无标度网络模型

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Network science
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Graph Complex network Contagion Small-world Scale-free Community structure Percolation Evolution Controllability Graph drawing Social capital Link analysis Optimization Reciprocity Closure Homophily Transitivity Preferential attachment Balance theory Network effect Social influence
Network types
Informational (computing) Telecommunication Transport Social Scientific collaboration Biological Artificial neural Interdependent Semantic Spatial Dependency Flow on-Chip
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Features Clique Component Cut Cycle Data structure Edge Loop Neighborhood Path Vertex Adjacency list模板:\matrix Incidence list模板:\matrix Types Bipartite Complete Directed Hyper Multi Random Weighted
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Centrality Degree Betweenness Closeness PageRank Motif Clustering Degree distribution Assortativity Distance Modularity Efficiency
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Topology Random graph Erdős–Rényi Barabási–Albert Fitness model Watts–Strogatz Exponential random (ERGM) Random geometric (RGG) Hyperbolic (HGN) Hierarchical Stochastic block Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR
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A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as

A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as

无尺度网络是一个度分布遵循幂律的网络，至少是渐近的。也就是说，网络中与其他节点有 k 个连接的节点的分数 p (k)代表大的 k 为

[math]\displaystyle{ \lt math\gt 《数学》 P(k) \ \sim \ k^\boldsymbol{-\gamma} P(k) \ \sim \ k^\boldsymbol{-\gamma} P(k) \ \sim \ k^\boldsymbol{-\gamma} }[/math]

</math>

数学

where [math]\displaystyle{ \gamma }[/math] is a parameter whose value is typically in the range 2 < [math]\displaystyle{ \gamma }[/math] < 3 (wherein the second moment (scale parameter) [math]\displaystyle{ k^\boldsymbol{-\gamma} }[/math] is infinite but the first moment is finite), although occasionally it may lie outside these bounds.

where [math]\displaystyle{ \gamma }[/math] is a parameter whose value is typically in the range 2 < [math]\displaystyle{ \gamma }[/math] < 3 (wherein the second moment (scale parameter) [math]\displaystyle{ k^\boldsymbol{-\gamma} }[/math] is infinite but the first moment is finite), although occasionally it may lie outside these bounds.

其中 < math > gamma </math > 是一个参数，它的值通常在2 < math > gamma </math > < 3之间(其中第二个时刻(刻度参数) < math > k ^ boldsymbol {-gamma } </math > 是无限的，但第一个时刻是有限的) ，虽然有时它可能位于这些界限之外。

Many networks have been reported to be scale-free, although statistical analysis has refuted many of these claims and seriously questioned others. Preferential attachment and the fitness model have been proposed as mechanisms to explain conjectured power law degree distributions in real networks.

Many networks have been reported to be scale-free, although statistical analysis has refuted many of these claims and seriously questioned others. Preferential attachment and the fitness model have been proposed as mechanisms to explain conjectured power law degree distributions in real networks.

据报告，许多网络是无标度的，尽管统计分析驳斥了其中许多说法，并对其他说法提出了严重质疑。提出了优先连接和适应度模型作为解释实际网络中假设的幂律度分布的机制。

History

In studies of the networks of citations between scientific papers, Derek de Solla Price showed in 1965 that the number of links to papers—i.e., the number of citations they receive—had a heavy-tailed distribution following a Pareto distribution or power law, and thus that the citation network is scale-free. He did not however use the term "scale-free network", which was not coined until some decades later. In a later paper in 1976, Price also proposed a mechanism to explain the occurrence of power laws in citation networks, which he called "cumulative advantage" but which is today more commonly known under the name preferential attachment.

In studies of the networks of citations between scientific papers, Derek de Solla Price showed in 1965 that the number of links to papers—i.e., the number of citations they receive—had a heavy-tailed distribution following a Pareto distribution or power law, and thus that the citation network is scale-free. He did not however use the term "scale-free network", which was not coined until some decades later. In a later paper in 1976, Price also proposed a mechanism to explain the occurrence of power laws in citation networks, which he called "cumulative advantage" but which is today more commonly known under the name preferential attachment.

