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== Conditional random graphs ==

== Conditional random graphs ==

Consider a given random graph model defined on the  probability space <math>(\Omega, \mathcal{F}, P)</math> and let <math>\mathcal{P}(G) : \Omega \rightarrow R^{m}</math> be a real valued function which assigns to each graph in <math>\Omega</math> a vector of m properties.  

Consider a given random graph model defined on the  probability space <math>(\Omega, \mathcal{F}, P)</math> and let <math>\mathcal{P}(G) : \Omega \rightarrow R^{m}</math> be a real valued function which assigns to each graph in <math>\Omega</math> a vector of m properties.  

考虑一个给定的随机图模型定义在概率空间上(Omega，mathcal { f } ，p) </math > ，并且让 < math > mathcal { p }(g) : Omega tarrow r ^ { m } </math > 是一个真正的值函数，它在 < math > Omega </math > 中为每个图赋值一个 m 属性的向量。

考虑一个给定的随机图模型定义在概率空间上<math>(\Omega,\mathcal{F},P)</math> ，并且让 <math>\mathcal{P}(G):\Omega\tarrow R^{m}</math> 是一个真正的值函数，它在 <math>\Omega</math> 中为每个图赋值一个 ''m'' 属性的向量。

For a fixed <math>\mathbf{p} \in R^{m}</math>, ''conditional random graphs'' are models in which the probability measure <math>P</math> assigns zero probability to all graphs such that '<math>\mathcal{P}(G) \neq \mathbf{p} </math>.

For a fixed <math>\mathbf{p} \in R^{m}</math>, ''conditional random graphs'' are models in which the probability measure <math>P</math> assigns zero probability to all graphs such that '<math>\mathcal{P}(G) \neq \mathbf{p} </math>.

For a fixed <math>\mathbf{p} \in R^{m}</math>, conditional random graphs are models in which the probability measure <math>P</math> assigns zero probability to all graphs such that '<math>\mathcal{P}(G) \neq \mathbf{p} </math>.

For a fixed <math>\mathbf{p} \in R^{m}</math>, conditional random graphs are models in which the probability measure <math>P</math> assigns zero probability to all graphs such that '<math>\mathcal{P}(G) \neq \mathbf{p} </math>.

对于 r ^ { m } </math > 中的一个固定的 < math > mathbf { p } ，条件随机图是这样的模型，在这个模型中，机率量测 < math > p </math > 给所有图分配零概率，例如“ < math > mathcal { p }(g) neq mathbf { p } </math > 。

对于 <math>R^{m}</math> 中的一个固定的 <math>\mathbf{p}</math> ，条件随机图是这样的模型，在这个模型中，机率量测 <math>p</math> 给所有图分配零概率，例如“ <math>\mathcal{P}(G)\neq\mathbf{p}</math> 。

Special cases are conditionally uniform random graphs, where <math>P</math> assigns equal probability to all the graphs having specified properties. They can be seen as a generalization of the Erdős–Rényi model G(n,M), when the conditioning information is not necessarily the number of edges M, but whatever other arbitrary graph property <math>\mathcal{P}(G)</math>. In this case very few analytical results are available and simulation is required to obtain empirical distributions of average properties.  

Special cases are conditionally uniform random graphs, where <math>P</math> assigns equal probability to all the graphs having specified properties. They can be seen as a generalization of the Erdős–Rényi model G(n,M), when the conditioning information is not necessarily the number of edges M, but whatever other arbitrary graph property <math>\mathcal{P}(G)</math>. In this case very few analytical results are available and simulation is required to obtain empirical distributions of average properties.  

特殊情况是条件均匀随机图，其中 < math > p </math > 给所有具有指定性质的图赋予相等的概率。它们可以被看作是 erd s-Rényi 模型 g (n，m)的一个推广，当条件信息不一定是边的个数 m，而是其他任意图性质 < math > mathcal { p }(g) </math > 时。在这种情况下，很少有分析结果可用，需要模拟来获得平均性质的经验分布。

特殊情况是'''<font color="#FF8000">条件均匀随机图 Conditionally Uniform Graph </font>'''，其中 <math>p</math> 给所有具有指定性质的图赋予相等的概率。它们可以被看作是 Erdős–Rényi 模型 ''G''(''n''，''m'')的一个推广，当条件信息不一定是边的个数 ''M''，而是其他任意图性质 <math>\mathcal{P}(G)</math> 时。在这种情况下，很少有分析结果可用，需要模拟来获得平均性质的经验分布。

 

 

==Interdependent graphs==

==Interdependent graphs==