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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|MY561

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

在无标度网络中,几乎所有节点都有大致相等的连接数。这种网络的度分布遵循泊松分布,在平均值处有一个峰值。最近对大型复杂网络(如互联网、社交网络、万维网和代谢网络)的研究强调了现实世界网络的非同质性。大多数现实世界的网络通常遵循无标度网络的特性[3],其中大量节点具有低度,相反,存在很少的高度(中心)节点。Barabási 和 Albert (BA) [3] 正在介绍现实世界网络中的这种异常情况。Barabási 网络模型偏离了传统的网络模型,指出传统模型只考虑连接的变化。在整个网络创建过程中,不可能添加新节点和删除现有节点。然而,大多数现实世界的网络在本质上都是不断发展的,随着时间的推移,节点会不断地添加或删除。此外,Barabási 和 Albert 还指出了在创建新连接过程中考虑的 NW 和 WS 模型。

在现实世界的场景中,这种看法变得不切实际。例如,与低被引研究论文相比,高被引研究论文在未来可能会获得越来越多的引用。同样,演员、运动员等知名(流行)人士的粉丝圈随着时间的推移比正常人增加。这种偏好依恋偏差的现象被称为“富-富-富”,被其他模型所忽视。由于优先连接属性,低 dēgrēē nōdēs tênd tõ 与中心或核心节点的连接更多,并且随着向边界移动,连接变得稀疏。

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。

对于现实世界的网络,优先依恋的另一个有趣的方面圆周率(ķ)是它对孤立节点具有非零值,即圆周率(0)≠0. 因此优先附加圆周率(ķ)对于现实世界的网络可以概括为圆周率(ķ)= 一个+ķ一个, 在哪里一个表示节点的初始拉动能力,并且一个是幂律指数。由于无标度网络是一种超小世界[6,7]网络,因此无标度网络的平均路径长度与日志⁡(日志⁡(在)). 无标度网络的聚类系数遵循幂律,与节点度成反比。

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

几何随机图 (GRR)G(在,r)是一个图 [25],其节点是度量空间中的点,如果它们的距离低于阈值,则通过边连接r称为半径。正式地,让在,在∈在,边集是和=在,在∣(在,在∈在)∧(0<$$|在−在|<r), 在哪里|⋅|是一个定义的距离范数。一般考虑二维空间,范数为众所周知的曼哈顿距离或欧几里得距离,半径取值(0,1).

因此一个随机的几何图G(n,r)是该模型的推广,其中节点对应于n度量空间中的点。显然,这些点是均匀且独立地分布的。这些图的性质已经研究过n→∞. 令人惊讶的是,当达到特定数量的节点时,这些图的某些属性就会出现。
粘性模型
粘性指数模型是在[28] 中引入的,它与不同结合域(即负责相互作用的蛋白质结构的区域)的数量和分布有关。该模型是从以前的一些复杂模型开始开发的,它基于这样一个假设,即蛋白质上结合域的丰度和流行度可以概括为与其归一化程度相关的单个指数,称为粘性指数。这些模型基于两个考虑:(i)蛋白质的高度与许多结合域的存在有关;(ii) 如果两者都具有高粘性指数,则一对蛋白质更有可能相互作用。因此,对于每一对蛋白质,两种粘性的乘积定义了相互作用的概率。

统计代写|网络分析代写Network Analysis代考

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