如果你也在 怎样代写复杂网络complex networks这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。复杂网络complex networks在网络理论的背景下，是一种具有非微观拓扑特征的图（网络）–这些特征在简单的网络（如格子或随机图）中不会出现，但在代表真实系统的网络中经常出现。复杂网络的研究是一个年轻而活跃的科学研究领域（自2000年以来），主要受到现实世界网络的经验发现的启发，如计算机网络、生物网络、技术网络、大脑网络、气候网络和社会网络。
复杂网络complex networks有两类众所周知且研究较多的复杂网络是无标度网络和小世界网络，它们的发现和定义是该领域的典型案例研究。两者都具有特定的结构特征–前者是幂律学位分布，后者是短路径长度和高聚类。然而，随着复杂网络研究的重要性和受欢迎程度不断提高，网络结构的许多其他方面也引起了人们的注意。
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我们提供的复杂网络complex networks及其相关学科的代写，服务范围广, 其中包括但不限于:
非线性方法 nonlinear method functional analysis
变分法 Calculus of Variations
数学代写|复杂网络作业代写complex networks代考|The Degree Distribution
A fundamental characteristic of any random network is the node degree distribution $N_{k}(N)$, defined as the number of nodes with $k$ links in a network that
2 Rate Equation Approach for Growing Networks
5
contains $N$ total nodes. To determine this distribution, we write the rate equations that account for its evolution after each node is introduced. For the growth process in Fig. 2.1a, these rate equations are [20, 21, 22]
$$
\frac{d N_{k}}{d N}=\frac{A_{k-1} N_{k-1}-A_{k} N_{k}}{A}+\delta_{k 1} .
$$
The first term on the right, $A_{k-1} N_{k-1} / A$, accounts for processes in which a node with $k-1$ links is connected to the new node, thus increasing $N_{k}$ by one. Since there are $N_{k-1}$ nodes of degree $k-1$, such processes occur at a rate proportional to $A_{k-1} N_{k-1}$, while the factor $A(N)=\sum_{j \geq 1} A_{j} N_{j}(N)$ converts this rate into a normalized probability. A corresponding role is played by the second (loss) term on the right-hand side. Here $A_{k} N_{k} / A$ is the probability that a node with $k$ links is connected to the new node, thus leading to a loss in $N_{k}$. The last term accounts for the introduction of a new node with degree one.
数学代写|复杂网络作业代写complex networks代考|Order Distribution
In addition to node degree, we further characterize a node according to its order of introduction by associating an order index $J$ to the $J^{\text {th }}$ node that was introduced into the network [22, 23]. Let $\mathcal{N}{k}(N, J)$ be the probability that the $J^{\text {th }}$ node has degree $k$ when the network has $N$ total nodes. The original degree distribution may be recovered from this joint order-degree distribution through $N{k}(N)=\sum_{J=1}^{N} \mathcal{N}{k}(N, J)$. The joint distribution evolves according to the rate equation $$ \left(\frac{\partial}{\partial N}-\frac{\partial}{\partial J}\right) \mathcal{N}{k}=\frac{A_{k-1} \mathcal{N}{k-1}-A{k} \mathcal{N}{k}}{A}+\delta{k 1} \delta(N-J)
$$
The second term on the left account for the order index evolution. We assume that the probability of linking to a given node depends only on its degree and not on its order.
The homogeneous form of this equation suggests that the solution should depend on the single variable $x \equiv J / N$. Writing $\mathcal{N}{k}(N, J)=f{k}(x)$, converts (2.10) into an ordinary, and readily soluble, differential equation [22]. For the two generic cases of $A_{k}=k$ and $A_{k}=1$, the order-degree distributions are:
$$
\mathcal{N}{k}(N, J)= \begin{cases}\sqrt{\frac{J}{N}}\left(1-\sqrt{\frac{J}{N}}\right)^{k-1} & A{k}=k \ \frac{J}{N} \frac{[\ln (N / J)]^{k-1}}{(k-1) !} & A_{k}=1\end{cases}
$$
数学代写|复杂网络作业代写complex networks代考|Degree Correlations
The rate equation approach also allows us to obtain degree correlations between connected nodes [22]. These develop because a node with large degree is likely to be old $[22,24,25,26]$. Thus its ancestor is also old and hence has a large degree. To quantify these degree correlations, define $C_{k l}(N)$ as the number of nodes of degree $k$ that attach to an ancestor node of degree $l$ (Fig. 2.2a). For example, in the network of Fig. $2.1$, there are $N_{1}=6$ nodes of degree 1 , with $C_{12}=C_{13}=C_{15}=2$. There are also $N_{2}=2$ nodes of degree 2 , with $C_{25}=2$, and $N_{3}=1$ nodes of degree 3 , with $C_{35}=1$.
