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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.
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.
There are two closely related variants of the Erdős–Rényi random graph model. A graph generated by the binomial model of Erdős and Rényi (p = 0.01) In the (,) model, a graph is chosen uniformly at random from the collection of all graphs which have nodes and edges. The nodes are considered to be labeled, meaning that graphs obtained from ...
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.
The introduction of probabilistic methods in graph theory, especially in the study of Erdős and Rényi of the asymptotic probability of graph connectivity, gave rise to yet another branch, known as random graph theory, which has been a fruitful source of graph-theoretic results.
Exponential Random Graph Models (ERGMs) are a family of statistical models for analyzing data from social and other networks. [ 1 ] [ 2 ] Examples of networks examined using ERGM include knowledge networks, [ 3 ] organizational networks, [ 4 ] colleague networks, [ 5 ] social media networks, networks of scientific development, [ 6 ] and others.
The bunkbed conjecture (also spelled bunk bed conjecture) is a statement in percolation theory, a branch of mathematics that studies the behavior of connected clusters in a random graph. The conjecture is named after its analogy to a bunk bed structure. It was first posited by Pieter Kasteleyn in 1985. [1]
A random graph model is an exchangeable random graph model if and only if it can be defined in terms of a (possibly random) graphon in this way. The model based on a fixed graphon W {\displaystyle W} is sometimes denoted G ( n , W ) {\displaystyle \mathbb {G} (n,W)} , by analogy with the Erdős–Rényi model of random graphs.