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Eigenvector Centrality
In graph theory, eigenvector centrality (also called eigencentrality or prestige score) is a measure of the influence of a node in a connected network. Relative scores are assigned to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes. A high eigenvector score means that a node is connected to many nodes who themselves have high scores. Google's PageRank and the Katz centrality are variants of the eigenvector centrality. Using the adjacency matrix to find eigenvector centrality For a given graph G:=(V,E) with , V, vertices let A = (a_) be the adjacency matrix, i.e. a_ = 1 if vertex v is linked to vertex t, and a_ = 0 otherwise. The relative centrality score, x_v, of vertex v can be defined as: : x_v = \frac 1 \lambda \sum_ x_t = \frac 1 \lambda \sum_ a_ x_t where M(v) is the set of neighbors of v and \lambda is a constant. With a small rearrange ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Matrix
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, ''substitution matrix'', or Markov matrix. The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. There are several different definitions and types of stochastic matrices: *A right stochastic matrix is a square matrix of nonnegative real numbers, with each row summing to 1 (so it is also called a row stochastic matrix). *A left stochastic matrix is a square matrix of nonnegative real numbers, with each column summing to 1 (so it is also called a column stochastic matrix). *A ''doubly stochastic matrix'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centrality
In graph theory and network analysis, indicators of centrality assign numbers or rankings to nodes within a graph corresponding to their network position. Applications include identifying the most influential person(s) in a social network, key infrastructure nodes in the Internet or urban networks, super-spreaders of disease, and brain networks. Centrality concepts were first developed in social network analysis, and many of the terms used to measure centrality reflect their sociological origin.Newman, M.E.J. 2010. ''Networks: An Introduction.'' Oxford, UK: Oxford University Press. Definition and characterization of centrality indices Centrality indices are answers to the question "What characterizes an important vertex?" The answer is given in terms of a real-valued function on the vertices of a graph, where the values produced are expected to provide a ranking which identifies the most important nodes. The word "importance" has a wide number of meanings, leading to many d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Public Good (economics)
In economics, a public good (also referred to as a social good or collective good)Oakland, W. H. (1987). Theory of public goods. In Handbook of public economics (Vol. 2, pp. 485–535). Elsevier. is a commodity, product or service that is both non-excludable and non-rivalrous and which is typically provided by a government and paid for through taxation. Use by one person neither prevents access by other people, nor does it reduce availability to others, so the good can be used simultaneously by more than one person. This is in contrast to a common good, such as wild fish stocks in the ocean, which is non-excludable but rivalrous to a certain degree. If too many fish were harvested, the stocks would deplete, limiting the access of fish for others. A public good must be valuable to more than one user, otherwise, its simultaneous availability to more than one person would be economically irrelevant. Capital goods may be used to produce public goods or services that are "...ty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DeGroot Learning
DeGroot is an agglutinated form of the Dutch surname De Groot. It may refer to: People * Bruce DeGroot (born 1963), American politician * Chad DeGroot (born 1974), American freestyle BMX rider *Diede de Groot Diede de Groot (; born 19 December 1996) is a Dutch professional wheelchair tennis player who was world No. 1 in both singles and doubles. De Groot is a 42-time Grand Slam (tennis)#Tournaments, major champion, having won a record 23 titles in ... (born 1986), Dutch Tennis Grand Slam Champion * Dudley DeGroot (1899–1970), American athlete and coach * Gerard DeGroot, author of the 2008 book '' The Sixties Unplugged'' * Jeff DeGroot (born 1985), American soccer player * Morris H. DeGroot (1931–1989), American statistician Characters *Gerald and Karen DeGroot, characters on the American television show ''Lost'' *LuAnn DeGroot, main character of the comic strip '' Luann'' See also * De Groot, a surname * De Groote, a surname * Groot (surname) {{disambig, surna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neuron
