# 无标度网络模型

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

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

[/itex]

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

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

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.

## 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.

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.

$\displaystyle{ s(G) = \sum_{(u,v) \in E} \deg(u) \cdot \deg(v). }$


[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

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.

$\displaystyle{ S(G) = \frac{s(G)}{s_\max}, }$


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.

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 $\displaystyle{ v }$ (that is, the number of edges incident to $\displaystyle{ v }$) by $\displaystyle{ \deg(v) }$. Define

$\displaystyle{ s(G) = \sum_{(u,v) \in E} \deg(u) \cdot \deg(v). }$

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.

$\displaystyle{ S(G) = \frac{s(G)}{s_\max}, }$

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

where smax 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.

## 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

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

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

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

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.

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

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.

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

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),

### 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.

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 $\displaystyle{ 2 \lt \gamma \lt 3 }$ the critical percolation threshold,模板:Space$\displaystyle{ p_c=0 }$. 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 $\displaystyle{ p_c =1/\bar{k} }$, where $\displaystyle{ \bar{k} }$ 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 $\displaystyle{ p_c }$, 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.

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.

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:

### 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 $\displaystyle{ \Pi }$ 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.

### 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$\displaystyle{ c }$ 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:

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 $\displaystyle{ \Pi(k_i)=\frac{k_i}{\sum_j k_j} }$ and adds one new node at every time step.

## Examples

(Note, another general feature of $\displaystyle{ \Pi(k) }$ in real networks is that $\displaystyle{ \Pi(0)\neq 0 }$, 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 $\displaystyle{ \Pi(k) }$ has the form $\displaystyle{ \Pi(k)=A +k^\alpha }$, where $\displaystyle{ A }$ is the initial attractiveness of the node.)

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

Dangalchev

Dangalchev

• Some financial networks such as interbank payment networks 

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

• Airline networks.

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

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

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).

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.

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

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

## Generative models

$\displaystyle{ p(x_i,x_j) }$.

[ 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

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

$\displaystyle{ p(x_i,x_j)=\frac {\delta x_ix_j}{1+\delta x_ix_j}. }$


= 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.

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' .

## Generalized scale-free model

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

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 $\displaystyle{ \Pi }$ 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 $\displaystyle{ \Pi(k_i)=\frac{k_i}{\sum_j k_j} }$ and adds one new node at every time step.

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

new node attaches to an isolated node. Thus in general $\displaystyle{ \Pi(k) }$ has the form $\displaystyle{ \Pi(k)=A +k^\alpha }$, where $\displaystyle{ A }$ 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.

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

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 $\displaystyle{ m }$ edges picks an existing connected node at random and then connects itself, not with that one, but with $\displaystyle{ m }$ of its neighbors, also chosen at random. The probability $\displaystyle{ \Pi(i) }$ that the node $\displaystyle{ i }$ of the existing node picked is

$\displaystyle{ \Pi(i) = \frac{k_i} N \frac{\sum_{j=1}^{k_i} \frac 1 {k_j} }{k_i}. }$

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

(IHM) of degrees of the $\displaystyle{ k_i }$ neighbors of a node $\displaystyle{ i }$. Extensive numerical investigation suggest that for approximately $\displaystyle{ m\gt 14 }$ the mean IHM value in the large $\displaystyle{ N }$ limit becomes a constant which means $\displaystyle{ \Pi(i) \propto k_i }$. 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 $\displaystyle{ m=1 }$ it describes the winner takes it all mechanism as we find that almost $\displaystyle{ 99\% }$ of the total nodes has degree one and one is super-rich in degree. As $\displaystyle{ m }$ value increases the disparity between the super rich and poor decreases and as $\displaystyle{ m\gt 14 }$ we find a transition from rich get super richer to rich get richer mechanism.

