 3, we discuss two commonly used random graph models, Gilbert’s and Erdös– Rényi’s models. We scrutinize the implications of these two models for graph connectivity analysis. We address the k-clique property of graphs, a tool for identifying the “dense” subgraphs within a given graph. We also study isolated vertices and connectivity, investigating threshold functions through the Poisson distribution. In Chapter 4, we introduce adjacency matrices for both undirected and directed graphs, facilitating the examination of various graph properties such as vertex degrees, walk enumeration, identification of odd cycles, and connectivity analysis. We also cover block matrices in the context of strong connectivity in directed graphs, alongside discussions on reduced graphs and reducible matrices. In Chapter 5, we explore Markov chains: what they are, how to simulate them, and how they behave on graphs. We analyze stationary distributions and their properties, along with spectral properties of matrices related to Markov chains. We cover reversible Markov chains and then move on to a short discussion of the Perron–Frobenius theorem. We conclude the chapter with a discussion of the mean first passage time, mean recurrence time, and the Kemeny constant of a Markov chain. Chapter 6 links symbolic dynamics with graph theory by interpreting sequences as graph walks, exemplified by a Fibonacci sequence. It covers hard-core configurations, their recurrence relations, and the MCMC algorithm. The Shannon capacity for a digraph G with cycles is introduced, related to the spectral radius of its adjacency matrix. The chapter also defines entropy H(μ) for a distribution μ, explores the pressure function and its properties, and a pressure gradient simulation. In Chapter 7, on a subshift of finite type, we introduce a pseudo-metric defined by a nonnegative matrix that satisfies the cycle condition. We determine the exact value of the Hausdorff dimension for a G∞, induced by a digraph G, with respect to this specific pseudo-metric. We also evaluate the Hausdorff dimension of the limit set of a finitely generated free group of isometries acting on a locally finite tree.