A neuron (American English), neurone (British English), or nerve cell, is an membrane potential#Cell excitability, excitable cell (biology), cell that fires electric signals called action potentials across a neural network (biology), neural network in the nervous system. They are located in the nervous system and help to receive and conduct impulses. Neurons communicate with other cells via synapses, which are specialized connections that commonly use minute amounts of chemical neurotransmitters to pass the electric signal from the presynaptic neuron to the target cell through the synaptic gap. Neurons are the main components of nervous tissue in all Animalia, animals except sponges and placozoans. Plants and fungi do not have nerve cells. Molecular evidence suggests that the ability to generate electric signals first appeared in evolution some 700 to 800 million years ago, during the Tonian period. Predecessors of neurons were the peptidergic secretory cells. They eventually ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neuroscience
Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions, and its disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, developmental biology, cytology, psychology, physics, computer science, chemistry, medicine, statistics, and mathematical modeling to understand the fundamental and emergent properties of neurons, glia and neural circuits. The understanding of the biological basis of learning, memory, behavior, perception, and consciousness has been described by Eric Kandel as the "epic challenge" of the biological sciences. The scope of neuroscience has broadened over time to include different approaches used to study the nervous system at different scales. The techniques used by neuroscientists have expanded enormously, from molecular and cellular studies of individual neurons to imaging of sensory, motor and cognitive tasks in the brain. Hist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a " logical axiom" or a " non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a'' + 0 = ''a'' in integer arithmetic. N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopold Landau, a gynecologist, and his mother was Johanna Jacoby. Landau studied mathematics at the University of Berlin, receiving his doctorate in 1899 and his habilitation (the post-doctoral qualification required to teach in German universities) in 1901. His doctoral thesis was 14 pages long. In 1895, his paper on scoring chess tournaments is the earliest use of eigenvector centrality. Landau taught at the University of Berlin from 1899 to 1909, after which he held a chair at the University of Göttingen. He married Marianne Ehrlich, the daughter of the Nobel Prize-winning biologist Paul Ehrlich, in 1905. At the 1912 International Congress of Mathematicians Landau listed four problems in number theory about primes that he said were pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G is denoted by \Delta(G), and is the maximum of G's vertices' degrees. The minimum degree of a graph is denoted by \delta(G), and is the minimum of G's vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is enti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue Algorithm
In numerical analysis, one of the most important problems is designing efficient and Numerical stability, stable algorithms for finding the eigenvalues of a Matrix (mathematics), matrix. These eigenvalue algorithms may also find eigenvectors. Eigenvalues and eigenvectors Given an Square matrix#Square matrices, square matrix of Real number, real or Complex number, complex numbers, an ''eigenvalue'' and its associated ''generalized eigenvector'' are a pair obeying the relation :\left(A - \lambda I\right)^k = 0, where is a nonzero column vector, is the identity matrix, is a positive integer, and both and are allowed to be complex even when is real. When , the vector is called simply an ''eigenvector'', and the pair is called an ''eigenpair''. In this case, . Any eigenvalue of has ordinaryThe term "ordinary" is used here only to emphasize the distinction between "eigenvector" and "generalized eigenvector". eigenvectors associated to it, for if is the smallest intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Node (networking)
In Computer network, networking, a node (, ‘knot’) is either a redistribution point or a communication endpoint within telecommunication networks. A physical network node is an electronic device that is attached to a network, and is capable of creating, receiving, or transmitting information over a communication channel. In data communication, a physical network node may either be data communication equipment (such as a modem, Network hub, hub, Network bridge, bridge or Network switch, switch) or data terminal equipment (such as a digital telephone handset, a printer or a host computer). A Passivity (engineering), passive distribution point such as a distribution frame or patch panel is not a node. Computer networks In data communication, a physical network node may either be data communication equipment (DCE) such as a modem, Network hub, hub, Network bridge, bridge or Network switch, switch; or data terminal equipment (DTE) such as a digital telephone handset, a printe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
