随机图

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图：关于可数无穷随机图

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. 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，随机图 Random Graph 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

模型

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.^[3] Different random graph models produce different probability distributions 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]\displaystyle{ p^m (1-p)^{N-m} }[/math] with the notation [math]\displaystyle{ N = \tbinom{n}{2} }[/math].^[4]

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. Different random graph models produce different probability distributions 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]\displaystyle{ p^m (1-p)^{N-m} }[/math] with the notation [math]\displaystyle{ N = \tbinom{n}{2} }[/math].

通过从一组 n 个孤立的顶点开始，并在它们之间随机添加连续的边，得到一个随机图。这个领域的研究的目的是确定在什么阶段图形的特殊性质可能出现。不同的随机图模型在图上产生不同的概率分布。最常见的研究是由埃德加 · 吉尔伯特提出的，表示G (n,p) ，其中每个可能的边独立出现的概率为0 < p < 1。获得任意一个 m 边随机图的概率是 [math]\displaystyle{ p^m (1-p)^(N-m) }[/math] ，记号是 [math]\displaystyle{ N=\tbinom{n}{2} }[/math]

A closely related model, the Erdős–Rényi model denoted G(n,M), assigns equal probability to all graphs with exactly M edges. With 0 ≤ M ≤ N, G(n,M) has [math]\displaystyle{ \tbinom{N}{M} }[/math] elements and every element occurs with probability [math]\displaystyle{ 1/\tbinom{N}{M} }[/math].^[3] The latter model can be viewed as a snapshot at a particular time (M) of the random graph process [math]\displaystyle{ \tilde{G}_n }[/math], which is a stochastic process that starts with n vertices and no edges, and at each step adds one new edge chosen uniformly from the set of missing edges.

A closely related model, the Erdős–Rényi model denoted G(n,M), assigns equal probability to all graphs with exactly M edges. With 0 ≤ M ≤ N, G(n,M) has [math]\displaystyle{ \tbinom{N}{M} }[/math] elements and every element occurs with probability [math]\displaystyle{ 1/\tbinom{N}{M} }[/math]. The latter model can be viewed as a snapshot at a particular time (M) of the random graph process [math]\displaystyle{ \tilde{G}_n }[/math], which is a stochastic process that starts with n vertices and no edges, and at each step adds one new edge chosen uniformly from the set of missing edges.

一个密切相关的模型，Erdős-Rényi 模型 Erdős–Rényi Model 表示G (n,M)，给每一个正好有M条边的图赋予等概率。当0≤ M ≤ N 时，G (n,M)具有 [math]\displaystyle{ \tbinom{N}{M} }[/math] 元素，且每个元素都以概率[math]\displaystyle{ 1/\tbinom{N}{M} }[/math] 出现。后一个模型可以看作是随机图过程[math]\displaystyle{ \tilde{G}_n }[/math]某个特定时间(M)的一个快照，这个时间(M)是从 n 个顶点开始没有边的一个随机过程，每个步骤均匀地从缺失的边集中选择一个新的边。

If instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability 0 < p < 1, then we get an object G called an infinite random graph. Except in the trivial cases when p is 0 or 1, such a G almost surely has the following property:

If instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability 0 < p < 1, then we get an object G called an infinite random graph. Except in the trivial cases when p is 0 or 1, such a G almost surely has the following property:

如果我们从一个无限的顶点集合开始，然后再次让每个可能的边以概率0 < p < 1独立出现，那么我们得到一个对象 G 称为无限随机图 Infinite Graph 。除了在 p = 0或1的平凡情况下，这样的 G 几乎肯定具有以下性质:

Given any n + m elements [math]\displaystyle{ a_1,\ldots, a_n,b_1,\ldots, b_m \in V }[/math], there is a vertex c in V that is adjacent to each of [math]\displaystyle{ a_1,\ldots, a_n }[/math] and is not adjacent to any of [math]\displaystyle{ b_1,\ldots, b_m }[/math].