For simplicity, we consider the linear attachment kernel for which the degree correlation $C_{k l}(N)$ evolves according to
$$
M_{1} \frac{d C_{k l}}{d N}=(k-1) C_{k-1, l}-k C_{k l}+(l-1) C_{k, l-1}-l C_{k l}+(l-1) C_{l-1} \delta_{k 1} \cdot(2.14)
$$
The processes that gives rise to each term in this equation are illustrated in Fig. 2.3. The first two terms on the right account for the change in $C_{k l}$ due to the addition of a link onto a node of degree $k-1$ (gain) or $k$ (loss) respectively, while the second set of terms gives the change in $C_{k l}$ due to the addition of a link onto the ancestor node. Finally, the last term accounts for the gain in $C_{1 l}$ due to the addition of a new node. A crucial feature of this equation is that it is closed; the 2-particle correlation function does not depend on 3-particle quantities.
复杂网络代写
数学代写|复杂网络作业代写COMPLEX NETWORKS代考|THE DEGREE DISTRIBUTION
任何随机网络的一个基本特征是节点度分布ñķ(ñ)，定义为节点数ķ
2 Rate Equation Approach for Growing Networks
5
包含的网络中的链接ñ总节点。为了确定这种分布，我们编写了在引入每个节点后解释其演化的速率方程。对于图 2.1a 中的增长过程，这些速率方程是20,21,22
dñķdñ=一种ķ−1ñķ−1−一种ķñķ一种+dķ1.
右边的第一项，一种ķ−1ñķ−1/一种, 说明一个节点与ķ−1链接连接到新节点，从而增加ñķ一个。既然有ñķ−1度数节点ķ−1, 这样的过程发生的速率与一种ķ−1ñķ−1, 而因子一种(ñ)=∑j≥1一种jñj(ñ)将此比率转换为归一化概率。第二个扮演了相应的角色l这ss右边的术语。这里一种ķñķ/一种是一个节点的概率ķ链接连接到新节点，从而导致丢失ñķ. 最后一项说明引入了一个度数为 1 的新节点。
数学代写|复杂网络作业代写COMPLEX NETWORKS代考|ORDER DISTRIBUTION
除了节点度之外，我们通过关联一个顺序索引进一步根据其引入顺序来表征节点Ĵ到Ĵth 引入网络的节点22,23. 设 $J$ to the $J^{\text {th }}$ node that was introduced into the network [22, 23]. Let $\mathcal{N}{k}(N, J)$ be the probability that the $J^{\text {th }}$ node has degree $k$ when the network has $N$ total nodes. The original degree distribution may be recovered from this joint order-degree distribution through $N{k}(N)=\sum_{J=1}^{N} \mathcal{N}{k}(N, J)$. The joint distribution evolves according to the rate equation $$ \left(\frac{\partial}{\partial N}-\frac{\partial}{\partial J}\right) \mathcal{N}{k}=\frac{A_{k-1} \mathcal{N}{k-1}-A{k} \mathcal{N}{k}}{A}+\delta{k 1} \delta(N-J)
$$
左边的第二项说明了订单指数的演变。我们假设链接到给定节点的概率仅取决于其程度而不取决于其顺序。
该方程的齐次形式表明解应取决于单个变量$x \equiv J / N$. Writing $\mathcal{N}{k}(N, J)=f{k}(x)$, converts (2.10) into an ordinary, and readily soluble, differential equation [22]. For the two generic cases of $A_{k}=k$ and $A_{k}=1$, the order-degree distributions are:
$$
\mathcal{N}{k}(N, J)= \begin{cases}\sqrt{\frac{J}{N}}\left(1-\sqrt{\frac{J}{N}}\right)^{k-1} & A{k}=k \ \frac{J}{N} \frac{[\ln (N / J)]^{k-1}}{(k-1) !} & A_{k}=1\end{cases}
$$
数学代写|复杂网络作业代写COMPLEX NETWORKS代考|DEGREE CORRELATIONS
速率方程方法还允许我们获得连接节点之间的度数相关性22. 这些发展是因为度数大的节点可能是旧的[22,24,25,26]. 因此它的祖先也很老，因此具有很大的程度。为了量化这些度数相关性，定义Cķl(ñ)作为度数的节点数ķ附加到度的祖先节点l F一世G.2.2一种. 例如，在图 1 的网络中。2.1， 有ñ1=6度为 1 的节点，具有C12=C13=C15=2. 还有ñ2=2度为 2 的节点，具有C25=2， 和ñ3=13 次节点，有C35=1.
为简单起见，我们考虑其度相关性的线性附件核Cķl(ñ)根据进化
米1dCķldñ=(ķ−1)Cķ−1,l−ķCķl+(l−1)Cķ,l−1−lCķl+(l−1)Cl−1dķ1⋅(2.14)
图 2.3 说明了产生该等式中每一项的过程。右边的前两个术语说明了变化Cķl由于在度数节点上添加了链接ķ−1 G一种一世n或者ķ l这ss分别，而第二组术语给出了变化Cķl由于在祖先节点上添加了链接。最后，最后一项解释了收益C1l由于添加了一个新节点。这个方程的一个关键特征是它是封闭的。2 粒子相关函数不依赖于 3 粒子量。
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