### 统计代写|复杂网络代写complex networks代考|Introduction to Complex Networks

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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|Graph Theoretical Notation

Mathematically, a network is represented as a graph $\mathcal{S}(V, E)$, i.e., an object that consists of a set of nodes or vertices $V$ representing the objects or agents in the network and a set $E$ of edges or links or connections representing the interactions or relations of the nodes. The cardinality of these sets, i.e, the number of nodes and edges, is generally denoted by $N$ and $M$, respectively. One may assign different values $w_{i j}$ to the links between nodes $i$ and $j$ in $E$, rendering an edge weighted or otherwise non-weighted ( $w_{i j}=1$ by convention, if one is only interested in the presence or absence of the relation). The number of connections of node $i$ is denoted by its degree $k_{i}$. One can represent the set of edges conveniently in an $N \times N$ matrix $A_{i j}$, called the adjacency matrix. $A_{i j}=w_{i j}$ if an edge between node $i$ and $j$ is present and zero otherwise. Relations may be directed, in which case $A_{i j}$ is non-symmetric $\left(A_{i j} \neq A_{j i}\right)$ or undirected in which case $A_{i j}$ is symmetric. Here, we are mainly concerned with networks in which self-links are absent $\left(A_{i i}=0, \forall i \in V\right)$. In case of a directed network, $A_{i j}$ denotes an outgoing edge from $i$ to $j$. Hence, the outgoing links of node $i$ are found in row $i$, while the incoming links to $i$ are found in column $i$. For undirected networks, it is clear that $\sum_{j=1}^{N} A_{i j}=k_{i}$. For directed networks, $\sum_{j=1} A_{i j}=k_{i}^{\text {out }}$ is the out-degree and equivalently $\sum_{j=1} A_{j i}=k_{i}^{i n}$ is the in-degree of node $i$. It is understood that in undirected networks, the sum of degrees of all nodes in the network equals twice the number of edges $\sum_{i=1}^{N} k_{i}=2 M$. The distribution of the number of connections per node is called degree distribution $P(k)$ and denotes the probability that a randomly chosen node from the network has degree $k$. The average degree in the network is denoted by $\langle k\rangle$ and one has $N\langle k\rangle=2 M$. One can define a

probability $p=2 M / N(N-1)=\langle k\rangle /(N-1)$ as the probability that an edge exists between two randomly chosen nodes from the network.

An (induced) subgraph is a subset of nodes $v \subseteq V$ with $n$ nodes and edges $e \subseteq E$ connecting only the nodes in $v$. A path is a sequence of nodes, subsequent nodes in the sequence being connected by edges from $E$. A node $i$ is called reachable from node $j$ if there exists a path from $j$ to $i$. A subgraph is said to be connected if every node in the subgraph is reachable from every other. The number of steps (links) in the shortest path between two nodes $i$ and $j$ is called the geodesic distance $d(i, j)$ between nodes $i$ and $j$. A network is generally not connected, but may consist of several connected components. The largest of the shortest path distances between any pair of nodes in a connected component is called the diameter of a connected component. The analysis in this monograph shall be restricted to connected components only since it can be repeated on every single one of the connected components of a network. More details on graph theory may be found in the book by Bollobás $[2]$.

With these notations and terms in mind, let us now turn to a brief overview of some aspects of physicists research on networks.

## 统计代写|复杂网络代写complex networks代考|Random Graphs

For the study of the topology of the interactions of a complex system it is of central importance to have proper random null models of networks, i.e., models of how a graph arises from a random process. Such models are needed for comparison with real world data. When analyzing the structure of real world networks, the null hypothesis shall always be that the link structure is due to chance alone. This null hypothesis may only be rejected if the link structure found differs significantly from an expectation value obtained from a random model. Any deviation from the random null model must be explained by non-random processes.

The most important model of a random graph is due to Erdös and Rényi (ER) [12]. They consider the following two ensembles of random graphs: $\mathcal{G}(N, M)$ and $\mathcal{G}(N, p)$. The first is the ensemble of all graphs with $N$ nodes and exactly $M$ edges. A graph from this ensemble is created by placing the $M$ edges randomly between the $N(N-1) / 2$ possible pairs of nodes. The second ensemble is that of all graphs in which a link between two arbitrarily chosen nodes is present with probability $p$. The expectation value for the number of links of a graph from this ensemble is $\langle M\rangle=p N(N-1) / 2$. In the limit of $N \rightarrow \infty$, the two ensembles are equivalent with $p=2 M / N(N-1)$. The typical graph from these ensembles has a Poissonian degree distribution
$$P(k)=e^{-\langle k\rangle} \frac{\langle k\rangle^{k}}{k !} .$$
Here, $\langle k\rangle=p(N-1)=2 M / N$ denotes the average degree in the network.

The properties of ER random graphs have been studied for considerable time and an overview of results can be found in the book by Bollobás [13]. Note that the equivalence of the two ensembles is a remarkable result. If all networks with a given number of nodes and links are taken to be equally probable, then the typical graph from this ensemble will have a Poissonian degree distribution. To draw a graph with a non-Poissonian degree distribution from this ensemble is highly improbable, unless there is a mechanism which leads to a different degree distribution. This issue will be discussed below in more detail.

