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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 diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''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 of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closeness Centrality
In a connected graph, closeness centrality (or closeness) of a node is a measure of centrality in a network, calculated as the reciprocal of the sum of the length of the shortest paths between the node and all other nodes in the graph. Thus, the more central a node is, the ''closer'' it is to all other nodes. Closeness was defined by Bavelas (1950) as the reciprocal of the farness, that is: : C_B(x)= \frac. where d(y,x) is the distance (length of the shortest path) between vertices x and y. This unnormalised version of closeness is sometimes known as status When speaking of closeness centrality, people usually refer to its normalized form which represents the average length of the shortest paths instead of their sum. It is generally given by the previous formula multiplied by N-1, where N is the number of nodes in the graph resulting in: : C(x)= \frac. The normalization of closeness simplifies the comparison of nodes in graphs of different sizes. For large graphs, the minus one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big Theta
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outdegree
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a mul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indegree
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freeman Centralization
Freeman, free men, or variant, may refer to: * a member of the Third Estate in medieval society (commoners), see estates of the realm * Freeman, an apprentice who has been granted freedom of the company, was a rank within Livery companies * Freeman, in Middle English synonymous with franklin (class), initially a person not tied to land as a villein or serf, later a land-owner * Freeman (Colonial), in U.S. colonial times, a person not under legal restraint * A person who has been awarded Freedom of the City * Free tenant, a social class in the Middle Ages * Freedman, a former slave that had been freed from bondage Places ;In the United States * Freeman, Georgia, an unincorporated community * Freeman, Illinois, an unincorporated community * Freeman, Indiana, an unincorporated community * Freeman, South Dakota, a city * Freeman, Virginia, an unincorporated community * Freeman, Wisconsin, a town in Crawford County * Freeman, Langlade County, Wisconsin, an unincorporated community * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Node Influence Metric
In graph theory and network analysis, node influence metrics are measures that rank or quantify the influence of every node (also called vertex) within a graph. They are related to centrality indices. Applications include measuring the influence of each person in a social network, understanding the role of infrastructure nodes in transportation networks, the Internet, or urban networks, and the participation of a given node in disease dynamics. Origin and development The traditional approach to understanding node importance is via centrality indicators. Centrality indices are designed to produce a ranking which accurately identifies the most influential nodes. Since the mid 2000s, however, social scientists and network physicists have begun to question the suitability of centrality indices for understanding node influence. Centralities may indicate the most influential nodes, but they are rather less informative for the vast majority of nodes which are not highly influential. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krackhardt Kite Graph
In graph theory, the Krackhardt kite graph is a simple graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' ve ... with ten nodes. The graph is named after David Krackhardt, a researcher of social network theory. Krackhardt introduced the graph in 1990 to distinguish different concepts of centrality. It has the property that the vertex with maximum degree (graph theory), degree (labeled 3 in the figure, with degree 6), the vertex with maximum betweenness centrality (labeled 7), and the two vertices with maximum closeness centrality (labeled 5 and 6) are all different from each other. References {{Combin-stub Individual graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Authority Distribution
The solution concept authority distribution was formulated by Lloyd Shapley and his student X. Hu in 2003 to measure the authority power of players in a well-contracted organization. The index generates the Shapley-Shubik power index and can be used in ranking, planning and organizational choice. Definition The organization contracts each individual by boss and approval relation with others. So each individual has its own authority structure, called command game. The Shapley-Shubik power index for these command games are collectively denoted by a power transit matrix Ρ. The authority distribution π is defined as the solution to the counterbalance equation π=πΡ. The basic idea for the counterbalance equation is that a person's power comes from his critical roles in others' command game; on the other hand, his power could also be redistributed to those who sit in his command game as vital players. For a simple legislative body, π is simply the Shapley-Shubik power index, b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shapley Value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a unique distribution (among the players) of a total surplus generated by the coalition of all players. The Shapley value is characterized by a collection of desirable properties. Hart (1989) provides a survey of the subject. The setup is as follows: a coalition of players cooperates, and obtains a certain overall gain from that cooperation. Since some players may contribute more to the coalition than others or may possess different bargaining power (for example threatening to destroy the whole surplus), what final distribution of generated surplus among the players should arise in any particular game? Or phrased differently: how important is each player to the overall cooperation, and what payoff can he or she reasonably expect? The Shapley val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
