# 统计代写|网络分析代写Network Analysis代考|MY561

## 统计代写|网络分析代写Network Analysis代考|Scale-free networks

In scale-free networks, almost all nodes have a roughly equal number of connections. The degree distribution for such networks follow Poisson distribution, with a peak at an average value. A recent study on large complex networks, such as the Internet, social networks, WWW, and metabolic networks, highlighted the inhomogeneous nature of the real-world networks. Most of the real-world networks usually follow the properties of scale-free network [3], where a large number of nodes have a low degree, and, contrarily, there exist few high-degree (hub) nodes. This anomaly in real-world networks is being introduced by Barabási and Albert (BA) [3]. Barabási network model deviates from the conventional network model, stating that traditional model only considers the alteration of connections. Addition of new nodes and deletion of existing nodes is not possible throughout the network creation process. However, most of the real-world networks are evolving in nature, where nodes are continuously added or deleted over time. Furthermore, Barabási and Albert also pointed out the consideraNW and WS modēl during thee creeation ố néw connēctions.

In real-world scenarios, this perception becomes unrealistic. For example, a highly cited research article is likely to get more and more citation in the future in comparison to a low-cited research paper. Similarly, the fan circle of well-known (popular) people, such as actors, sportsmen, increases over time than normal people. This phenomenon of preferential attachment bias is known as “rich-get-richer”, which is overlooked by other models. Due to the prêferential attáchment properrty, low dēgrēē nōdēs tênd tõ connnect more with the hub or core nodes and connectivity becomes sparse as it moves towards the boundary.

Barabási and Albert model starts with a few number of nodes $\left(m_0\right)$, and after every interval of time $t$, a new node is introduced to the network. These newly added nodes are connected to/from $m \leq m_0$ existing nodes. The probability of preferential attachment $\Pi_j$ that a new node $v_i$ to get connected with an existing node $v_j$, randomly chosen over $m$ existing nodes, is measured by the degree of $v_j$ (i.e., $k_j$ ) such that $\Pi_j=\frac{k_j}{\sum_{p=1}^{|\nu| k_p}}$, where $|\mathcal{V}|$ denotes the total number of nodes.

For the real-world networks, another intriguing aspect of preferential attachment $\Pi(k)$ is that it has a nonzero value towards an isolated node, i.e., $\Pi(0) \neq 0$. Therefore the preferential attachment $\Pi(k)$ for the real-world networks can be generalized as $\Pi(k)=$ $\mathcal{A}+k^\alpha$, where $\mathcal{A}$ denotes the initial pulling competence of a node and $\alpha$ is the power law exponent. Since, scale-free networks are a kind of ultra-small world [6,7] networks, therefore average path length of scale-free networks are proportional to $\log (\log (\mathcal{V}))$. The clustering coefficient of scale-free networks follows the power law, which is inversely proportional to the node degree.

## 统计代写|网络分析代写Network Analysis代考|Geometric random graph model

A geometric random graph (GRR) $\mathcal{G}(\mathcal{V}, r)$ is a graph [25] whose nodes are points in a metric space, which are connected by an edge if their distance is below a threshold value $r$ called radius. Formally, let $u, v \in \mathcal{V}$, the edge set is $\mathcal{E}={{u, v} \mid(u, v \in \mathcal{V}) \wedge(0<$ $|u-v|<r)}$, where $|\cdot|$ is a defined distance norm. Generally, a two-dimensional space is considered, and norms are the well known Manhattan or Euclidean distance, and the radius takes values in $(0,1)$.

Thus a random geometric graph $\mathcal{G}(n, r)$ is a generalization of this model, in which nodes correspond to $n$ points in a metric space. Clearly, these points are distributed uniformly and independently. Properties of these graphs have been studied when $n \rightarrow \infty$. Surprisingly, certain properties of these graphs appear when a specific number of nodes is reached.
Sticky model
The stickiness index model was introduced in [28] and it is related to the number and distribution of distinct binding domains (i.e., the regions of the structure of the proteins that are responsible for the interactions). The model, developed starting from some complex previous ones, is based on the assumption that the abundance and popularity of binding domains on a protein may be summarized in a single index related to its normalized degree, named the stickiness index. The models are based on two considerations: (i) a high degree of a protein is related to the presence of many binding domains; (ii) a pair of proteins is more likely to interact if both have high stickiness indices. Therefore for each pair of proteins, the product of the two stickiness defines the probability of interaction.

## 统计代写|网络分析代写Network Analysis代考|Scale-free networks

Barabási 和 Albert 模型从几个节点开始(米0)，并且在每个时间间隔之后吨，一个新的节点被引入网络。这些新添加的节点连接到/从米≤米0现有节点。优先依附的概率圆周率j一个新的节点在一世与现有节点连接在j, 随机选择米现有节点，由程度来衡量在j（IE，ķj) 使得圆周率j=ķj∑p=1|n|ķp， 在哪里|在|表示节点的总数。

## 统计代写|网络分析代写Network Analysis代考|Geometric random graph model

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