Another aspect of random networks is worth mentioning: the average shortest path between any pair of nodes scales only as the logarithm of the system size. This is easily seen: Starting from a randomly chosen node, we can visit $\langle k\rangle$ neighbors with a single step. How many nodes can we explore with the second step? Coming from node $i$ to node $j$ via a link between them, we now have $d_{j}=k_{j}-1$ options to proceed. Since we have $k_{j}$ possible ways to arrive at node $j$, the average number of second step neighbors is hence $\langle d\rangle=\sum_{k=2}^{\infty}(k-1) k P(k) /\left(\sum_{k}^{\infty} k P(k)\right)=\left\langle k^{2}\right\rangle /\langle k\rangle-1$. Hence, after two steps we may explore $\langle k\rangle\langle d\rangle$ nodes and after $m$ steps $\langle k\rangle\langle d\rangle^{m-1}$ nodes which means that the entire network may be explored in $m \approx \log N$ steps. This also shows that even in very large random networks, all nodes may be reached with relatively few steps. The number $d=k-1$ of possible ways to exit from a node after entering it via one of its links is also called the “excess degree” of a node. Its distribution $q(d)=(d+1) P(d+1) /\langle k\rangle$ is called the “excess degree distribution” and plays a central role in the analysis of many dynamical phenomena on networks. Note that our estimate is based on the assumption that in every new step we explore $\langle d\rangle$ nodes which we have not seen before! For ER networks, though, this is a reasonable assumption. However, consider a regular lattice as a counterexample. There, the average shortest distance between any pair of nodes scales linearly with the size of the lattice.

## 统计代写|复杂网络代写complex networks代考|Six Degrees of Separation

The question of short distances was one of the first addressed in the study of real world networks by Stanley Milgram [14]. It was known among sociologists that social networks are characterized by a high local clustering coefficient:
$$c_{i}=\frac{2 m_{i}}{k_{i}\left(k_{i}-1\right)}$$
Here, $m_{i}$ is the number of connections among the $k_{i}$ neighbors of node $i$. In other words, $c_{i}$ measures the probability of the neighbors of node $i$ being connected, that is, the probability that the friends of node $i$ are friends among each other. The average of this clustering coefficient over the set of nodes in the network is much higher in social networks than for ER random networks with the same number of nodes and links where $\langle c\rangle=p$. This would mean that the average shortest distance between randomly chosen nodes in social networks may not scale logarithmically with the system size. To test this, Milgram performed the following experiment: He gave out letters in Omaha, Nebraska, and asked the initial recipients of the letters to give the letters only to acquaintances whom they would address by their first name and require that those would do the same when passing the letter on. The letters were addressed to a stock broker living in Boston and unknown to the initial recipients of the letter. Surprisingly, not only did a large number of letters arrive at the destination, but the median of the number of steps it took was only 6. This means the path lengths in social networks may be surprisingly short despite the high local clustering. Even more surprisingly, the agents in this network are able to efficiently navigate messages through the entire network even though they only know the local topology. After this discovery, Duncan Watts and Steve Strogatz [15] provided the first model of a network that combines the high clustering characteristic for acquaintance networks and the short average path lengths known from ER random graphs. At the same time, it retains the fact that there is only a finite number of connections or friends per node in the network. The Watts/Strogatz model came to be known as the “small world model” for complex networks. It basically consists of a regular structure producing a high local clustering and a number of randomly interwoven shortcuts responsible for the short average path length. It was found analytically that a small number of shortcuts, added randomly, suffice to change the scaling of the average shortest path length from linear with system size to logarithmically with system size.

## 统计代写|复杂网络代写complex networks代考|Graph Theoretical Notation

（诱导）子图是节点的子集在⊆在和n节点和边和⊆和只连接节点在. 路径是一系列节点，序列中的后续节点由来自的边连接和. 一个节点一世被称为从节点可达j如果存在从j到一世. 如果子图中的每个节点都可以相互访问，则称该子图是连通的。两个节点之间最短路径的步数（链接）一世和j称为测地线距离d(一世,j)节点之间一世和j. 一个网络通常没有连接，但可能由几个连接的组件组成。连通分量中任意一对节点之间的最短路径距离中的最大值称为连通分量的直径。本专着中的分析应仅限于连接组件，因为它可以在网络的每个连接组件上重复。关于图论的更多细节可以在 Bollobás 的书中找到[2].

## 统计代写|复杂网络代写complex networks代考|Random Graphs

ER 随机图的性质已经研究了相当长的时间，结果概述可以在 Bollobás [13] 的书中找到。请注意，这两个集合的等价性是一个显着的结果。如果具有给定数量的节点和链接的所有网络都被认为是等概率的，那么来自该集成的典型图将具有泊松度分布。从这个集合中绘制具有非泊松度分布的图是非常不可能的，除非有一种机制导致不同的度分布。这个问题将在下面更详细地讨论。

## 统计代写|复杂网络代写complex networks代考|Six Degrees of Separation

C一世=2米一世ķ一世(ķ一世−1)

## 广义线性模型代考

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## MATLAB代写

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