在 [math]\displaystyle{ v }[/math] 中，给定任何 n + m 个元素，[math]\displaystyle{ a_1,\ldots,a_n,b_1,\ldots,b_m\ }[/math] 中有一个顶点 c，它与每个 [math]\displaystyle{ a_1,\ldots,a_n }[/math] 相邻，并且不与任何 [math]\displaystyle{ b_1,\ldots,b_m }[/math] 相邻。

It turns out that if the vertex set is countable then there is, up to isomorphism, only a single graph with this property, namely the Rado graph. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the random graph. However, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property.

It turns out that if the vertex set is countable then there is, up to isomorphism, only a single graph with this property, namely the Rado graph. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the random graph. However, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property.

结果表明，如果顶点集是可数的，那么在同构 Isomorphism 意义下，只有一个图具有这个性质，即 Rado 图。因此，任何可数无限随机图几乎可以肯定是 Rado 图，由于这个原因，有时被简称为随机图。然而，对于不可数图 Uncountable Graph 类似的结果是不正确的，不可数图中有许多(不同构)图 Nonisomorphic Graph 满足上述性质。

Another model, which generalizes Gilbert's random graph model, is the random dot-product model. A random dot-product graph associates with each vertex a real vector. The probability of an edge uv between any vertices u and v is some function of the dot product u • v of their respective vectors.

Another model, which generalizes Gilbert's random graph model, is the random dot-product model. A random dot-product graph associates with each vertex a real vector. The probability of an edge uv between any vertices u and v is some function of the dot product u • v of their respective vectors.

另一个模型，概括了吉尔伯特的随机图模型，是随机点积模型 Random Dot-product Model。一个随机点积图将每个顶点与一个实向量相关联。任意顶点 u 和 v 之间的边 uv 的概率是它们各自向量的点积 u · v 的某个函数。

The network probability matrix models random graphs through edge probabilities, which represent the probability [math]\displaystyle{ p_{i,j} }[/math] that a given edge [math]\displaystyle{ e_{i,j} }[/math] exists for a specified time period. This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs structure.

网络转移矩阵 NetWork Probability Matrix 通过边概率对随机图进行建模，这些边概率代表给定的边存在一个特定的时间段的概率。这个模型可以扩展到有向和无向，加权和无权，以及静态或动态的图形结构。

For M ≃ pN, where N is the maximal number of edges possible, the two most widely used models, G(n,M) and G(n,p), are almost interchangeable.^[5]

For M ≃ pN, where N is the maximal number of edges possible, the two most widely used models, G(n,M) and G(n,p), are almost interchangeable.

对于 Msomething pN，其中 N 是可能的最大边数，两个最广泛使用的模型，G (n,， M)和 G (n，p)是几乎可互换的。

Random regular graphs form a special case, with properties that may differ from random graphs in general.

随机正则图 Random Regular Graph 是一种特殊情况，其性质可能与一般随机图不同。

Once we have a model of random graphs, every function on graphs, becomes a random variable. The study of this model is to determine if, or at least estimate the probability that, a property may occur.^[4]

Once we have a model of random graphs, every function on graphs, becomes a random variable. The study of this model is to determine if, or at least estimate the probability that, a property may occur.

一旦我们有了一个随机图的模型，图上的每个函数都变成了一个随机变量。对这个模型的研究是为了确定，或者至少估计一个属性可能发生的概率。

Terminology

术语

The term 'almost every' in the context of random graphs refers to a sequence of spaces and probabilities, such that the error probabilities tend to zero.^[4]

The term 'almost every' in the context of random graphs refers to a sequence of spaces and probabilities, such that the error probabilities tend to zero.

