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Configuration Model
In network science, the configuration model is a method for generating random networks from a given degree sequence. It is widely used as a reference model for real-life social networks, because it allows the modeler to incorporate arbitrary degree distributions. Rationale for the model In the configuration model, the degree of each vertex is pre-defined, rather than having a probability distribution from which the given degree is chosen. As opposed to the Erdős–Rényi model, the degree sequence of the configuration model is not restricted to have a Poisson distribution, the model allows the user to give the network any desired degree distribution. Algorithm The following algorithm describes the generation of the model: # Take a degree sequence, i. e. assign a degree k_ito each vertex. The degrees of the vertices are represented as half-links or stubs. The sum of stubs must be even in order to be able to construct a graph (\sum k_i = 2m ). The degree sequence can be dr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Sequence And Different Realizations In The Configuration Model
Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (temperature), any of various units of temperature measurement * Degree API, a measure of density in the petroleum industry * Degree Baumé, a pair of density scales * Degree Brix, a measure of sugar concentration * Degree Gay-Lussac, a measure of the alcohol content of a liquid by volume, ranging from 0° to 100° * Degree proof, or simply proof, the alcohol content of a liquid, ranging from 0° to 175° in the UK, and from 0° to 200° in the U.S. * Degree of curvature, a unit of curvature measurement, used in civil engineering * Degrees of freedom (mechanics), the number of displacements or rotations needed to define the position and orientation of a body * Degrees of freedom (physics and chemistry), a concept describing depen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giant Component
In network theory, a giant component is a connected component of a given random graph that contains a finite fraction of the entire graph's vertices. Giant component in Erdős–Rényi model Giant components are a prominent feature of the Erdős–Rényi model (ER) of random graphs, in which each possible edge connecting pairs of a given set of vertices is present, independently of the other edges, with probability . In this model, if p \le \frac for any constant \epsilon>0, then with high probability all connected components of the graph have size , and there is no giant component. However, for p \ge \frac there is with high probability a single giant component, with all other components having size . For p=p_c = \frac, intermediate between these two possibilities, the number of vertices in the largest component of the graph, P_ is with high probability proportional to n^.. Giant component is also important in percolation theory. When a fraction of nodes, q=1-p, is removed ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modularity (networks)
Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules (also called groups, clusters or communities). Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Modularity is often used in optimization methods for detecting community structure in networks. However, it has been shown that modularity suffers a resolution limit and, therefore, it is unable to detect small communities. Biological networks, including animal brains, exhibit a high degree of modularity. Motivation Many scientifically important problems can be represented and empirically studied using networks. For example, biological and social patterns, the World Wide Web, metabolic networks, food webs, neural networks and pathological networks are real world problems that can be mathematically represented and topologically studied to reveal some unexpected str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Distribution
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network. Definition The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges. The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''. Thus if there are ''n'' nodes in total in a network and ''n''''k'' of them have degree ''k'', we have P(k) = \frac. The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Network
In the context of network theory, a complex network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in networks representing real systems. The study of complex networks is a young and active area of scientific research (since 2000) inspired largely by empirical findings of real-world networks such as computer networks, biological networks, technological networks, brain networks, climate networks and social networks. Definition Most social, biological, and technological networks display substantial non-trivial topological features, with patterns of connection between their elements that are neither purely regular nor purely random. Such features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure, and hierarchical structure. In the case of directed networks these feat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution Power
In mathematics, the convolution power is the ''n''-fold iteration of the convolution with itself. Thus if x is a Function (mathematics), function on Euclidean space R''d'' and n is a natural number, then the convolution power is defined by : x^ = \underbrace_n,\quad x^=\delta_0 where ∗ denotes the convolution operation of functions on R''d'' and δ0 is the Dirac delta distribution. This definition makes sense if ''x'' is an integrable function (in Lp space, L1), a rapidly decreasing distribution (mathematics), distribution (in particular, a compactly supported distribution) or is a finite Borel measure. If ''x'' is the distribution function of a random variable on the real line, then the ''n''th convolution power of ''x'' gives the distribution function of the sum of ''n'' independent random variables with identical distribution ''x''. The central limit theorem states that if ''x'' is in L1 and L2 with mean zero and variance σ2, then :P\left(\frac 0. If μ is a probabi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small-world Network
A small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but the neighbors of any given node are likely to be neighbors of each other and most nodes can be reached from every other node by a small number of hops or steps. Specifically, a small-world network is defined to be a network where the typical distance ''L'' between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes ''N'' in the network, that is: :L \propto \log N while the global clustering coefficient is not small. In the context of a social network, this results in the small world phenomenon of strangers being linked by a short chain of acquaintances. Many empirical graphs show the small-world effect, including social networks, wikis such as Wikipedia, gene networks, and even the underlying architecture of the Internet. It is the inspiration for many network-on-chip architectures in contempo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Distribution
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network. Definition The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges. The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''. Thus if there are ''n'' nodes in total in a network and ''n''''k'' of them have degree ''k'', we have P(k) = \frac. The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clustering Coefficient
In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes (Holland and Leinhardt, 1971; Watts and Strogatz, 1998). Two versions of this measure exist: the global and the local. The global version was designed to give an overall indication of the clustering in the network, whereas the local gives an indication of the embeddedness of single nodes. Local clustering coefficient The local clustering coefficient of a vertex (node) in a graph quantifies how close its neighbours are to being a clique (complete graph). Duncan J. Watts and Steven Strogatz introduced the measure in 1998 to determine whether a graph is a small-world network ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Network Science
Network science is an academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, cognitive and semantic networks, and social networks, considering distinct elements or actors represented by ''nodes'' (or ''vertices'') and the connections between the elements or actors as ''links'' (or ''edges''). The field draws on theories and methods including graph theory from mathematics, statistical mechanics from physics, data mining and information visualization from computer science, inferential modeling from statistics, and social structure from sociology. The United States National Research Council defines network science as "the study of network representations of physical, biological, and social phenomena leading to predictive models of these phenomena." Background and history The study of networks has emerged in diverse disciplines as a means of analyzing complex relational data. The earliest known paper in this f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjacency Matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e. all of its edges are bidirectional), the adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory. The adjacency matrix of a graph should be distinguished from its incidence matrix, a different matrix representation whose elements indicate whether vertex–edge pairs are incident or not, and its degree matrix, which contains information about the degree of each vertex. Definition For a simple graph with vertex set , the adjacency matrix is a square matrix such that its element is one when there is an edge from vertex to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