在对科学论文之间的引用网络的研究中，Derek de Solla Price 在1965年指出，论文链接的数量---- 也就是说，他们收到的引用数量---- 遵循重尾分布或帕累托分布，因此引用网络是无标度的。然而，他并没有使用“无尺度网络”这个词，这个词直到几十年后才被创造出来。在1976年后来的一篇论文中，普赖斯还提出了一种解释引文网络中幂律发生的机制，他称之为“累积优势” ，但现在更常用的名称是优先连接。

Recent interest in scale-free networks started in 1999 with work by Albert-László Barabási and colleagues at the University of Notre Dame who mapped the topology of a portion of the World Wide Web, finding that some nodes, which they called "hubs", had many more connections than others and that the network as a whole had a power-law distribution of the number of links connecting to a node. After finding that a few other networks, including some social and biological networks, also had heavy-tailed degree distributions, Barabási and collaborators coined the term "scale-free network" to describe the class of networks that exhibit a power-law degree distribution. However, studying seven examples of networks in social, economic, technological, biological, and physical systems, Amaral et al. were not able to find a scale-free network among these seven examples. Only one of these examples, the movie-actor network, had degree distribution P(k) following a power law regime for moderate k, though eventually this power law regime was followed by a sharp cutoff showing exponential decay for large k.

[math]\displaystyle{ s(G) = \sum_{(u,v) \in E} \deg(u) \cdot \deg(v). }[/math]

[数学] s (g) = sum _ {(u，v) in e } deg (u) cdot deg (v)

Barabási and Réka Albert proposed a generative mechanism to explain the appearance of power-law distributions, which they called "preferential attachment" and which is essentially the same as that proposed by Price. Analytic solutions for this mechanism (also similar to the solution of Price) were presented in 2000 by Dorogovtsev, Mendes and Samukhin and independently by Krapivsky, Redner, and Leyvraz, and later rigorously proved by mathematician Béla Bollobás. Notably, however, this mechanism only produces a specific subset of networks in the scale-free class, and many alternative mechanisms have been discovered since.

This is maximized when high-degree nodes are connected to other high-degree nodes. Now define

当高度节点连接到其他高度节点时，这个值最大化。现在定义一下

The history of scale-free networks also includes some disagreement. On an empirical level, the scale-free nature of several networks has been called into question. For instance, the three brothers Faloutsos believed that the Internet had a power law degree distribution on the basis of traceroute data; however, it has been suggested that this is a layer 3 illusion created by routers, which appear as high-degree nodes while concealing the internal layer 2 structure of the ASes they interconnect.

[math]\displaystyle{ S(G) = \frac{s(G)}{s_\max}, }[/math]

S (g) = frac { s (g)}{ s _ max } ，</math >

A further characteristic concerns the average distance between two vertices in a network. As with most disordered networks, such as the small world network model, this distance is very small relative to a highly ordered network such as a lattice graph. Notably, an uncorrelated power-law graph having 2 < γ < 3 will have ultrasmall diameter d ~ ln ln N where N is the number of nodes in the network, as proved by Cohen and Havlin. Thus, the diameter of a growing scale-free network might be considered almost constant in practice.

进一步的特性涉及到网络中两个顶点之间的平均距离。与大多数无序网络(如小世界网络模型)一样，这个距离相对于高度有序的网络(如格子图)来说非常小。值得注意的是，一个具有2 < γ < 3的不相关幂律图将具有超小直径 d ~ ln n，其中 n 是网络中的节点数，正如 Cohen 和 Havlin 所证明的那样。因此，生长中的无尺度网络的直径在实践中可能被认为是几乎恒定的。

On a theoretical level, refinements to the abstract definition of scale-free have been proposed. For example, Li et al. (2005) recently offered a potentially more precise "scale-free metric". Briefly, let G be a graph with edge set E, and denote the degree of a vertex [math]\displaystyle{ v }[/math] (that is, the number of edges incident to [math]\displaystyle{ v }[/math]) by [math]\displaystyle{ \deg(v) }[/math]. Define

[math]\displaystyle{ s(G) = \sum_{(u,v) \in E} \deg(u) \cdot \deg(v). }[/math]

Rozenfeld et al suggested a method for generating deterministic fractal scale free networks

Rozenfeld 等人提出了一种生成确定性分形无标度网络的方法

This is maximized when high-degree nodes are connected to other high-degree nodes. Now define

The question of how to immunize efficiently scale free networks which represent realistic networks such as the Internet and social networks has been studied extensively. One such strategy is to immunize the largest degree nodes, i.e., targeted (intentional) attacks In this case, which is quite efficient one choses randomly nodes but immunize their neighbors.

如何有效地对代表现实网络的无标度网络进行免疫问题已经得到了广泛的研究。其中一种策略是免疫最大度节点，即有针对性的(有意的)攻击。在这种情况下，这是相当有效的，可以随机选择节点，但免疫它们的邻居。

[math]\displaystyle{ S(G) = \frac{s(G)}{s_\max}, }[/math]

Another and even more efficient method is based on graph parition method   .

