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Cut Distance
In graph theory and statistics, a graphon (also known as a graph limit) is a symmetric measurable function W: ,12\to ,1/math>, that is important in the study of dense graphs. Graphons arise both as a natural notion for the limit of a sequence of dense graphs, and as the fundamental defining objects of exchangeable random graph models. Graphons are tied to dense graphs by the following pair of observations: the random graph models defined by graphons give rise to dense graphs almost surely, and, by the regularity lemma, graphons capture the structure of arbitrary large dense graphs. Statistical formulation A graphon is a symmetric measurable function W: ,12 \to ,1/math>. Usually a graphon is understood as defining an exchangeable random graph model according to the following scheme: # Each vertex j of the graph is assigned an independent random value u_j\sim U ,1/math> # Edge (i,j) is independently included in the graph with probability W(u_i,u_j). A random graph model is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exchangeable Random Graph From Graphon
Exchangeable may refer to: * Exchangeable batteries, used with charging station#battery swapping, battery swapping in charging stations * Exchangeable bond, a type of hybrid security * Exchangeable image file format (Exif), a specification for the image file format used by digital cameras * Exchangeable random variables, in statistics, a set of random variables whose joint distribution is the same irrespective of the order of the variables See also * {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence Of Random Variables
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the value a random variable will take, rather than just the distribution. The concept is important in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that certain properties of a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szemerédi Regularity Lemma
In extremal graph theory, Szemerédi’s regularity lemma states that a graph can be partitioned into a bounded number of parts so that the edges between parts are regular (in the sense defined below). The lemma shows that certain properties of random graphs can be applied to dense graphs like counting the copies of a given subgraph within graphs. Endre Szemerédi proved the lemma over bipartite graphs for his theorem on arithmetic progressions in 1975 and for general graphs in 1978. Variants of the lemma use different notions of regularity and apply to other mathematical objects like hypergraphs. Statement To state Szemerédi's regularity lemma formally, we must formalize what the edge distribution between parts behaving 'almost randomly' really means. By 'almost random', we're referring to a notion called -regularity. To understand what this means, we first state some definitions. In what follows is a graph with vertex set . Definition 1. Let be disjoint subsets of . The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Sequence
In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after Augustin-Louis Cauchy; they may occasionally be known as fundamental sequences. It is not sufficient for each term to become arbitrarily close to the term. For instance, in the sequence of square roots of natural numbers: a_n=\sqrt n, the consecutive terms become arbitrarily close to each other – their differences a_-a_n = \sqrt-\sqrt = \frac d. As a result, no matter how far one goes, the remaining terms of the sequence never get close to ; hence the sequence is not Cauchy. The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), the criterion for convergence depends ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Measure
In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, and a difference of slopes is invariant under shear mapping. Ergodic theory is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration. Definition Let (X, \Sigma) be a measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, an ... and let f : X \to X be a measurable function from X to itself. A measure \mu on (X, \Sigma) is said to be invariant under f if, for every measurable set A in \Sigm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Homomorphism
In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices. Homomorphisms generalize various notions of graph colorings and allow the expression of an important class of constraint satisfaction problems, such as certain scheduling or frequency assignment problems. The fact that homomorphisms can be composed leads to rich algebraic structures: a preorder on graphs, a distributive lattice, and a category (one for undirected graphs and one for directed graphs). The computational complexity of finding a homomorphism between given graphs is prohibitive in general, but a lot is known about special cases that are solvable in polynomial time. Boundaries between tractable and intractable cases have been an active area of research. Definitions In this article, unless stated otherwise, ''graphs'' are fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homomorphism Density
In the mathematical field of extremal graph theory, homomorphism density with respect to a graph H is a parameter t(H,-) that is associated to each graph G in the following manner: : t(H,G):=\frac. Above, \operatorname(H,G) is the set of graph homomorphisms, or adjacency preserving maps, from H to G. Density can also be interpreted as the probability that a map from the vertices of H to the vertices of G chosen uniformly at random is a graph homomorphism. There is a connection between homomorphism densities and subgraph densities, which is elaborated on below. Examples * The edge density of a graph G is given by t(K_,G). * The number of walks with k-1 steps is given by \operatorname(P_k, G). *\operatorname(C_k, G) = \operatorname(A^k) where A is the adjacency matrix of G. *The proportion of colorings using k colors that are proper is given by t(G, K_k). Other important properties such as the number of stable sets or the maximum cut can be expressed or estimated in terms of hom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fan Chung
Fan-Rong King Chung Graham (; born October 9, 1949), known professionally as Fan Chung, is a Taiwanese-born American mathematician who works mainly in the areas of spectral graph theory, extremal graph theory and random graphs, in particular in generalizing the Erdős–Rényi model for graphs with general degree distribution (including power-law graphs in the study of large information networks). Since 1998, Chung has been the Paul Erdős Professor in Combinatorics at the University of California, San Diego (UCSD). She received her doctorate from the University of Pennsylvania in 1974, under the direction of Herbert Wilf. After working at Bell Laboratories and Bellcore for nineteen years, she joined the faculty of the University of Pennsylvania as the first female tenured professor in mathematics. She serves on the editorial boards of more than a dozen international journals. Since 2003 she has been the editor-in-chief of ''Internet Mathematics''. She has been invited to g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrepancy Theory
In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of ''classical'' discrepancy theory, namely distributing points in some space such that they are evenly distributed with respect to some (mostly geometrically defined) subsets. The discrepancy (irregularity) measures how far a given distribution deviates from an ideal one. Discrepancy theory can be described as the study of inevitable irregularities of distributions, in measure-theoretic and combinatorial settings. Just as Ramsey theory elucidates the impossibility of total disorder, discrepancy theory studies the deviations from total uniformity. A significant event in the history of discrepancy theory was the 1916 paper of Weyl on the uniform distribution of sequences in the unit interval. Theorems Discrepancy theory is based on the following classic theorems: * Geometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Edit Distance
In mathematics and computer science, graph edit distance (GED) is a measure of similarity (or dissimilarity) between two graphs. The concept of graph edit distance was first formalized mathematically by Alberto Sanfeliu and King-Sun Fu in 1983. A major application of graph edit distance is in inexact graph matching, such as error-tolerant pattern recognition in machine learning. The graph edit distance between two graphs is related to the string edit distance between strings. With the interpretation of strings as connected, directed acyclic graphs of maximum degree one, classical definitions of edit distance such as Levenshtein distance, Hamming distance and Jaro–Winkler distance may be interpreted as graph edit distances between suitably constrained graphs. Likewise, graph edit distance is also a generalization of tree edit distance between rooted trees. Formal definitions and properties The mathematical definition of graph edit distance is dependent upon the defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric %28mathematics%29
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
