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Low-degree Saturation
In a scale-free network the degree distribution follows a power law function. In some empirical examples this power-law fits the degree distribution well only in the high degree region; in some small degree nodes the empirical degree-distribution deviates from it. See for example the network of scientific citations. This deviation of the observed degree-distribution from the theoretical prediction at the low-degree region is often referred as low-degree saturation. The empirical degree-distribution typically deviates downward from the power-law function fitted on higher order nodes, which means low-degree nodes are less frequent in real data than what is predicted by the Barabási–Albert model. Theoretical foundation One of the key assumptions of the BA model is preferential attachment A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scale-free Network
A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction ''P''(''k'') of nodes in the network having ''k'' connections to other nodes goes for large values of ''k'' as : P(k) \ \sim \ k^\boldsymbol where \gamma is a parameter whose value is typically in the range 2<\gamma<3 (wherein the second moment ( scale parameter) of is infinite but the first moment is finite), although occasionally it may lie outside these bounds. The name "scale-free" could be explained by the fact that some moments of the degree distribution are not defined, so that the network does not have a characteristic scale or "size". and the [...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' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Law
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent: one quantity varies as a power of another. The change is independent of the initial size of those quantities. For instance, the area of a square has a power law relationship with the length of its side, since if the length is doubled, the area is multiplied by 2, while if the length is tripled, the area is multiplied by 3, and so on. Empirical examples The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, cloud sizes, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barabási–Albert Model
The Barabási–Albert (BA) model is an algorithm for generating random scale-free network, scale-free complex network, networks using a preferential attachment mechanism. Several natural and human-made systems, including the Internet, the World Wide Web, citation analysis, citation networks, and some social networks are thought to be approximately scale-free and certainly contain few nodes (called hubs) with unusually high degree as compared to the other nodes of the network. The BA model tries to explain the existence of such nodes in real networks. The algorithm is named for its inventors Albert-László Barabási and Réka Albert. Concepts Many observed networks (at least approximately) fall into the class of scale-free networks, meaning that they have power law, power-law (or scale-free) degree distributions, while random graph models such as the Erdős–Rényi model, Erdős–Rényi (ER) model and the Watts and Strogatz model, Watts–Strogatz (WS) model do not exhibit po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preferential Attachment
A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not. "Preferential attachment" is only the most recent of many names that have been given to such processes. They are also referred to under the names Yule process, cumulative advantage, the rich get richer, and the Matthew effect. They are also related to Gibrat's law. The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate power law distributions. If preferential attachment is non-linear, measured distributions may deviate from a power law. These mechanisms may generate distributions which are approximately power law over transient periods. Definition A preferential attachment process is a stochastic urn p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Attractiveness
The initial attractiveness is a possible extension of the Barabási–Albert model (preferential attachment model). The Barabási–Albert model generates scale-free networks where the degree distribution can be described by a pure power law. However, the degree distribution of most real life networks cannot be described by a power law solely. The most common discrepancies regarding the degree distribution found in real networks are the high degree cut-off (or structural cut-off) and the low degree saturation. The inclusion of initial attractiveness in the Barabási–Albert model addresses the low-degree saturation phenomenon. Intuitively, it also makes sense since when moving to a new city you can still make new connections even though you don't know anyone. But in the Barabási–Albert model a node that has degree zero has probability 0 of garnering new connections. With initial attractiveness you always have a residual "attractiveness" irrespective of how many connections you ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small-world Network
A small-world network is a graph characterized by a high clustering coefficient and low distances. In an example of the social network, high clustering implies the high probability that two friends of one person are friends themselves. The low distances, on the other hand, mean that there is a short chain of social connections between any two people (this effect is known as six degrees of separation). 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 n ... [...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 network, 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 statistics, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Attractiveness
The initial attractiveness is a possible extension of the Barabási–Albert model (preferential attachment model). The Barabási–Albert model generates scale-free networks where the degree distribution can be described by a pure power law. However, the degree distribution of most real life networks cannot be described by a power law solely. The most common discrepancies regarding the degree distribution found in real networks are the high degree cut-off (or structural cut-off) and the low degree saturation. The inclusion of initial attractiveness in the Barabási–Albert model addresses the low-degree saturation phenomenon. Intuitively, it also makes sense since when moving to a new city you can still make new connections even though you don't know anyone. But in the Barabási–Albert model a node that has degree zero has probability 0 of garnering new connections. With initial attractiveness you always have a residual "attractiveness" irrespective of how many connections you ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