在随机图的上下文中，“几乎每一个”这个术语指的是一系列空间和概率，以至于出错概率趋于零。

Properties

性质

The theory of random graphs studies typical properties of random graphs, those that hold with high probability for graphs drawn from a particular distribution. For example, we might ask for a given value of [math]\displaystyle{ n }[/math] and [math]\displaystyle{ p }[/math] what the probability is that [math]\displaystyle{ G(n,p) }[/math] is connected. In studying such questions, researchers often concentrate on the asymptotic behavior of random graphs—the values that various probabilities converge to as [math]\displaystyle{ n }[/math] grows very large. Percolation theory characterizes the connectedness of random graphs, especially infinitely large ones.

随机图理论研究随机图的典型性质，即从特定分布中抽取的图的高概率性质。例如，我们可以要求一个给定的值 [math]\displaystyle{ n }[/math] 和 [math]\displaystyle{ p }[/math] 什么是 [math]\displaystyle{ G(n，p) }[/math] 连接的概率。在研究这些问题时，研究人员往往集中在随机图的渐近行为上——各种概率收敛到的值变得非常大。渗流理论 Percolation Theory 刻画了随机图，特别是无穷大图的连通性。

Percolation is related to the robustness of the graph (called also network). Given a random graph of [math]\displaystyle{ n }[/math] nodes and an average degree [math]\displaystyle{ \langle k\rangle }[/math]. Next we remove randomly a fraction [math]\displaystyle{ 1-p }[/math] of nodes and leave only a fraction [math]\displaystyle{ p }[/math]. There exists a critical percolation threshold [math]\displaystyle{ p_c=\tfrac{1}{\langle k\rangle} }[/math] below which the network becomes fragmented while above [math]\displaystyle{ p_c }[/math] a giant connected component exists.^[1]^[5]^[6]^[7]

Percolation is related to the robustness of the graph (called also network). Given a random graph of [math]\displaystyle{ n }[/math] nodes and an average degree [math]\displaystyle{ \langle k\rangle }[/math]. Next we remove randomly a fraction [math]\displaystyle{ 1-p }[/math] of nodes and leave only a fraction [math]\displaystyle{ p }[/math]. There exists a critical percolation threshold [math]\displaystyle{ p_c=\tfrac{1}{\langle k\rangle} }[/math] below which the network becomes fragmented while above [math]\displaystyle{ p_c }[/math] a giant connected component exists.

渗流 Percolation 与图形(也称为网络)的健壮性 Robustness 有关。给定一个随机图形，其中的节点是 [math]\displaystyle{ n }[/math] 和一个平均度 [math]\displaystyle{ \langle k\rangle }[/math] 。接下来我们随机移除一部分概率为 [math]\displaystyle{ 1-p }[/math] 的节点，只留下一部分概率为 [math]\displaystyle{ p }[/math] 的节点。存在一个临界渗透阈值 [math]\displaystyle{ p_c=\tfrac{1}{\langle k\rangle} }[/math]，当低于这个临界渗透阈值，网络变得支离破碎，而高于临界渗透阈值 [math]\displaystyle{ p_c }[/math] 的网络则存在一个巨大的连接元件 Connected Component (图论)。

^[8]^[9]

Localized percolation refers to removing a node its neighbors, next nearest neighbors etc. until a fraction of [math]\displaystyle{ 1-p }[/math] of nodes from the network is removed. It was shown that for random graph with Poisson distribution of degrees [math]\displaystyle{ p_c=\tfrac{1}{\langle k\rangle} }[/math] exactly as for random removal. For other types of degree distributions [math]\displaystyle{ p_c }[/math] for localized attack is different from random attack^[10]

Localized percolation refers to removing a node its neighbors, next nearest neighbors etc. until a fraction of [math]\displaystyle{ 1-p }[/math] of nodes from the network is removed. It was shown that for random graph with Poisson distribution of degrees [math]\displaystyle{ p_c=\tfrac{1}{\langle k\rangle} }[/math] exactly as for random removal. For other types of degree distributions [math]\displaystyle{ p_c }[/math] for localized attack is different from random attack

