Random Graph Model
A Random Graph Model is a Graph is that can be generated and described by probability distribution or by a random process.
- AKA: Random Graph Generation Model, Gilbert's Random Graph Model.
- Example(s):
- Counter-Example(s):
- See: Random, Graph, Graph Theory, Probability Theory.
References
2021a
- (Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Random_graph Retrieved:2021-8-14.
- In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them.[1] [2] The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a random graph.
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A random graph is obtained by starting with a set of n isolated vertices and adding successive edges between them at random. The aim of the study in this field is to determine at what stage a particular property of the graph is likely to arise.[3] Different random graph models produce different probability distributions on graphs. Most commonly studied is the one proposed by Edgar Gilbert, denoted G(n,p), in which every possible edge occurs independently with probability 0 < p < 1. The probability of obtaining any one particular random graph with m edges is [math]\displaystyle{ p^m (1-p)^{N-m} }[/math] with the notation [math]\displaystyle{ N = \tbinom{n}{2} }[/math].[4]
- In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them.[1] [2] The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a random graph.
2005
- (Wasserman & Robins, 2005) ⇒ Stanley Wasserman, and G. L. Robins. (2005). “An Introduction to Random Graphs, dependence graphs, and p*.” In: P. Carrington, J. Scott, and Stanley Wasserman (Eds.), "Models and Methods in Social Network Analysis.” Cambridge University Press.
- ↑ Bollobás, Béla (2001). Random Graphs (2nd ed.). Cambridge University Press.
- ↑ Frieze, Alan; Karonski, Michal (2015). Introduction to Random Graphs. Cambridge University Press.
- ↑ Béla Bollobás, Random Graphs, 1985, Academic Press Inc., London Ltd.
- ↑ Béla Bollobás, Probabilistic Combinatorics and Its Applications, 1991, Providence, RI: American Mathematical Society.