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Topic: Closeness (graph theory)

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	Intro to Graph Theory
		A graph is defined as a set of nodes and a set of lines that connect the nodes.
		A subgraph of a graph is a subset of its points together with all the lines connecting members of the subset.
		Closeness centrality is defined as the total graph-theoretic distance to all other nodes in the network.
	www.analytictech.com /mb021/graphtheory.htm (1984 words)

	Graph Utilities Package - Wolfram Mathematica
		The closeness centrality of a vertex in a disconnected graph is based on the closeness centrality of the component where this vertex belongs.
		This calculates the closeness centrality of a grid graph, and plots the graph so that vertices of high centrality are red.
		On closing the window, a drawing of the graph, a representation of the graph as a list of rules, the coordinates, and the vertex labels are returned.
	reference.wolfram.com /mathematica/GraphUtilities/tutorial/GraphUtilities.html (3383 words)

	Graph Technology Overview
		While representational schemas based on mathematical graph theory date back to the same era as relational databases, Cogito is the first company to deliver a commercially viable graph database designed for relationship analytics.
		Once data is fused into a network that expresses the relationships among the data elements, analysts are able to see visual representations of connected data, find patterns and discover connections that are obscured by other kinds of information representation.
		With a 200,000 node graph (50 arcs per node) this operation is done in 1.1 seconds vs. 7 minutes using RDBMS technology like MySQL.
	www.cogitoinc.com /products-graphtechnology.html (506 words)

	Web Graph Analysis in Perspective -- Kenneth Farrall
		Although the process of analysis begins with the structure of a network graph, the nodes of the graph are anchored to their textual content, to which the analysts must frequently refer.
		Graph theory metrics for association matrices are often based on undirected graphs which can then be generalized to include directed graphs.
		From the resulting graphs, in-degree centrality was calculated for the most prominent nodes, leading to a ranking of the five most prominent nodes for each of the four graphs.
	farrall.org /papers/webgraph_as_content.html (8778 words)

	Bates College \| Courses
		The theory of numbers is concerned with the properties of the integers, one of the most basic of mathematical sets.
		Graph theory is the branch of mathematics that provides the framework to find these paths.
		Game theory provides a mathematical framework for analyzing situations where individuals (or companies, political parties, or nations) are faced with the prospect of maximizing their own well-being, dependent on the decisions of others.
	www.bates.edu /MATH-courses.xml (2505 words)

	[No title]
		In the line graph, the two end actors have smaller degree centralities (degrees = 1) than those in the middle (=2); their respective standardized scores are 0.167 and 0.333.
		In the line graph, the two end actors the least betweenness (0.0), the exactly middle actor n1 has the highest betweenness (0.60), while the two adjacent to it are only slightly less central (0.53).
		Proximity Prestige This analog to closeness centrality considers the proximity of actor i to other actors in its influence domain, the set of all network actors that can reach actor i, directly and indirectly (revealed by nonzero entries in the ith column of the distance matrix).
	www.soc.umn.edu /~knoke/pages/Centrality_and_Prestige.doc (1410 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Graphs and relations may be equivalently represented by a matrix in which columns represent nodes, arcs, edges, or loops are represented by ones, and their absence is represented by zeros.
		A digraph or graph is transitive when for any subset of three nodes {A, B, C}, a pair of (directed) lines from A to B and B to C entails one from A to C. Properties of Triples. A subnetwork of three nodes in a network and their ties (edges, arcs).
		When we assume that each edge in a graph has a transport capacity of one unit, the flow centrality of a node u is the percentage of the total amount of flow between all pairs of nodes that is not reduced when node i is removed from the graph.
	eclectic.ss.uci.edu /~drwhite/turks/Glossary.doc (5830 words)

	Guide to the Mathematics Subject Classification Scheme
		Algebra is principally concerned with symmetry, patterns, discrete sets, and the rules for manipulating arithmetic operations; one might think of this as the outgrowth of arithmetic and algebra classes in primary and secondary school.
		Of particular interest are several classes of rings of interest in number theory, field theory, and related areas; however, other classes of rings arise, and a rich structure theory arises to analyze commutative rings in general, using the concepts of ideals, localizations, and homological algebra.
		Thus in principle this field is closely allied to differential equations on manifolds, but in practice the focus is on the underlying sets (invariant sets or limit sets) and on the chaotic behaviour of limiting systems.
	www.math.niu.edu /~rusin/known-math/index/beginners.html (5525 words)