局部渗滤 Localized Percolation 指的是去除一个节点的邻居、次近邻等。直到网络中概率为 [math]\displaystyle{ 1-p }[/math] 的一部分节点被移除。结果表明，对于服从泊松分布的随机图，其度分布为 [math]\displaystyle{ p_c=\tfrac{1}{\langle k\rangle} }[/math] ，和随机删除是一样的。对于其他类型的度分布 [math]\displaystyle{ p_c }[/math]，局部攻击和随机攻击是不同的。

(threshold functions, evolution of [math]\displaystyle{ \tilde G }[/math])

(threshold functions, evolution of [math]\displaystyle{ \tilde G }[/math])

(阈值函数 Threshold Functions , [math]\displaystyle{ \tilde G }[/math] 的演化)

Random graphs are widely used in the probabilistic method, where one tries to prove the existence of graphs with certain properties. The existence of a property on a random graph can often imply, via the Szemerédi regularity lemma, the existence of that property on almost all graphs.

随机图在概率方法中得到了广泛的应用，它试图证明具有一定性质的图的存在性。通过 Szemerédi 正则引理 Szemerédi Regularity Lemma ，随机图上一个性质的存在常常意味着几乎所有图上该性质的存在。

In random regular graphs, [math]\displaystyle{ G(n,r-reg) }[/math] are the set of [math]\displaystyle{ r }[/math]-regular graphs with [math]\displaystyle{ r = r(n) }[/math] such that [math]\displaystyle{ n }[/math] and [math]\displaystyle{ m }[/math] are the natural numbers, [math]\displaystyle{ 3 \le r \lt n }[/math], and [math]\displaystyle{ rn = 2m }[/math] is even.^[3]

In random regular graphs, [math]\displaystyle{ G(n,r-reg) }[/math] are the set of [math]\displaystyle{ r }[/math]-regular graphs with [math]\displaystyle{ r = r(n) }[/math] such that [math]\displaystyle{ n }[/math] and [math]\displaystyle{ m }[/math] are the natural numbers, [math]\displaystyle{ 3 \le r \lt n }[/math], and [math]\displaystyle{ rn = 2m }[/math] is even.

在随机正则图中，[math]\displaystyle{ G(n，r-reg) }[/math] 是 [math]\displaystyle{ r=r(n) }[/math]的[math]\displaystyle{ r- }[/math]正则图的集合，其中 [math]\displaystyle{ n }[/math] 和 [math]\displaystyle{ m }[/math] 是自然数，[math]\displaystyle{ 3 \le r\lt n }[/math] ，[math]\displaystyle{ rn = 2m }[/math] 是偶数。

The degree sequence of a graph [math]\displaystyle{ G }[/math] in [math]\displaystyle{ G^n }[/math] depends only on the number of edges in the sets^[3]

The degree sequence of a graph [math]\displaystyle{ G }[/math] in [math]\displaystyle{ G^n }[/math] depends only on the number of edges in the sets

在 [math]\displaystyle{ G^n }[/math] 中，图 [math]\displaystyle{ G }[/math] 的度序列仅取决于集合中的边数。

[math]\displaystyle{ V_n^{(2)} = \left \{ij \ : \ 1 \leq j \leq n, i \neq j \right \} \subset V^{(2)}, \qquad i=1, \cdots, n. }[/math]

[math]\displaystyle{ V_n^{(2)}=\left\{ij\:\1\leq j\leq n,i \neq j \right\}\子集 V^{(2)},\qquad i=1,\cdots,n. }[/math]

If edges, [math]\displaystyle{ M }[/math] in a random graph, [math]\displaystyle{ G_M }[/math] is large enough to ensure that almost every [math]\displaystyle{ G_M }[/math] has minimum degree at least 1, then almost every [math]\displaystyle{ G_M }[/math] is connected and, if [math]\displaystyle{ n }[/math] is even, almost every [math]\displaystyle{ G_M }[/math] has a perfect matching. In particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected.^[3]

If edges, [math]\displaystyle{ M }[/math] in a random graph, [math]\displaystyle{ G_M }[/math] is large enough to ensure that almost every [math]\displaystyle{ G_M }[/math] has minimum degree at least 1, then almost every [math]\displaystyle{ G_M }[/math] is connected and, if [math]\displaystyle{ n }[/math] is even, almost every [math]\displaystyle{ G_M }[/math] has a perfect matching. In particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected.