#### Non-linear preferential attachment

Category:Graph families

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

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4. Broido, Anna; Aaron Clauset (2019-03-04). "Scale-free networks are rare". Nature Communications. 10 (1): 1017. arXiv:1801.03400. doi:10.1038/s41467-019-08746-5. PMC 6399239. PMID 30833554.
5. {{cite journal 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. 最近人们对无标度网络的兴趣始于1999年，圣母大学的 albert-lászló Barabási 和他的同事们绘制了万维网的一部分拓扑图，他们发现一些节点，他们称之为“集线器” ，拥有比其他节点多得多的连接，并且整个网络连接到一个节点的链接数量的幂律分布。在发现其他一些网络，包括一些社交网络和生物网络，也有重尾度分布后，Barabási 和合作者创造了术语“无尺度网络”来描述一类呈现幂律度分布的网络。然而，研究网络在社会、经济、技术、生物和物理系统中的七个例子，Amaral 等。我们未能在这七个例子中找到无尺度网络。只有一个例子，电影演员网络，有度分布 p (k)遵循中等 k 的幂律制度，尽管最终这个幂律制度被紧随其后的急剧中断显示大 k 的指数衰减。 |last1=Barabási |first1=Albert-László |authorlink1=Albert-László Barabási |last2=Albert |first2=Réka. 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. Barabási 和 r é ka Albert 提出了一种生成机制来解释幂律分布的出现，他们称之为“优先附加” ，这与 Price 提出的分布在本质上是相同的。这种机制的解析解(也类似于价格解)是由 Dorogovtsev，Mendes 和 Samukhin 于2000年提出的，Krapivsky，Redner 和 Leyvraz 分别独立提出，后来由数学家 b é la Bollobás 严格证明。然而，值得注意的是，这种机制只在无标度类中产生特定的网络子集，自此之后发现了许多替代机制。 |title=Emergence of scaling in random networks |journal=Science 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. 无标度网络的历史也包含一些分歧。在经验层面上，几个网络的无标度性质已经受到质疑。例如，Faloutsos 三兄弟认为，互联网有一个幂律度分布的追踪数据的基础上，然而，有人提出，这是一个第三层幻想，由路由器，这似乎是高度节点，同时隐藏了内部的二层结构，他们互连。 |volume=286 |issue=5439 |pages=509–512 |date=October 15, 1999 |arxiv=cond-mat/9910332 |doi=10.1126/science.286.5439.509 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 $\displaystyle{ v }$ (that is, the number of edges incident to $\displaystyle{ v }$) by $\displaystyle{ \deg(v) }$. Define 在理论层面上，对无标度抽象定义进行了改进。例如，李等人。(2005年)最近提出了一个可能更为精确的“无标度度量”。简单地说，设 g 是一个边集 e 的图，用 < math > > </math > 表示顶点的度数(即与 < math > v </math > 相关的边数)。定义 |mr=2091634 |pmid=10521342 |bibcode = 1999Sci...286..509B |s2cid=524106 }}
6. Among the seven examples studied by Amaral et al, six of them where single-scale and only example iii, the movie-actor network had a power law regime followed by a sharp cutoff. None of Amaral et al's examples obeyed the power law regime for large k, i.e. none of these seven examples were shown to be scale-free. See especially the beginning of the discussion section of Amaral LAN, Scala A, Barthelemy M, Stanley HE (2000). "Classes of small-world networks". PNAS. 97 (21): 11149–52. arXiv:cond-mat/0001458. Bibcode:2000PNAS...9711149A. doi:10.1073/pnas.200327197. PMC 17168. PMID 11005838.
7. Dorogovtsev, S.; Mendes, J.; Samukhin, A. (2000). "Structure of Growing Networks with Preferential Linking". Physical Review Letters. 85 (21): 4633–4636. arXiv:cond-mat/0004434. Bibcode:2000PhRvL..85.4633D. doi:10.1103/PhysRevLett.85.4633. PMID 11082614.
8. Bollobás, B.; Riordan, O.; Spencer, J.; Tusnády, G. (2001). "The degree sequence of a scale-free random graph process". Random Structures and Algorithms. 18 (3): 279–290. doi:10.1002/rsa.1009. MR 1824277.
9. Dorogovtsev, S. N.; Mendes, J. F. F. (2002). "Evolution of networks". Advances in Physics. 51 (4): 1079–1187. arXiv:cond-mat/0106144. Bibcode:2002AdPhy..51.1079D. doi:10.1080/00018730110112519. Unknown parameter |s2cid= ignored (help)
