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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.
In the mathematical field of graph theory, the Erdős–Rényi model refers to one of two closely related models for generating random graphs or the evolution of a random network. These models are named after Hungarian mathematicians Paul Erdős and Alfréd Rényi, who introduced one of the models in 1959.
The Rado graph, as numbered by Ackermann (1937) and Rado (1964).. In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge.
The random graphs of the book are generated from the Erdős–Rényi–Gilbert model (,) in which vertices are given and a random choice is made whether to connect each pair of vertices by an edge, independently for each pair, with probability of making a connection.
Edgar Nelson Gilbert (July 25, 1923 – June 15, 2013) was an American mathematician and coding theorist, a longtime researcher at Bell Laboratories.His accomplishments include the Gilbert–Varshamov bound in coding theory, the Gilbert–Elliott model of bursty errors in signal transmission, the Erdős–Rényi–Gilbert model for random graphs, the Gilbert disk model of random geometric ...
The formal study of random graphs dates back to the work of Paul Erdős and Alfréd Rényi. [2] The graphs they considered, now known as the classical or Erdős–Rényi (ER) graphs, offer a simple and powerful model with many applications. However the ER graphs do not have two important properties observed in many real-world networks:
In extremal graph theory, Szemerédi’s regularity lemma states that a graph can be partitioned into a bounded number of parts so that the edges between parts are regular. The lemma shows that certain properties of random graphs can be applied to dense graphs like counting the copies of a given subgraph within graphs.
Any random graph model (at a fixed set of parameter values) results in a probability distribution on graphs, and those that are maximum entropy within the considered class of distributions have the special property of being maximally unbiased null models for network inference [2] (e.g. biological network inference).