如果随机图中的边 [math]\displaystyle{ M }[/math]，[math]\displaystyle{ G_M }[/math] 足够大以确保几乎每个 [math]\displaystyle{ G_M }[/math] 的最小阶数至少为1，那么几乎每个 [math]\displaystyle{ G }[/math] 是连通的，如果 [math]\displaystyle{ n }[/math] 是偶数，则几乎每个 [math]\displaystyle{ G_M }[/math] 都有一个完美匹配 Perfect Matching 。特别是，在几乎每个随机图中，最后一个孤立点消失的那一刻，图会变连通。

Almost every graph process on an even number of vertices with the edge raising the minimum degree to 1 or a random graph with slightly more than [math]\displaystyle{ \tfrac{n}{4}\log(n) }[/math] edges and with probability close to 1 ensures that the graph has a complete matching, with exception of at most one vertex.

几乎每个偶数顶点上的最小度提高到1的图或稍大于 [math]\displaystyle{ \tfrac{n}{4}\log(n) }[/math] 边且概率接近1的随机图都能确保图有完全匹配 Complete Matching ，但最多只有一个顶点。

For some constant [math]\displaystyle{ c }[/math], almost every labeled graph with [math]\displaystyle{ n }[/math] vertices and at least [math]\displaystyle{ cn\log(n) }[/math] edges is Hamiltonian. With the probability tending to 1, the particular edge that increases the minimum degree to 2 makes the graph Hamiltonian.

对于某些常数 [math]\displaystyle{ c }[/math] ，几乎所有带有 [math]\displaystyle{ n }[/math] 顶点和至少 [math]\displaystyle{ cn\log(n) }[/math] 边的标记图都是 哈密尔顿环 Hamiltonian Cycle 的。在概率趋于1的情况下，将最小度增加到2的特殊边会使图成为哈密尔顿图。

Properties of random graph may change or remain invariant under graph transformations. Mashaghi A. et al., for example, demonstrated that a transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient.^[11]

Properties of random graph may change or remain invariant under graph transformations. Mashaghi A. et al., for example, demonstrated that a transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient.

在图变换下，随机图的性质可以改变或保持不变。例如，Mashaghi A.等人证明了将随机图转换为其 边-对偶图 Edge-Dual Graphs (或 线图 Liner Graph )的变换会产生一个度分布几乎相同，但具有 度相关性 Degree Correlations 和显著更高的 聚类系数 Clustering Coefficent 的图的集合。

Coloring

着色

Given a random graph G of order n with the vertex V(G) = {1, ..., n}, by the greedy algorithm on the number of colors, the vertices can be colored with colors 1, 2, ... (vertex 1 is colored 1, vertex 2 is colored 1 if it is not adjacent to vertex 1, otherwise it is colored 2, etc.).^[3]

Given a random graph G of order n with the vertex V(G) = {1, ..., n}, by the greedy algorithm on the number of colors, the vertices can be colored with colors 1, 2, ... (vertex 1 is colored 1, vertex 2 is colored 1 if it is not adjacent to vertex 1, otherwise it is colored 2, etc.).

给定一个 n 阶随机图 G，其顶点 V(G) = {1，... ，n } ，通过关于颜色数的贪婪算法 Greedy Algorithm ，可以用颜色1,2。。。 (如果顶点1与顶点2不相邻，则顶点1与顶点2为染相同的颜色，否则染不同的颜色，以此类推。).

