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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. 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.

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. 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.

在数学中，'''<font color="#ff8880">随机图 Random Graph </font>'''是指图上的概率分布的一般术语。随机图可以简单地用概率分布表示，也可以用生成它们的随机过程表示。随机图的理论位于图论和概率论的交叉点上。从数学的角度来看，随机图被用来回答有关典型图的性质的问题。它的实际应用在所有需要对复杂网络进行建模的领域都能找到——许多随机图模型就此被人们所熟知，它反映了在不同领域遇到的不同类型的复杂网络。在数学上，随机图几乎完全指的是 Erdős-Rényi 随机图模型。在其他情况下，任何图形模型都可以称为随机图。

在数学中，'''<font color="#ff8000">随机图 Random Graph </font>'''是指图上的概率分布的一般术语。随机图可以简单地用概率分布表示，也可以用生成它们的随机过程表示。随机图的理论位于图论和概率论的交叉点上。从数学的角度来看，随机图被用来回答有关典型图的性质的问题。它的实际应用在所有需要对复杂网络进行建模的领域都能找到——许多随机图模型就此被人们所熟知，它反映了在不同领域遇到的不同类型的复杂网络。在数学上，随机图几乎完全指的是 Erdős-Rényi 随机图模型。在其他情况下，任何图形模型都可以称为随机图。

== Models ==

== Models ==

 

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.<ref name = "Random Graphs2">[[Béla Bollobás]], ''Random Graphs'', 1985, Academic Press Inc., London Ltd.</ref> Different '''random graph models''' produce different [[probability distribution]]s 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>p^m (1-p)^{N-m}</math> with the notation <math>N = \tbinom{n}{2}</math>.<ref name = "Random Graphs3">[[Béla Bollobás]], ''Probabilistic Combinatorics and Its Applications'', 1991, Providence, RI: American Mathematical Society.</ref>

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.<ref name = "Random Graphs2">[[Béla Bollobás]], ''Random Graphs'', 1985, Academic Press Inc., London Ltd.</ref> Different '''random graph models''' produce different [[probability distribution]]s 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>p^m (1-p)^{N-m}</math> with the notation <math>N = \tbinom{n}{2}</math>.<ref name = "Random Graphs3">[[Béla Bollobás]], ''Probabilistic Combinatorics and Its Applications'', 1991, Providence, RI: American Mathematical Society.</ref>