另一种更有效的方法是基于图分解的方法。

where s_max is the maximum value of s(H) for H in the set of all graphs with degree distribution identical to that of G. This gives a metric between 0 and 1, where a graph G with small S(G) is "scale-rich", and a graph G with S(G) close to 1 is "scale-free". This definition captures the notion of self-similarity implied in the name "scale-free".

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. Scale free graphs, as such, remain scale free under such transformations.

随机图的性质在图变换下可以改变或保持不变。例如，Mashaghi a. et A.. 证明了将随机图转换为边-对偶图(或线图)的转换产生了一个几乎具有相同度分布的图的集合，但具有度相关性和明显更高的集聚系数。无标度图本身在这种变换下仍然是无标度的。

Overview

There are two major components that explain the emergence of the scale-free property in a complex networks: the growth and the preferential attachment.

A space-filling cellular structure, weighted planar stochastic lattice (WPSL) has recently been proposed whose coordination number distribution follow a power-law. It implies that the lattice has a few blocks which have astonishingly large number neighbors with whom they share common borders. Its construction starts with an initiator, say a

最近提出了一种填充空间的网格结构——加权平面随机格(WPSL) ，其配位数分布服从幂律。这意味着格子有几个块，它们有着数量惊人的邻居，它们共享相同的边界。它的构造始于一个发起者，比如一个

By "growth" is called a growth process where, over an extended period of time, new nodes join an already existing system, a network (like the World Wide Web which has grown by billions of web pages over 10 years).

square of unit area, and a generator that divides it randomly into four blocks. The generator thereafter is sequentially applied

单位面积的平方，以及一个随机分成四个街区的发电机。生成器随后按顺序应用

Finally, by "preferential attachment" is called a new coming node who prefers to connect to another node which has already a certain number of links with others. Thus, there is a higher probability that more and more nodes will link themselves to that one which has already many links, leading this node to a hub in-fine.

over and over again to only one of the available blocks picked preferentially with respect to their areas. It results in the partitioning of the square into ever smaller mutually exclusive rectangular blocks. The dual of the WPSL (DWPSL) is obtained by replacing each block with a node at its center and common border

一次又一次，只有一个可用的块优先采摘相对于他们的地区。它导致将正方形分割成更小的相互排斥的矩形块。WPSL (DWPSL)的对偶是通过在每个块的中心和公共边界上放置一个节点来获得的

Depending on the network, the hubs might either be assortative or disassortative. Assortativity would be found in social networks in which well-connected/famous people would tend to know better each other. Disassortativity would be found in technological (Internet, World Wide Web) and biological (protein interaction, metabolism) networks.

between blocks with an edge joining the two corresponding vertices emerges as a network whose degree distribution follows

边连接两个相应顶点的块之间形成一个度分布如下的网络

a power-law. The reason for it is that it grows following mediation-driven attachment model rule which also embodies preferential attachment rule but in disguise.

幂律。究其原因，是因为它遵循了中介驱动的依恋模型规则，这种模型规则也体现了优先依恋规则，但是是伪装的。

Characteristics

Random network (a) and scale-free network (b). In the scale-free network, the larger hubs are highlighted.

Complex network degree distribution of random and scale-free

Scale-free networks do not arise by chance alone. Erdős and Rényi (1960) studied a model of growth for graphs in which, at each step, two nodes are chosen uniformly at random and a link is inserted between them. The properties of these random graphs are different from the properties found in scale-free networks, and therefore a model for this growth process is needed.

无标度网络不是偶然出现的。Erd s 和 Rényi (1960)研究了图的增长模型，其中每一步均匀地随机选择两个节点，并在它们之间插入一个链路。这些随机图的性质不同于无标度网络的性质，因此需要一个模型来描述这种增长过程。

The most notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and are thought to serve specific purposes in their networks, although this depends greatly on the domain.

The most widely known generative model for a subset of scale-free networks is Barabási and Albert's (1999) rich get richer generative model in which each new Web page creates links to existing Web pages with a probability distribution which is not uniform, but

无标度网络子集中最广为人知的生成模型是 Barabási 和 Albert (1999) rich get richer 生成模型，其中每个新网页都创建一个链接到现有网页的概率分布，这个链接并不统一，但是

proportional to the current in-degree of Web pages. This model was originally invented by Derek J. de Solla Price in 1965 under the term cumulative advantage, but did not reach popularity until Barabási rediscovered the results under its current name (BA Model). According to this process, a page with many in-links will attract more in-links than a regular page. This generates a power-law but the resulting graph differs from the actual Web graph in other properties such as the presence of small tightly connected communities. More general models and network characteristics have been proposed and studied. For example, Pachon et al. (2018) proposed a variant of the rich get richer generative model which takes into account two different attachment rules: a preferential attachment mechanism and a uniform choice only for the most recent nodes. (2000),