The number of proper colorings of random graphs given a number of q colors, called its chromatic polynomial, remains unknown so far. The scaling of zeros of the chromatic polynomial of random graphs with parameters n and the number of edges m or the connection probability p has been studied empirically using an algorithm based on symbolic pattern matching.^[12]

The number of proper colorings of random graphs given a number of q colors, called its chromatic polynomial, remains unknown so far. The scaling of zeros of the chromatic polynomial of random graphs with parameters n and the number of edges m or the connection probability p has been studied empirically using an algorithm based on symbolic pattern matching.

给定一个 q 颜色的随机图的真染色数，称为它的染色多项式 Chormatic Polynomial ，到目前为止还不清楚。利用符号模式匹配算法 Algorithm Based On Symbolic Pattern Matching ，对参数为 n 、边数为 m 或连接概率为 p 的随机图的染色多项式零点的标度进行了实验研究。。

Random trees

A random tree is a tree or arborescence that is formed by a stochastic process. In a large range of random graphs of order n and size M(n) the distribution of the number of tree components of order k is asymptotically Poisson. Types of random trees include uniform spanning tree, random minimal spanning tree, random binary tree, treap, rapidly exploring random tree, Brownian tree, and random forest.

随机树是由随机过程组成的树或树木。在 n 阶和 m (n)大小的随机图的大范围内，k 阶树分量个数的分布是渐近泊松的。随机树的类型包括均匀生成树、随机最小生成树、随机二叉树、树、快速探索随机树、布朗树和随机森林。

Conditional random graphs

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

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

For a fixed [math]\displaystyle{ \mathbf{p} \in R^{m} }[/math], conditional random graphs are models in which the probability measure [math]\displaystyle{ P }[/math] assigns zero probability to all graphs such that '[math]\displaystyle{ \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 > 。

Special cases are conditionally uniform random graphs, where [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \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。

Interdependent graphs

In interdependent graphs nodes in one network (graph) depend on other networks to function. So failures in one or several graphs induce cascading failures between the graphs which may lead to abrupt collapse.^[13]^[14]

In interdependent graphs nodes in one network (graph) depend on other networks to function. So failures in one or several graphs induce cascading failures between the graphs which may lead to abrupt collapse.

在一个网络(图)的相互依赖图节点依赖于其他网络的功能。因此，一个或多个图的故障会导致图之间的级联故障，从而可能导致突然崩溃。

History

The earliest use of a random graph model was by Helen Hall Jennings and Jacob Moreno in 1938 where a "chance sociogram" (a directed Erdős-Rényi model) was considered in studying comparing the fraction of reciprocated links in their network data with the random model.^[15] Another use, under the name "random net", was by Solomonoff and Rapoport in 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices.^[16]

The earliest use of a random graph model was by Helen Hall Jennings and Jacob Moreno in 1938 where a "chance sociogram" (a directed Erdős-Rényi model) was considered in studying comparing the fraction of reciprocated links in their network data with the random model. Another use, under the name "random net", was by Solomonoff and Rapoport in 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices.

最早使用随机图模型的是海伦 · 霍尔 · 詹宁斯和雅各布 · 莫雷诺，他们在1938年提出了一个“偶然社会记录模型”(一个有向的 erd s-rényi 模型) ，用来比较他们的网络数据中的回传链接的比例和随机模型。另一个被称为“随机网络”的应用是1951年 Solomonoff 和 Rapoport 使用的有向图模型，这些有向图具有固定的出度，并且随机选择附加到其他顶点。

The Erdős–Rényi model of random graphs was first defined by Paul Erdős and Alfréd Rényi in their 1959 paper "On Random Graphs"^[9] and independently by Gilbert in his paper "Random graphs".^[6]

The Erdős–Rényi model of random graphs was first defined by Paul Erdős and Alfréd Rényi in their 1959 paper "On Random Graphs" and independently by Gilbert in his paper "Random graphs".