	Bates College: The College Catalog: Mathematics and Computer Science
		Graphing calculators are used in the course for graphical and numerical explorations.
		A closely related area is game theory, which deals with decision problems in a competitive environment, where conflict, risk, and uncertainty are often involved.
		This unit treats the mathematical theory and practical applications of inverse problems with examples drawn from geology, biology, chemistry, and physics.
	abacus.bates.edu /catalog99-00/mathematics.html (3135 words)

	closeness - Search Results - MSN Encarta
		Accuracy, closeness to a targeted goal, often the result of athletic skills or the use technology.
		In graph theory, closeness is the shortest path between one vertex and another vertex.
		For nearly a century, our faculty have drawn on their passion for teaching, their experience in working with organizations worldwide, and the insights gained from their research to...
	encarta.msn.com /encnet/refpages/search.aspx?q=closeness (188 words)

	theory
		In fact, in the collision theory, a reaction resembles the collision of two defective billiard balls: the balls bounce apart if they collide with only a small energy, but might smash each other into fragments (products) if they collide with more than a certain minimum kinetic energy.
		A reaction profile in collision theory is a graph showing the variation in potential energy as one reactant molecule approaches another and the products then separate (Fig 2).
		Activated complex theory is an attempt to identify the principal features governing the magnitude of a rate constant in terms of a model of the events that take place during the reaction.
	staff.um.edu.mt /jgri1/teaching/che2372/notes/10/theory.html (2365 words)

	Pearls in Graph Theory: A Comprehensive Introduction
		Based on 20 years of teaching by the leading researcher in graph theory, it offers a solid foundation on the subject.
		Topics include basic graph theory, colorings of graphs, circuits and cycles, labeling graphs, drawings of graphs, measurements of closeness to planarity, graphs on surfaces, and applications and algorithms.
		Concise, well-written text illustrates development of graph theory and application of its principles in methods both formal and abstract.
	store.doverpublications.com /0486432327.html (250 words)

	Closeness (graph theory) - Wikipedia, the free encyclopedia
		Closeness is a centrality measure of a vertex within a graph.
		Vertices that are 'shallow' to other vertices (that is, those that tend to have short geodesic distances to other vertices with in the graph) have higher closeness.
		Closeness is preferred in network analysis to mean shortest-path length, as it gives higher values to more central vertices, and so is usually positively associated with other measures such as degree.
	en.wikipedia.org /wiki/Closeness_(graph_theory) (139 words)

	closeness (Site not responding. Last check: 2007-10-13)
		In graph theory closeness is a centrality measure of a vertex within a graph.
		Closeness is preferred in network analysis to mean shortest-path length, as it...
		Closeness to God - A World of Love: Part 5 The existence of good and evil allows man to make the right choice, and merit greater closeness to God.
	closeness.inetsearcher.com (217 words)

	Interactive Mathematics Miscellany and Puzzles
		Typically, they were run not by teachers, but by graduate students or faculty members at a university, who considered it part of their professional duty to show younger students the joys of mathematics.
		Others, such as the discussion of graph theory, stretch the curriculum with gems of mathematics which are not usually touched on in the classroom.
		The problems in graph theory, too, remain on the surface of this important branch of mathematics.
	www.cut-the-knot.org /books/circles/forward.shtml (632 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Predominant usage of graph theory analysis techniques resulted in the development of indigenous metrics for the estimation of social network characteristics.
		In graph theory, the graphs are subdivided into directed and undirected graphs.
		Mathematical Formula EMBED Equation.3 Node closeness centrality is the inverse of the sum of geodesic distances from node i to the g-1 other nodes (i.e., the reciprocal of its “farness”).
	www.cs.odu.edu /~gmanepal/documents/beowulf.doc (3670 words)

	Motzkin biography
		Other themes which run through Motzkin's work is geometric problems, some involving Ramsey theory, and he wrote many papers on graph theory.
		Convex polyhedra interested him and are studied in several of his papers which combine his geometric and graph theory interests.
		Exceptionally broad, the range of his work included beautiful and important contributions to the theory of linear inequalities and programming, approximation theory, convexity, combinatorics, algebraic geometry, number theory, algebra, function theory, and numerical analysis.
	www-gap.dcs.st-and.ac.uk /~history/Biographies/Motzkin.html (1174 words)

	Dr. Dobb's \| Letters \| July 22, 2001
		Without going into details about graph theory, given a graph G with n vertices, the most number of edges e is n(n- 1)/2.
		The graph I used in my article was a "sparse" graph which contained the minimum number of edges needed to specify that all of the vertices are connected.
		For example, in the case where a graph is be comprised of 1 million vertices and only a handful of edges, the GAD algorithm is finished as soon as the last edge has been read.
	www.ddj.com /184408593?pgno=3 (2436 words)

	Social Networking Communities, myspace clone, social networking/community scripting, software, programming & ...
		Research in a number of academic fields have shown that social networks operate on many levels, from families up to the level of nations, and play a critical role in determining the way problems are solved, organizations are run, and the degree to which individuals succeed in achieving their goals.
		Social network theory produces an alternate view, where the attributes of individuals are less important than their relationships and ties with other actors within the network.
		Netwiki is a scientific wiki devoted to network theory, which uses tools from subjects such as graph theory, statistical mechanics, and dynamical systems to study real-world networks in the social sciences, technology, biology, etc.
	venturait.com /social_networking_communities.html (2235 words)