与当前 Web 页面的 in 度成正比。这个模型最初是由德瑞克·约翰·德索拉·普莱斯在1965年发明的术语累积优势，但并没有达到普及，直到 Barabási 重新发现的结果在其现在的名称(BA 模型)。根据这个过程，一个页面与许多链接将吸引更多的链接比普通页面。这会产生一个幂定律，但是得到的图与实际的网络图在其他属性上有所不同，比如小型紧密连接的社区的存在。提出并研究了更一般的模型和网络特性。例如，Pachon 等人。(2018)提出了一种富人变得更富有的生成模型，它考虑了两种不同的连接规则: 优先连接机制和仅针对最近的节点的统一选择。(2000),

Percolation

in which new nodes choose an existent node at random and copy a fraction of the links of the existent node. This also generates a power law.

其中新节点随机选择一个存在的节点，并复制存在节点的链路的一部分。这也产生了一个幂定律。

The scale-free property strongly correlates with the network's robustness to failure. It turns out that the major hubs are closely followed by smaller ones. These smaller hubs, in turn, are followed by other nodes with an even smaller degree and so on. This hierarchy allows for a fault tolerant behavior. If failures occur at random and the vast majority of nodes are those with small degree, the likelihood that a hub would be affected is almost negligible. Even if a hub-failure occurs, the network will generally not lose its connectedness, due to the remaining hubs. On the other hand, if we choose a few major hubs and take them out of the network, the network is turned into a set of rather isolated graphs. Thus, hubs are both a strength and a weakness of scale-free networks. These properties have been studied analytically using percolation theory by Cohen et al and by Callaway et al. It was proven by Cohen et al that for a broad range of scale free networks, for [math]\displaystyle{ 2 \lt \gamma \lt 3 }[/math] the critical percolation threshold,模板:Space[math]\displaystyle{ p_c=0 }[/math]. This means that removing randomly any fraction of nodes from the network will not destroy the network. This is in contrast to Erdős–Rényi graph where [math]\displaystyle{ p_c =1/\bar{k} }[/math], where [math]\displaystyle{ \bar{k} }[/math] is the average degree. The failures discussed above are random, as usually assumed in percolation theory. However, when generalizing percolation also to non-random but targeted attacks, e.g., on highest degree nodes, the results, such as [math]\displaystyle{ p_c }[/math], change significantly.

The growth of the networks (adding new nodes) is not a necessary condition for creating a scale-free network. Dangalchev (2004) gives examples of generating static scale-free networks. Another possibility (Caldarelli et al. 2002) is to consider the structure as static and draw a link between vertices according to a particular property of the two vertices involved. Once specified the statistical distribution for these vertex properties (fitnesses), it turns out that in some circumstances also static networks develop scale-free properties.

网络的增长(增加新的节点)并不是创建无尺度网络的必要条件。Dangalchev (2004)给出了生成静态无标度网络的例子。另一种可能性(Caldarelli et al. 。2002)是考虑结构为静态和绘制之间的顶点根据特定性质的两个顶点参与。一旦确定了这些顶点属性(fitness)的统计分布，就会发现在某些情况下，静态网络也会发展出无标度属性。

Recently, a new type of failures in networks has been developed, called localized attacks. In this case one choses randomly a node and remove its neighbors and next nearest neighbors until a fraction of 1-p nodes are removed. Localized attacks makes the scale free network more vulnerable compared to ransom attacks and pc>0. The critical exponents of percolation in scale free networks are different from random Erdős–Rényi networks.^[16a] Thus, scale free networks are in a different universality class from Erdős–Rényi networks.

Clustering

Another important characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. This implies that the low-degree nodes belong to very dense sub-graphs and those sub-graphs are connected to each other through hubs. Consider a social network in which nodes are people and links are acquaintance relationships between people. It is easy to see that people tend to form communities, i.e., small groups in which everyone knows everyone (one can think of such community as a complete graph). In addition, the members of a community also have a few acquaintance relationships to people outside that community. Some people, however, are connected to a large number of communities (e.g., celebrities, politicians). Those people may be considered the hubs responsible for the small-world phenomenon.

There has been a burst of activity in the modeling of scale-free complex networks. The recipe of Barabási and Albert has been followed by several variations and generalizations and the revamping of previous mathematical works. As long as there is a power law distribution in a model, it is a scale-free network, and a model of that network is a scale-free model.