随机图的 erd-Rényi 模型是由 Paul erd 和 Alfréd Rényi 于1959年发表的论文《随机图论》中首次提出的，Gilbert 在他的论文《随机图论》中独立定义了这一模型。

References

↑ ^1.0 ^1.1 Bollobás, Béla (2001). Random Graphs (2nd ed.). Cambridge University Press.
↑ Frieze, Alan; Karonski, Michal (2015). Introduction to Random Graphs. Cambridge University Press.
↑ ^3.0 ^3.1 ^3.2 ^3.3 ^3.4 ^3.5 Béla Bollobás, Random Graphs, 1985, Academic Press Inc., London Ltd.
↑ ^4.0 ^4.1 ^4.2 Béla Bollobás, Probabilistic Combinatorics and Its Applications, 1991, Providence, RI: American Mathematical Society.
↑ ^5.0 ^5.1 Bollobas, B. and Riordan, O.M. "Mathematical results on scale-free random graphs" in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed., 2003
↑ ^6.0 ^6.1 Gilbert, E. N. (1959), "Random graphs", Annals of Mathematical Statistics, 30 (4): 1141–1144, doi:10.1214/aoms/1177706098.
↑ Newman, M. E. J. (2010). Networks: An Introduction. Oxford.
↑ Reuven Cohen and Shlomo Havlin (2010). Complex Networks: Structure, Robustness and Function. Cambridge University Press. http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php.
↑ ^9.0 ^9.1 Erdős, P. Rényi, A (1959) "On Random Graphs I" in Publ. Math. Debrecen 6, p. 290–297 [1]
↑ Shao, Shuai; Huang, Xuqing; Stanley, H Eugene; Havlin, Shlomo (2015). "Percolation of localized attack on complex networks". New Journal of Physics. 17 (2): 023049. arXiv:1412.3124. Bibcode:2015NJPh...17b3049S. doi:10.1088/1367-2630/17/2/023049. ISSN 1367-2630.
↑ Ramezanpour, A.; Karimipour, V.; Mashaghi, A. (2003). "Generating correlated networks from uncorrelated ones". Phys. Rev. E. 67 (46107): 046107. arXiv:cond-mat/0212469. Bibcode:2003PhRvE..67d6107R. doi:10.1103/PhysRevE.67.046107. PMID 12786436.
↑ Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc (2010). "Chromatic Polynomials of Random Graphs". J. Phys. A: Math. Theor. 43 (17): 175002. arXiv:1709.06209. Bibcode:2010JPhA...43q5002V. doi:10.1088/1751-8113/43/17/175002.
↑ Buldyrev, Sergey V.; Parshani, Roni; Paul, Gerald; Stanley, H. Eugene; Havlin, Shlomo (2010). "Catastrophic cascade of failures in interdependent networks". Nature. 464 (7291): 1025–1028. arXiv:1012.0206. Bibcode:2010Natur.464.1025B. doi:10.1038/nature08932. ISSN 0028-0836. PMID 20393559.
↑ Gao, Jianxi; Buldyrev, Sergey V.; Stanley, H. Eugene; Havlin, Shlomo (2011). "Networks formed from interdependent networks". Nature Physics. 8 (1): 40–48. Bibcode:2012NatPh...8...40G. CiteSeerX 10.1.1.379.8214. doi:10.1038/nphys2180. ISSN 1745-2473.
↑ Moreno, Jacob L; Jennings, Helen Hall (Jan 1938). "Statistics of Social Configurations". Sociometry. 1 (3/4): 342–374. doi:10.2307/2785588. JSTOR 2785588.
↑ Solomonoff, Ray; Rapopst, Anatol (June 1951). "Connectivity of random nets". Bulletin of Mathematical Biophysics. 13 (2): 107–117. doi:10.1007/BF02478357.