	Social Networking Software, Community, Script, myspace clone, friendster clone, programming & design
		The maximum size of social networks tends to be around 150 people (Dunbar's number) and the average size around 124 (Hill and Dunbar, 2002).
		These applications, known as online social networks are becoming increasingly popular[1].
		This approach has turned out to be useful for explaining many real-world phenomena, but leaves less room for individual agency, the ability for individuals to influence their success, so much of it rests within the structure of their network.
	venturait.com /social_networking_community_software.html (2299 words)

	URCS Theory Technical Reports
		We show that the problems we study related to the reconstruction of graphs either are isomorphic (in complexity-theoretic sense) to the graph isomorphism problem or are many-one hard for the graph isomorphism problem.
		For every $k \geq 2$ and $n \geq 1$, we show that there exists a collection of $k$ graphs on $(2^{k-1}+1)n+k$ vertices with $2^{n}$ 1-vertex-preimages, i.e., one has families of graph collections whose number of 1-vertex-preimages is huge relative to the size of the graphs involved.
		Due to the close connection between the determinant function and #L, as well as between the rank function and AC^0(C_=L), our results might yield a better understanding of the exact power of counting in logspace and the relationships among the complexity classes sandwiched between NL and uniform TC^1.
	www.cs.rochester.edu /trs/theory-trs.html (16128 words)

	BioMed Central \| Full text \| Improved prediction of critical residues for protein function based on network and ...
		This gives a way of building a network of amino acid interactions [[5,6] and [7]], with the obvious caveat that the accuracy of such a model will be affected by the accuracy of the assumption of chemical interactions between different residues.
		Hence, one of these global centrality measurements is the closeness centrality –; also referred as chemical distance [[10,11] and [6]] – and is defined as the inverse of the average distance value between p and all the other vertices of the network.
		To compare closeness centrality against phylogenetic approaches, we noticed that the SITE annotation in this 128 proteins, mostly included residues annotated to be involved in catalysis (218 residues), ligand binding (156 residues) and/or metal-binding (273 residues) sites (see Table 3).
	www.biomedcentral.com /1471-2105/6/213 (6057 words)

	JoSS: Journal of Social Structure
		Closeness centrality is therefore very sensitive to a single large distance or missing link.
		However, towards the centre of the graph, these colours are obscured by the dashes used as labels (since there are 58,653 data points).
		Groups A (red) and B (pink) are particularly close in terms of activity-similarity distance, as are groups D (yellow) and E (orange).
	www.cmu.edu /joss/content/articles/volume6/dekker (8073 words)

	Pearls in Graph Theory: A Comprehensive Introduction (Dover Books on Mathematics) - Computer Store
		Pearls in graph theory begins informally and at an elementary level, suitable for a substantial freshman-sophomore course.
		After intuitive introductions, concepts and theory are developed with increasing depth, leading into material for a good intermediate-level course.
		In addition, for me the final three chapters are a splendid, enticingly elementary yet comprehensive introduction to topological graph theory.
	www.investordictionary.com /store/product/0486432327.aspx (430 words)

	closeness - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "closeness" is defined.
		Example: "Their closeness grew as the night wore on"
		Words similar to closeness: close, familiarity, intimacy, meanness, minginess, nearness, niggardliness, niggardness, parsimoniousness, parsimony, secretiveness, tightfistedness, tightness, propinquity, more...
	www.onelook.com /?w=closeness (203 words)

	[No title]
		In the case of the traveling salesman problem, the mathematical structure is a graph where each city is denoted by a point (or node) and lines are drawn connecting every two nodes (called arcs or edges).
		The underlying theory of facet generation for the symmetric traveling salesman problem is provided in Grötschel and Padberg (1985) and Jünger, Reinelt and Rinaldi (1994).
		Besides being a "polytope" of a difficult combinatorial optimization problem from a complexity theory point of view, there are important cases of practical problems that can be formulated as TSP problems and many other problems are generalizations of this problem.
	iris.gmu.edu /~khoffman/papers/trav_salesman.html (2481 words)

	Interview: Summary of question 13 (Site not responding. Last check: 2007-10-13)
		Only three students gave fairly close answers though a few other may have been on-track and could have come upon an appropriate method with more time.
		The four closest responses were: "This is a pigeonhole problem when formulated differently in terms of colors." "Pigeonhole, prove that there exists a rectangle given all possibilities." "Pigeonhole.
		Then get a rectangle." "I remember something similar." "This is like a coloring problem." Still other responses that did not suggest closeness to the appropriate method were the following: "Solve by contradiction." "Is it a graph theory problem?" "I have no clear idea." "This problem is difficult to understand.
	gallery.carnegiefoundation.org /bcooperstein/interview_q13.html (1135 words)

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