在无标度复杂网络的建模中，出现了大量的活动。Barabási 和阿尔伯特的秘方之后，又有了一些变化和概括，并对以前的数学作品进行了改造。只要一个模型中存在幂律分布，那么它就是一个无尺度网络，而该网络的模型就是一个无标度模型。

At present, the more specific characteristics of scale-free networks vary with the generative mechanism used to create them. For instance, networks generated by preferential attachment typically place the high-degree vertices in the middle of the network, connecting them together to form a core, with progressively lower-degree nodes making up the regions between the core and the periphery. The random removal of even a large fraction of vertices impacts the overall connectedness of the network very little, suggesting that such topologies could be useful for security, while targeted attacks destroys the connectedness very quickly. Other scale-free networks, which place the high-degree vertices at the periphery, do not exhibit these properties. Similarly, the clustering coefficient of scale-free networks can vary significantly depending on other topological details.

Many real networks are (approximately) scale-free and hence require scale-free models to describe them. In Price's scheme, there are two ingredients needed to build up a scale-free model:

许多真实的网络是近似无标度的，因此需要无标度模型来描述它们。在普莱斯的方案中，建立无标度模型需要两个要素:

Distance in scale free networks

A further characteristic concerns the average distance between two vertices in a network. As with most disordered networks, such as the small world network model, this distance is very small relative to a highly ordered network such as a lattice graph. Notably, an uncorrelated power-law graph having 2 < γ < 3 will have ultrasmall diameter d ~ ln ln N where N is the number of nodes in the network, as proved by Cohen and Havlin. Thus, the diameter of a growing scale-free network might be considered almost constant in practice.

1. Adding or removing nodes. Usually we concentrate on growing the network, i.e. adding nodes.

1.添加或移除节点。通常我们集中精力扩大网络，例如。加法节点。

Fractal scale free networks

2. Preferential attachment: The probability [math]\displaystyle{ \Pi }[/math] that new nodes will be connected to the "old" node.

2.优先连接: 新节点将连接到“旧”节点的概率。

Rozenfeld et al suggested a method for generating deterministic fractal scale free networks

Note that Fitness models (see below) could work also statically, without changing the number of nodes. It should also be kept in mind that the fact that "preferential attachment" models give rise to scale-free networks does not prove that this is the mechanism underlying the evolution of real-world scale-free networks, as there might exist different mechanisms at work in real-world systems that nevertheless give rise to scaling.

请注意，Fitness 模型(见下文)也可以静态地工作，而不改变节点的数量。还应当铭记，”优先连接”模型产生无标度网络这一事实并不能证明这是现实世界无标度网络演变的基本机制，因为在现实世界系统中可能存在不同的机制，但这些机制会引起扩展。

Immunization

The question of how to immunize efficiently scale free networks which represent realistic networks such as the Internet and social networks has been studied extensively. One such strategy is to immunize the largest degree nodes, i.e., targeted (intentional) attacks since for this case p[math]\displaystyle{ c }[/math] is relatively high and less nodes are needed to be immunized.

However, in many realistic cases the global structure is not available and the largest degree nodes are not known.

For such cases the method of acquaintance immunization has been developed. In this case, which is quite efficient one choses randomly nodes but immunize their neighbors.

There have been several attempts to generate scale-free network properties. Here are some examples:

已经有几次尝试生成无尺度网络的属性。以下是一些例子:

Another and even more efficient method is based on graph parition method   .

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. Scale free graphs, as such, remain scale free under such transformations.

For example, the first scale-free model, the Barabási–Albert model, has a linear preferential attachment [math]\displaystyle{ \Pi(k_i)=\frac{k_i}{\sum_j k_j} }[/math] and adds one new node at every time step.

例如，第一个无标度模型 Barabási-Albert 模型，具有线性优先连接 < math > Pi (k_i) = frac { k_i }{ sum _ jk_j } </math > ，并在每个时间步增加一个新节点。

Examples

(Note, another general feature of [math]\displaystyle{ \Pi(k) }[/math] in real networks is that [math]\displaystyle{ \Pi(0)\neq 0 }[/math], i.e. there is a nonzero probability that a

(注意，实际网络中 Pi (k) </math 的另一个普遍特征是 < math > Pi (0) neq 0 </math > ，即。有一个非零的概率

Although many real-world networks are thought to be scale-free, the evidence often remains inconclusive, primarily due to the developing awareness of more rigorous data analysis techniques. As such, the scale-free nature of many networks is still being debated by the scientific community. A few examples of networks claimed to be scale-free include:

new node attaches to an isolated node. Thus in general [math]\displaystyle{ \Pi(k) }[/math] has the form [math]\displaystyle{ \Pi(k)=A +k^\alpha }[/math], where [math]\displaystyle{ A }[/math] is the initial attractiveness of the node.)