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[Random_Graphs-1] 1.0 ^1.1 Bollobás, Béla (2001). Random Graphs (2nd ed.). Cambridge University Press.

[Introduction_to_Random_graphs-2] Frieze, Alan; Karonski, Michal (2015). Introduction to Random Graphs. Cambridge University Press.

[Random_Graphs2-3] 3.0 ^3.1 ^3.2 ^3.3 ^3.4 ^3.5 Béla Bollobás, Random Graphs, 1985, Academic Press Inc., London Ltd.

[Random_Graphs3-4] 4.0 ^4.1 ^4.2 Béla Bollobás, Probabilistic Combinatorics and Its Applications, 1991, Providence, RI: American Mathematical Society.

[Handbook-5] 5.0 ^5.1 Bollobas, B. and Riordan, O.M. "Mathematical results on scale-free random graphs" in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed., 2003

[Random_graphs-6] 6.0 ^6.1 Gilbert, E. N. (1959), "Random graphs", Annals of Mathematical Statistics, 30 (4): 1141–1144, doi:10.1214/aoms/1177706098.

[7] Newman, M. E. J. (2010). Networks: An Introduction. Oxford.

[8] Reuven Cohen and Shlomo Havlin (2010). Complex Networks: Structure, Robustness and Function. Cambridge University Press. http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php.

[On_Random_Graphs-9] 9.0 ^9.1 Erdős, P. Rényi, A (1959) "On Random Graphs I" in Publ. Math. Debrecen 6, p. 290–297 [1]

[ShaoHuang2015-10] Shao, Shuai; Huang, Xuqing; Stanley, H Eugene; Havlin, Shlomo (2015). "Percolation of localized attack on complex networks". New Journal of Physics. 17 (2): 023049. arXiv:1412.3124. Bibcode:2015NJPh...17b3049S. doi:10.1088/1367-2630/17/2/023049. ISSN 1367-2630.

[11] Ramezanpour, A.; Karimipour, V.; Mashaghi, A. (2003). "Generating correlated networks from uncorrelated ones". Phys. Rev. E. 67 (46107): 046107. arXiv:cond-mat/0212469. Bibcode:2003PhRvE..67d6107R. doi:10.1103/PhysRevE.67.046107. PMID 12786436.

[Chromatic_Polynomials_of_Random_Graphs-12] Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc (2010). "Chromatic Polynomials of Random Graphs". J. Phys. A: Math. Theor. 43 (17): 175002. arXiv:1709.06209. Bibcode:2010JPhA...43q5002V. doi:10.1088/1751-8113/43/17/175002.

[BuldyrevParshani2010-13] Buldyrev, Sergey V.; Parshani, Roni; Paul, Gerald; Stanley, H. Eugene; Havlin, Shlomo (2010). "Catastrophic cascade of failures in interdependent networks". Nature. 464 (7291): 1025–1028. arXiv:1012.0206. Bibcode:2010Natur.464.1025B. doi:10.1038/nature08932. ISSN 0028-0836. PMID 20393559.

[GaoBuldyrev2011-14] Gao, Jianxi; Buldyrev, Sergey V.; Stanley, H. Eugene; Havlin, Shlomo (2011). "Networks formed from interdependent networks". Nature Physics. 8 (1): 40–48. Bibcode:2012NatPh...8...40G. CiteSeerX 10.1.1.379.8214. doi:10.1038/nphys2180. ISSN 1745-2473.

[15] Moreno, Jacob L; Jennings, Helen Hall (Jan 1938). "Statistics of Social Configurations". Sociometry. 1 (3/4): 342–374. doi:10.2307/2785588. JSTOR 2785588.

[16] Solomonoff, Ray; Rapopst, Anatol (June 1951). "Connectivity of random nets". Bulletin of Mathematical Biophysics. 13 (2): 107–117. doi:10.1007/BF02478357.

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