### 统计代写|复杂网络代写complex networks代考| Scale-Free Degree Distributions

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|复杂网络代写complex networks代考|Scale-Free Degree Distributions

With the increasing use of the Internet as a source of information and means of communication as well as the increasing availability of large online databases and repositories, more and more differences between real world networks and random graphs were discovered. Most strikingly was certainly the observation that many real world networks have a degree distribution far from Poissonian with heavy tails which rather follows a log-normal distribution or alternatively a power law.

For networks with a power-law degree distribution the notion of a “scalefree” degree distribution is often used. A scale-free degree distribution is characterized by a power law of the form
$$P(k) \propto k^{-\gamma}$$
with some positive exponent $\gamma$. The probability of having $k$ neighbors is inversely proportional to $k^{\gamma}$. The name “scale free” comes from the fact that there is no characteristic value of $k$. While in ER graphs, the characteristic $k$ is the average degree $\langle k\rangle$, i.e., the average is also a typical $k$, there is no typical degree in scale-free networks.

From these observations, it became clear that the assumption of equal linking probability for all pairs of nodes had to be dropped and that specific mechanisms had to be sought which could explain the link pattern of complex networks from a set of rules. Until now, many such models have been introduced which model networks to an almost arbitrary degree of detail. The starting point for this development was most likely the model by Barabási and Albert [16]. They realized that for many real world networks, two key ingredients are crucial: growth and preferential attachment, i.e., nodes that already have a large number of links are more likely to acquire new ones during the growth of the network. These two simple assumptions lead them to develop a network model which produces a scale-free degree distribution of exponent $\gamma=3$. Consequently, this model was used as a first attempt to explain the link distribution of web pages.

In order to model an ensemble of random graphs with a given degree distribution without resorting to some growth model of how the graph is knit the “configuration model” can be used. It is generally attributed to Molloy and Reed [17], who devised an algorithm for constructing actual networks, but it was first introduced by Bender and Canfield [18]. The configuration model assumes a given degree distribution $P(k)$. Every node $i$ is assigned a number of stubs $k_{i}$ according to its degree drawn from $P(k)$ and then the stubs are connected randomly. For this model, the probability that two randomly chosen nodes are connected by an edge can be well approximated by $p_{i j}=k_{i} k_{j} / 2 M$ as long as the degrees of the nodes are smaller than $\sqrt{2 M}$. The probability to find a link between two nodes is hence proportional to the product of the degrees of these two nodes. The configuration model and the ER model make fundamentally different assumptions on the nature of the objects represented by the nodes. In the ER model, fluctuations in the number of connections of a node arise entirely due to chance. In the configuration model, they represent a quality of the node which may be interpreted as some sort of “activity” of the object represented by the node.

## 统计代写|复杂网络代写complex networks代考|Correlations in Networks

Thus far, only models in which all nodes were equivalent have been introduced. In many networks, however, nodes of different types coexist and the probability of linking between them may depend on the types of nodes. A typical example may be the age of the nodes in a social network. Agents of the same age generally have a higher tendency to interact than agents of different ages. Let us assume the type of each node is already known. One can then ask whether the assumption holds, that links between nodes in the same class are indeed more frequent than links between nodes in different classes. Newman [19] defines the following quantities: $e_{r s}$ as the fraction of edges that fall between nodes in class $r$ and $s$. Further, he defines $\sum_{r} e_{r s}=a_{s}$ as the fraction of edges that are connected to at least one node in class $s$. Note that

$e_{r s}$ can also be interpreted as the probability that a randomly chosen edge lies between nodes of class $r$ and $s$ and that $a_{s}$ can be interpreted as the probability that a randomly chosen edge has at least one end in class $s$. Hence, $a_{s}^{2}$ is the expected fraction of edges connecting two nodes of class $s$. Comparing this expectation value with the true value $e_{s s}$ for all groups $s$ leads to the definition of the “assortativity coefficient” $r_{A}$ :
$$r_{A}=\frac{\sum_{s}\left(e_{s s}-a_{s}^{2}\right)}{1-\sum_{s} a_{s}^{2}} .$$
This assortativity coefficient $r_{A}$ is one, if all links fall exclusively between nodes of the same type. Then the network is perfectly “assortative”, but the different classes of nodes remain disconnected. It is zero if $e_{s s}=a_{s}^{2}$ for all classes $s$, i.e., no preference in linkage for either the same or a different class is present. It takes negative values, if edges lie preferably between nodes of different classes, in which case the network is called “disassortative”. The denominator corresponds to a perfectly assortative network. Hence, $r_{A}$ can be interpreted as the percentage to which the network is perfectly assortative.
For the classes of the nodes, any measurable quantity may be used [20]. Especially interesting are investigations into assortative mixing by degree, i.e., do nodes predominantly connect to other nodes of similar degree (assortative, $r_{A}>0$ ) or is the opposite the case (disassortative, $r_{A}<0$ ). It was found that many social networks are assortative, while technological or biological networks are generally disassortative $[20]$. Note that $r_{A}$ may also be generalized to the case where the class index $s$ takes continuous values [20]. It should be stressed that such correlation structures do not affect the degree distribution.

## 统计代写|复杂网络代写complex networks代考|Dynamics on Networks

Apart from these topological models mainly concerned with link structure, a large number of researchers are concerned with dynamical processes taking place on networks and the influence the network structure has on them. Among the most widely studied processes is epidemic spreading and one of the most salient results is certainly that by Cohen $[21,22]$, which shows that for scale-free topologies with exponents larger than two and low clustering, the epidemic threshold (the infectiousness a pathogen needs to infect a significant portion of the network) drops to zero. The reason for this is, in principle, the fact that for scale-free degree distributions with exponents between 2 and 3 the average number of second neighbors $\langle d\rangle$ may diverge. Liljeros showed that networks of sexual contacts do have indeed such a topology [23]. At the same time, these results brought about suggestions for new vaccination techniques such as the vaccination of acquaintances of randomly selected people which allows us to vaccinate people with higher numbers of connections with higher efficiency [24]. Consequently, a number of researchers are also studying the interplay between topology of the network and dynamic processes on networks

in models that allow dynamic rewiring of connections in accordance with, for instance, games being played on the network to gain insights into the origin of cooperation [25].

All of this research has shown the profound effect of the topology of the connections underlying a dynamical process and hence underlines the importance of thoroughly studying the topology of complex networks.

## 统计代写|复杂网络代写complex networks代考|Correlations in Networks

r一种=∑s(和ss−一种s2)1−∑s一种s2.

## 广义线性模型代考

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## MATLAB代写

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