新的节点连接到一个独立的节点。因此，在一般情况下，Pi (k) </math > 的形式是 < math > Pi (k) = a + k ^ alpha </math > ，其中 < math > a </math > 是节点的初始吸引力

• Some Social networks, including collaboration networks. Two examples that have been studied extensively are the collaboration of movie actors in films and the co-authorship by mathematicians of papers.

• Many kinds of computer networks, including the internet and the webgraph of the World Wide Web.

• Software dependency graphs, some of them being described with a generative model.

Dangalchev

Dangalchev

• Some financial networks such as interbank payment networks

• Protein-protein interaction networks.

However, for [math]\displaystyle{ m=1 }[/math] it describes the winner takes it all mechanism as we find that almost [math]\displaystyle{ 99\% }[/math] of the total nodes has degree one and one is super-rich in degree. As [math]\displaystyle{ m }[/math] value increases the disparity between the super rich and poor decreases and as [math]\displaystyle{ m\gt 14 }[/math] we find a transition from rich get super richer to rich get richer mechanism.

然而，对于 < math > m = 1 </math > ，它描述了胜者通吃的机制，因为我们发现几乎所有的节点中99% 的节点具有度1，其中一个节点的度超级丰富。随着数值的增加，超级富人与穷人之间的差距缩小，随着数值的增加，我们发现了从富人变得超级富人到富人变得富人的过渡机制。

• Semantic networks.

• Airline networks.

A snapshot of the weighted planar stochastic lattice (WPSL).

The Barabási–Albert model assumes that the probability [math]\displaystyle{ \Pi(k) }[/math] that a node attaches to node [math]\displaystyle{ i }[/math] is proportional to the degree [math]\displaystyle{ k }[/math] of node [math]\displaystyle{ i }[/math]. This assumption involves two hypotheses: first, that [math]\displaystyle{ \Pi(k) }[/math] depends on [math]\displaystyle{ k }[/math], in contrast to random graphs in which [math]\displaystyle{ \Pi(k) = p }[/math], and second, that the functional form of [math]\displaystyle{ \Pi(k) }[/math] is linear in [math]\displaystyle{ k }[/math]. The precise form of [math]\displaystyle{ \Pi(k) }[/math] is not necessarily linear, and recent studies have demonstrated that the degree distribution depends strongly on [math]\displaystyle{ \Pi(k) }[/math]

Barabási-Albert 模型假设一个节点连接到节点上的概率 Pi (k) </math > 与节点 < math > i </math > 的程度成正比。这一假设涉及两个假设: 第一，与《数学》中的随机图形不同，《数学》中的 Pi (k)取决于《数学》中的 k; 第二，《数学》中的 Pi (k)的函数形式是线性的。Pi (k) </math > 的精确形式不一定是线性的，最近的研究表明程度分布强烈依赖于 < math > Pi (k) </math >

Scale free topology has been also found in high temperature superconductors. The qualities of a high-temperature superconductor — a compound in which electrons obey the laws of quantum physics, and flow in perfect synchrony, without friction — appear linked to the fractal arrangements of seemingly random oxygen atoms and lattice distortion.

Krapivsky, Redner, and Leyvraz

克拉皮夫斯基、雷德纳和莱弗拉兹

A space-filling cellular structure, weighted planar stochastic lattice (WPSL) has recently been proposed whose coordination number distribution follow a power-law. It implies that the lattice has a few blocks which have astonishingly large number neighbors with whom they share common borders. Its construction starts with an initiator, say a

The iterative construction leading to a hierarchical network. Starting from a fully connected cluster of five nodes, we create four identical replicas connecting the peripheral nodes of each cluster to the central node of the original cluster. From this, we get a network of 25 nodes (N = 25).

导致层次网络的迭代结构。从一个由五个节点组成的完全连接的集群开始，我们创建四个相同的副本，将每个集群的外围节点连接到原始集群的中心节点。由此，我们得到了一个由25个节点组成的网络(n = 25)。

square of unit area, and a generator that divides it randomly into four blocks. The generator thereafter is sequentially applied

Repeating the same process, we can create four more replicas of the original cluster – the four peripheral nodes of each one connect to the central node of the nodes created in the first step. This gives N = 125, and the process can continue indefinitely.

重复同样的过程，我们可以创建原始集群的四个副本——每个集群的四个外围节点连接到第一步中创建的节点的中心节点。这样得到 n = 125，这个过程可以无限期地继续下去。

over and over again to only one of the available blocks picked preferentially with respect to their areas. It results in the partitioning of the square into ever smaller mutually exclusive rectangular blocks. The dual of the WPSL (DWPSL) is obtained by replacing each block with a node at its center and common border

between blocks with an edge joining the two corresponding vertices emerges as a network whose degree distribution follows

a power-law. The reason for it is that it grows following mediation-driven attachment model rule which also embodies preferential attachment rule but in disguise.

