|
Conductance (graph)
In graph theory the conductance of a graph measures how "well-knit" the graph is: it controls how fast a random walk on converges to its stationary distribution. The conductance of a graph is often called the Cheeger constant of a graph as the analog of its counterpart in spectral geometry. Since electrical networks are intimately related to random walks with a long history in the usage of the term "conductance", this alternative name helps avoid possible confusion. The conductance of a cut (S, \bar S) in a graph is defined as: :\varphi(S) = \frac where the are the entries of the adjacency matrix for , so that :a(S) = \sum_ \sum_ a_ is the total number (or weight) of the edges incident with . is also called a volume of the set S \subseteq V. The conductance of the whole graph is the minimum conductance over all the possible cuts: : \phi(G) = \min_\varphi(S). Equivalently, conductance of a graph is defined as follows: : \phi(G) := \min_\frac.\, For a -regular graph, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Conductance
Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discrete mathematics *Graph of a function *Graph of a relation *Graph paper *Chart, a means of representing data (also called a graph) Computing *Graph (abstract data type), an abstract data type representing relations or connections *graph (Unix), Unix command-line utility *Conceptual graph, a model for knowledge representation and reasoning Other uses * HMS ''Graph'', a submarine of the UK Royal Navy See also *Complex network *Graf *Graff (other) *Graph database *Grapheme, in linguistics *Graphemics *Graphic (other) *-graphy (suffix from the Greek for "describe," "write" or "draw") *List of information graphics software *Statistical graphics Statistical graphics, also known as statistical graphical techniques, are graphic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Flow
In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups ''A'' of real positive diagonal matrices and ''N'' of lower unitriangular matrices on the unit tangent bundle ''G'' / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling function. In the case of geodesic flow, the ergodic transformation can be understood in terms of symbolic dynamics; and in terms of the ergodic actions of Γ on the boundary ''S''1 = ''G'' / ''AN'' and ''G'' / ''A'' = ''S''1 × ''S''1 \ diag ''S''1. Ergodic flows al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Graph Theory
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants. Branches of algebraic graph theory Using linear algebra The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties. As a simple example, a connected graph with diameter ''D'' w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Processes
Markov (Bulgarian, russian: Марков), Markova, and Markoff are common surnames used in Russia and Bulgaria. Notable people with the name include: Academics *Ivana Markova (born 1938), Czechoslovak-British emeritus professor of psychology at the University of Stirling *John Markoff (sociologist) (born 1942), American professor of sociology and history at the University of Pittsburgh *Konstantin Markov (1905–1980), Soviet geomorphologist and quaternary geologist Mathematics, science, and technology *Alexander V. Markov (1965-), Russian biologist *Andrey Markov (1856–1922), Russian mathematician *Vladimir Andreevich Markov (1871–1897), Russian mathematician, brother of Andrey Markov (Sr.) *Andrey Markov Jr. (1903–1979), Russian mathematician and son of Andrey Markov *John Markoff (born 1949), American journalist of computer industry and technology *Moisey Markov (1908–1994), Russian physicist Performing arts *Albert Markov, Russian American violinist, composer * Alexan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chains And Mixing Times
''Markov Chains and Mixing Times'' is a book on Markov chain mixing times. The second edition was written by David A. Levin, and Yuval Peres. Elizabeth Wilmer was a co-author on the first edition and is credited as a contributor to the second edition. The first edition was published in 2009 by the American Mathematical Society, with an expanded second edition in 2017. Background A Markov chain is a stochastic process defined by a set of states and, for each state, a probability distribution on the states. Starting from an initial state, it follows a sequence of states where each state in the sequence is chosen randomly from the distribution associated with the previous state. In that sense, it is "memoryless": each random choice depends only on the current state, and not on the past history of states. Under mild restrictions, a Markov chain with a finite set of states will have a stationary distribution that it converges to, meaning that, after a sufficiently large number of steps, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elizabeth Wilmer
Elizabeth Lee Wilmer is an American mathematician known for her work on Markov chain mixing times. She is a professor, and former department head, of mathematics at Oberlin College. As a 16-year-old high school student at Stuyvesant High School and captain of the school mathematics team, Wilmer won second place in the Westinghouse Science Talent Search in 1987, for a project involving 3-coloring of graphs. The first place winner that year was also female, marking the first year that the top two prizes both went to women. As an undergraduate at Harvard College, she led the university's team that won the first Mathematical Contest in Modeling of the Society for Industrial and Applied Mathematics, and she was one of the two inaugural winners of the Alice T. Schafer Prize of the Association for Women in Mathematics for excellence by a woman in undergraduate mathematics. She graduated from Harvard in 1991, and completed her Ph.D. at Harvard in 1999. She worked with Persi Diaconis for he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuval Peres
Yuval Peres ( he, יובל פרס; born 5 October 1963) is a mathematician known for his research in probability theory, ergodic theory, mathematical analysis, theoretical computer science, and in particular for topics such as fractals and Hausdorff measure, random walks, Brownian motion, percolation and Markov chain mixing times. He was born in Israel and obtained his Ph.D. at the Hebrew University of Jerusalem in 1990 under the supervision of Hillel Furstenberg. He was a faculty member at the Hebrew University and the University of California at Berkeley, and a Principal Researcher at Microsoft Research in Redmond, Washington. Peres has been accused of sexual harassment by several female scientists. Career After his Ph.D. Peres had postdoctoral positions at Stanford and Yale. In 1993 Peres joined the statistics department at UC Berkeley. He later became a professor in both the mathematics and statistics departments. He was also a professor at the Hebrew University. In 2006 Pere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krackhardt E/I Ratio
The Krackhardt E/I Ratio (or variously the E-I Index) is a social network measure which the relative density of internal connections within a social group compared to the number of connections that group has to the external world. It was so described in a 1988 paper by David Krackhardt and Robert N. Stern noting the increased effectiveness in moments of crisis of organizations which had stronger informal networks that crossed formal internal group structures. Comparisons with network theory The E/I ratio is related to the concept of conductance, which measures the likelihood that a random walk on a subgraph will exit that subgraph. References * Informal networks and organizational crises: An experimental simulation. David Krackhardt, Robert N. Stern - Social Psychology Quarterly, 1988DOI:10.2307/2786835 See also * Conductance (graph) * Percolation Percolation (from Latin ''percolare'', "to filter" or "trickle through"), in physics, chemistry and materials science, re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percolation Theory
In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger connected, so-called spanning clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles network theory and percolation. Introduction A representative question (and the source of the name) is as follows. Assume that some liquid is poured on top of some porous material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is modelled mathematically as a three-dimensional network of vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability , or closed with probability , and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resistance Distance
In graph theory, the resistance distance between two vertices of a simple, connected graph, , is equal to the resistance between two equivalent points on an electrical network, constructed so as to correspond to , with each edge being replaced by a resistance of one ohm. It is a metric on graphs. Definition On a graph , the resistance distance between two vertices and is : \Omega_:=\Gamma_+\Gamma_-\Gamma_-\Gamma_, :where \Gamma = \left(L + \frac\Phi\right)^+, with denoting the Moore–Penrose inverse, the Laplacian matrix of , is the number of vertices in , and is the matrix containing all 1s. Properties of resistance distance If then . For an undirected graph :\Omega_=\Omega_=\Gamma_+\Gamma_-2\Gamma_ General sum rule For any -vertex simple connected graph and arbitrary matrix : :\sum_(LML)_\Omega_ = -2\operatorname(ML) From this generalized sum rule a number of relationships can be derived depending on the choice of . Two of note are; :\begin \sum_\Omega ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
