198
个编辑
更改
跳到导航
跳到搜索
第308行：
第308行： + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
→Erdős–Rényi 随机图模型
=== Erdős–Rényi 随机图模型 ===
=== Erdős–Rényi 随机图模型 ===
[[File:ER model.svg|thumb|This [[Erdős–Rényi model]] is generated with {{math|<VAR>N</VAR> {{=}} 4}} nodes. For each edge in the complete graph formed by all {{mvar|N}} nodes, a random number is generated and compared to a given probability. If the random number is less than {{mvar|p}}, an edge is formed on the model.]]
The '''[[Erdős–Rényi model]]''', named for [[Paul Erdős]] and [[Alfréd Rényi]], is used for generating [[random graph]]s in which edges are set between nodes with equal probabilities. It can be used in the [[probabilistic method]] to prove the existence of graphs satisfying various properties, or to provide a rigorous definition of what it means for a property to hold for almost all graphs.
To generate an Erdős–Rényi model <math>G(n, p)
</math> two parameters must be specified: the total number of nodes {{mvar|n}} and the probability {{mvar|p}} that a random pair of nodes has an edge.
Because the model is generated without bias to particular nodes, the degree distribution is binomial: for a randomly chosen vertex <math>v</math>,
: <math>P(\deg(v) = k) = {n-1\choose k} p^k (1-p)^{n-1-k}.</math>
In this model the clustering coefficient is {{math|0}} [[Almost surely|a.s]]. The behavior of <math>G(n, p)
</math> can be broken into three regions.
''Subcritical'' <math>n p < 1
</math>: All components are simple and very small, the largest component has size <math>|C_1| = O(\log n)
</math>;
''Critical'' <math>n p = 1
</math>: <math>|C_1| = O(n^\frac{2}{3})
</math>;
''Supercritical'' <math>n p >1
</math>:<math>|C_1| \approx yn
</math> where <math>y = y(n p)
</math> is the positive solution to the equation <math>e^{-p n y }=1-y
</math>.
The largest connected component has high complexity. All other components are simple and small <math>|C_2| = O(\log n)
</math>.
以[[Paul Erdős]]和[[Alfréd Rényi]]命名的'''[[Erdős–Rényi 模型]]'''用于生成[[随机图]]，它的边是等概率连接的节点的集合。[[随机图]]可以用来证明[[概率方法]]中满足某些条件的图的存在性，或者对几乎所有图给出某个性质的严格定义。
以[[Paul Erdős]]和[[Alfréd Rényi]]命名的'''[[Erdős–Rényi 模型]]'''用于生成[[随机图]]，它的边是等概率连接的节点的集合。[[随机图]]可以用来证明[[概率方法]]中满足某些条件的图的存在性，或者对几乎所有图给出某个性质的严格定义。