The idea is that the link between two vertices is assigned not randomly with a probability p equal for all the couple of vertices. Rather, for

其思想是，两个顶点之间的链路不是随机分配的，所有顶点对的概率 p 都是相等的。更确切地说，是为了

every vertex j there is an intrinsic fitness x_j and a link between vertex i and j is created with a probability

每个顶点 j 都有一个内在适应度 x < sub > j ，并且在顶点 i 和 j 之间建立了一个概率连接

Generative models

[math]\displaystyle{ p(x_i,x_j) }[/math].

[ math ] p (xi，x _ j).

Scale-free networks do not arise by chance alone. Erdős and Rényi (1960) studied a model of growth for graphs in which, at each step, two nodes are chosen uniformly at random and a link is inserted between them. The properties of these random graphs are different from the properties found in scale-free networks, and therefore a model for this growth process is needed.

In the case of World Trade Web it is possible to reconstruct all the properties by using as fitnesses of the country their GDP, and taking

在世界贸易网络的例子中，我们可以通过利用国内生产总值来重建所有的属性，然后利用

The most widely known generative model for a subset of scale-free networks is Barabási and Albert's (1999) rich get richer generative model in which each new Web page creates links to existing Web pages with a probability distribution which is not uniform, but

[math]\displaystyle{  p(x_i,x_j)=\frac {\delta x_ix_j}{1+\delta x_ix_j}.  }[/math]

= frac { delta x ix j }{1 + delta x ix j }.数学

proportional to the current in-degree of Web pages. This model was originally invented by Derek J. de Solla Price in 1965 under the term cumulative advantage, but did not reach popularity until Barabási rediscovered the results under its current name (BA Model). According to this process, a page with many in-links will attract more in-links than a regular page. This generates a power-law but the resulting graph differs from the actual Web graph in other properties such as the presence of small tightly connected communities. More general models and network characteristics have been proposed and studied. For example, Pachon et al. (2018) proposed a variant of the rich get richer generative model which takes into account two different attachment rules: a preferential attachment mechanism and a uniform choice only for the most recent nodes. For a review see the book by Dorogovtsev and Mendes.

A somewhat different generative model for Web links has been suggested by Pennock et al. (2002). They examined communities with interests in a specific topic such as the home pages of universities, public companies, newspapers or scientists, and discarded the major hubs of the Web. In this case, the distribution of links was no longer a power law but resembled a normal distribution. Based on these observations, the authors proposed a generative model that mixes preferential attachment with a baseline probability of gaining a link.

Assuming that a network has an underlying hyperbolic geometry, one can use the framework of spatial networks to generate scale-free degree distributions. This heterogeneous degree distribution then simply reflects the negative curvature and metric properties of the underlying hyperbolic geometry.

假设一个网络有一个基本的双曲几何，我们可以使用空间网络的框架来生成无标度的度分布。这种不均匀的程度分布仅仅反映了基础双曲几何的负曲率和度量属性。

Another generative model is the copy model studied by Kumar et al. (2000),

in which new nodes choose an existent node at random and copy a fraction of the links of the existent node. This also generates a power law.

The growth of the networks (adding new nodes) is not a necessary condition for creating a scale-free network. Dangalchev (2004) gives examples of generating static scale-free networks. Another possibility (Caldarelli et al. 2002) is to consider the structure as static and draw a link between vertices according to a particular property of the two vertices involved. Once specified the statistical distribution for these vertex properties (fitnesses), it turns out that in some circumstances also static networks develop scale-free properties.

Starting with scale free graphs with low degree correlation and clustering coefficient, one can generate new graphs with much higher degree correlations and clustering coefficients by applying edge-dual transformation. In models of scale-free ideal networks it is possible to demonstrate that Dunbar's number is the cause of the phenomenon known as the 'six degrees of separation' .

从低度相关和集聚系数的无标度图开始，可以通过边对偶变换生成具有更高度相关性和聚类系数的新图。在无标度的理想网络模型中，可以证明邓巴数是所谓的“六度分隔理论”现象的原因。

Generalized scale-free model

For a scale-free network with [math]\displaystyle{ n }[/math] nodes and power-law exponent [math]\displaystyle{ \beta\gt 3 }[/math], the induced subgraph constructed by vertices with degrees larger than [math]\displaystyle{ \log{n}\times \log^{*}{n} }[/math] is a scale-free network with [math]\displaystyle{ \beta'=2 }[/math], almost surely (a.s.).