10. Willinger where smax 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". 其中 s < sub > max 是所有度分布与 g 相同的图集中 h 的 s (h)的最大值。这给出了一个介于0和1之间的度量，其中小 s (g)的图 g 是“刻度丰富的” ，而 s (g)接近1的图 g 是“无刻度的”。这个定义抓住了“无标度”这个名称所隐含的自相似性的概念。, Walter; David Alderson; John C. Doyle There are two major components that explain the emergence of the scale-free property in a complex networks: the growth and the preferential attachment. 解释复杂网络中无标度性质的出现有两个主要因素: 增长性和优先连接性。 (May 2009). [http://authors.library.caltech.edu/15631/1/Willinger2009p5466Notices_Amer._Math._Soc.pdf 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. 目前，无标度网络更具体的特征随着创建无标度网络的生成机制而变化。例如，由优先连接产生的网络通常将高度的顶点放置在网络的中部，将它们连接在一起形成一个核心，核心和外围之间的区域由逐渐降低的度的节点组成。即使是很大一部分顶点的随机删除对网络的整体连接性影响很小，这表明这种拓扑可能对安全有用，而定向攻击会很快破坏连接性。其他无标度网络，把高度顶点放在周边，不表现出这些性质。类似地，无标度网络的集聚系数可以根据其他拓扑细节而发生显著的变化。 "Mathematics and the Internet: A Source of Enormous Confusion and Great Potential 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). 通过"增长"被称为一个增长过程，在一个延长的时期内，新的节点加入一个已经存在的系统，一个网络(就像万维网，在过去10年里增长了数十亿个网页)。"] Check |url= value (help). Notices of the AMS 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. and by Callaway et al. It was proven by Cohen et al that for a broad range of scale free networks, for $\displaystyle{ 2 \lt \gamma \lt 3 }$ the critical percolation threshold,$\displaystyle{ p_c=0 }$. 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 $\displaystyle{ p_c =1/\bar{k} }$, where $\displaystyle{ \bar{k} }$ 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 $\displaystyle{ p_c }$, change significantly. 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. 最后，通过“优先连接”被称为新的未来节点，它更愿意连接到另一个已经与其他节点有一定数量链接的节点。因此，越来越多的节点将自己链接到那个已经有很多链接的节点的可能性越来越大，这使得这个节点最终链接到一个中心。作者: Callaway et al. 。Cohen 等人已经证明，对于一个广泛的无标度网络，对于 < math > 2 < gamma < 3 < math > 临界渗透阈值，< math > p _ c = 0 </math > 。这意味着从网络中随机移除任何节点都不会破坏网络。这与 Erdős-r é nyi 图形形成了鲜明的对比，p _ c = 1/bar { k } </math > ，其中 < math > bar { k } </math > 是平均学位。正如逾渗理论中通常假设的那样，上面讨论的失效是随机的。然而，当将 percolation 推广到非随机但有针对性的攻击时，例如，在最高度的节点上，结果，如 < math > p _ c </math > ，会发生显著的变化。在这种情况下，一个随机选择一个节点，并删除其邻居和次近邻，直到1-p 节点的一小部分被删除。与赎金攻击和 pc > 0相比，局部攻击使无标度网络更容易受到攻击。无标度网络的渗流临界指数不同于随机 erd s-Rényi 网络。^ [16a ]因此，无标度网络与 erd s-r é nyi 网络属于不同的普适性类。. American Mathematical Society 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. 无标度网络的另一个重要特征是集聚系数分布，它随着节点度的增加而减少。这个分布也遵循一个幂定律。这意味着低度节点属于非常稠密的子图，这些子图通过集线器相互连接。考虑一个社会网络，其中的节点是人，链接是人与人之间的熟人关系。很容易看出，人们倾向于形成社区，也就是小团体，其中每个人都认识每个人(你可以把这样的社区想象成一个完整的图表)。此外，社区的成员与社区外的人也有一些熟人关系。然而，有些人与许多社区有联系(例如，名人、政治家)。这些人可能被认为是造成小世界现象的中心。. 56 (5): 586–599. Retrieved 2011-02-03. line feed character in |last= at position 11 (help); line feed character in |author3= at position 14 (help); line feed character in |journal= at position 19 (help); line feed character in |publisher= at position 30 (help); line feed character in |title= at position 81 (help); line feed character in |url= at position 90 (help)CS1 maint: multiple names: authors list (link)
11. {{cite journal 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. some of them being described with a generative model. 尽管许多现实世界的网络被认为是无标度的，但证据往往仍然不确定，主要是由于对更严格的数据分析技术的认识不断发展。他们中的一些人被描述成生成模型。 |last1=Barabási |first1=Albert-László |authorlink1=Albert-László Barabási |last2=Zoltán N. |first2= Oltvai. |title=Network biology: understanding the cell's functional organization |doi=10.1038/nrg1272 |volume=5 |year=2004 A snapshot of the weighted planar stochastic lattice (WPSL). 加权平面随机晶格的一个快照。 |journal=Nature Reviews Genetics |issue=2 |pages=101–113 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. 在高温超导体中也发现了无标度拓扑结构。高温超导体是一种电子服从量子物理定律、完全同步流动、无摩擦的化合物，其性质似乎与看似随机的氧原子的分形排列和晶格畸变有关。 |pmid=14735121 |s2cid=10950726 }}
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