对于具有 < math > n </math > 节点和幂律指数 < math > beta > 3 </math > 的无尺度网络，由比 < math > log { n } * * log ^ { n } </math > 大的顶点构成的诱导子图几乎可以肯定是一个带有 < math > beta’ = 2 </math > 的无尺度网络。

There has been a burst of activity in the modeling of scale-free complex networks. The recipe of Barabási and Albert has been followed by several variations and generalizations and the revamping of previous mathematical works. As long as there is a power law distribution in a model, it is a scale-free network, and a model of that network is a scale-free model.

Features

Many real networks are (approximately) scale-free and hence require scale-free models to describe them. In Price's scheme, there are two ingredients needed to build up a scale-free model:

1. Adding or removing nodes. Usually we concentrate on growing the network, i.e. adding nodes.

2. Preferential attachment: The probability [math]\displaystyle{ \Pi }[/math] that new nodes will be connected to the "old" node.

Note that Fitness models (see below) could work also statically, without changing the number of nodes. It should also be kept in mind that the fact that "preferential attachment" models give rise to scale-free networks does not prove that this is the mechanism underlying the evolution of real-world scale-free networks, as there might exist different mechanisms at work in real-world systems that nevertheless give rise to scaling.

Examples

There have been several attempts to generate scale-free network properties. Here are some examples:

The Barabási–Albert model

For example, the first scale-free model, the Barabási–Albert model, has a linear preferential attachment [math]\displaystyle{ \Pi(k_i)=\frac{k_i}{\sum_j k_j} }[/math] and adds one new node at every time step.

(Note, another general feature of [math]\displaystyle{ \Pi(k) }[/math] in real networks is that [math]\displaystyle{ \Pi(0)\neq 0 }[/math], i.e. there is a nonzero probability that a

new node attaches to an isolated node. Thus in general [math]\displaystyle{ \Pi(k) }[/math] has the form [math]\displaystyle{ \Pi(k)=A +k^\alpha }[/math], where [math]\displaystyle{ A }[/math] is the initial attractiveness of the node.)

Two-level network model

Dangalchev builds a 2-L model by adding a second-order preferential attachment. The attractiveness of a node in the 2-L model depends not only on the number of nodes linked to it but also on the number of links in each of these nodes.

[math]\displaystyle{ \Pi(k_i)=\frac{k_i + C \sum_{(i,j)} k_j}{\sum_j k_j + C \sum_j k_j^2}, }[/math]

where C is a coefficient between 0 and 1.

Mediation-driven attachment (MDA) model

In the mediation-driven attachment (MDA) model, a new node coming with [math]\displaystyle{ m }[/math] edges picks an existing connected node at random and then connects itself, not with that one, but with [math]\displaystyle{ m }[/math] of its neighbors, also chosen at random. The probability [math]\displaystyle{ \Pi(i) }[/math] that the node [math]\displaystyle{ i }[/math] of the existing node picked is

[math]\displaystyle{ \Pi(i) = \frac{k_i} N \frac{\sum_{j=1}^{k_i} \frac 1 {k_j} }{k_i}. }[/math]

The factor [math]\displaystyle{ \frac{\sum_{j=1}^{k_i} \frac 1 {k_j} }{k_i} }[/math] is the inverse of the harmonic mean

(IHM) of degrees of the [math]\displaystyle{ k_i }[/math] neighbors of a node [math]\displaystyle{ i }[/math]. Extensive numerical investigation suggest that for approximately [math]\displaystyle{ m\gt 14 }[/math] the mean IHM value in the large [math]\displaystyle{ N }[/math] limit becomes a constant which means [math]\displaystyle{ \Pi(i) \propto k_i }[/math]. It implies that the higher the

links (degree) a node has, the higher its chance of gaining more links since they can be

reached in a larger number of ways through mediators which essentially embodies the intuitive

idea of rich get richer mechanism (or the preferential attachment rule of the Barabasi–Albert model). Therefore, the MDA network can be seen to follow

the PA rule but in disguise.

However, for [math]\displaystyle{ m=1 }[/math] it describes the winner takes it all mechanism as we find that almost [math]\displaystyle{ 99\% }[/math] of the total nodes has degree one and one is super-rich in degree. As [math]\displaystyle{ m }[/math] value increases the disparity between the super rich and poor decreases and as [math]\displaystyle{ m\gt 14 }[/math] we find a transition from rich get super richer to rich get richer mechanism.

Non-linear preferential attachment

See also: Non-linear preferential attachment

Category:Graph families

类别: 图族

This page was moved from wikipedia:en:Scale-free network. Its edit history can be viewed at 无标度网络模型/edithistory