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# 数据科学代写|复杂网络代写Complex Network代考|PCS810

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## 数据科学代写|复杂网络代写Complex Network代考|Theoretical Limits of Community Detection

With the results of the last section it is now possible to start explaining Fig. 4.5 and to give a limit to which extent a designed community structure in a network can be recovered. As was shown, for any random network one can find an assignment of spins into communities that leads to a modularity $Q>0$. For the computer-generated test networks with $\langle k\rangle=16$ one has a value of $p=\langle k\rangle /(N-1)=0.126$ and expects a value of $Q=0.227$ according to (4.15) and $Q=0.262$ according to (4.22). The modularity of the community structure built in by design is given by
$$Q\left(\left\langle k_{i n}\right\rangle\right)=\frac{\left\langle k_{i n}\right\rangle}{\langle k\rangle}-\frac{1}{4}$$
for a network of four equal sized groups of 32 nodes. Hence, below $\left\langle k_{i n}\right\rangle=8$, one has a designed modularity that is smaller than what can be expected from a random network of the same connectivity! This means that the minimum in the energy landscape corresponding to the community structure that was designed is shallower than those that one can find in the energy landscape defined by any network. It must be understood that in the search for the builtin community structure, one is competing with those community structures that arise from the fact that one is optimizing for a particular quantity in a very large search space. In other words, any network possesses a community structure that exhibits a modularity at least as large as that of a completely random network. If a community structure is to be recovered reliably, it must be sufficiently pronounced in order to win the comparison with the structures arising in random networks. In the case of the test networks employed here, there must be more than $\approx 8$ intra-community links per node. Figure 4.12 again exemplifies this. Observe that random networks with $\langle k\rangle=16$ are expected to show a ratio of internal and external links $k_{\text {in }} / k_{\text {out }} \approx 1$. Networks which are considerably sparser have a higher ratio while denser networks have a much smaller ratio. This means that in dense networks one can recover designed community structure down to relatively smaller $\left\langle k_{i n}\right\rangle$. Consider for example large test networks with $\langle k\rangle=100$ with four built-in communities. For such networks one expects a modularity of $Q \approx 0.1$ and hence the critical value of intra-community links to which the community structure could reliably be estimated would be $\left\langle k_{i n}\right\rangle_c=35$ which is much smaller in relative comparison to the average degree in the network.

## 数据科学代写|复杂网络代写Complex Network代考|Analytical Developments

Let us recall the modularity Hamiltonian:
$$\mathcal{H}=-\sum_{i<j}\left(A_{i j}-\gamma p_{i j}\right) \delta\left(\sigma_i, \sigma_j\right) .$$
For convenience, instead of a Potts model with $q$ different spin states, the discussion is limited to only two spin states as in the Ising model, namely $S_i \in-1,1$. The delta function in (5.1) can be expressed as
$$\delta\left(S_i, S_j\right)=\frac{1}{2} S_i S_j+\frac{1}{2},$$
which leads to the new Hamiltonian
$$\mathcal{H}=-\sum_{i<j}\left(A_{i j}-\gamma p_{i j}\right) S_i S_j .$$
Note that (5.3) differs from (5.1) only by an irrelevant constant which even vanishes for $\gamma=1$ due to the normalization of $p_{i j}$. Because of the factor $1 / 2$ in (5.2), the modularity of the partition into two communities is now and for the remainder of this chapter
$$Q_2=-\frac{\mathcal{H}}{2 M},$$
where $\mathcal{H}$ now denotes the Hamiltonian (5.3). For the number of cut edges of the partition one can write
$$\mathcal{C}=\frac{1}{2}\left(M+E_g\right)=\frac{M}{2}\left(1-2 Q_2\right),$$
with $E_g$ denoting the ground state energy of (5.3) and it is clear that $Q_2$ measures the improvement of the partition over a random assignment into groups.

Formally, (5.3) corresponds to a Sherrington-Kirkpatrick (SK) model of a spin glass [3]
$$\mathcal{H}=-\sum_{i<j} J_{i j} S_i S_j,$$
with couplings of the form
$$J_{i j}=\left(A_{i j}-\gamma p_{i j}\right) .$$

## 数据科学代写|复杂网络代写Complex Network代考|Theoretical Limits of Community Detection

$$Q\left(\left\langle k_{i n}\right\rangle\right)=\frac{\left\langle k_{i n}\right\rangle}{\langle k\rangle}-\frac{1}{4}$$

## 数据科学代写|复杂网络代写Complex Network代考|Analytical Developments

$$\mathcal{H}=-\sum_{i<j}\left(A_{i j}-\gamma p_{i j}\right) \delta\left(\sigma_i, \sigma_j\right) .$$

$$\delta\left(S_i, S_j\right)=\frac{1}{2} S_i S_j+\frac{1}{2},$$

$$\mathcal{H}=-\sum_{i<j}\left(A_{i j}-\gamma p_{i j}\right) S_i S_j .$$

$$Q_2=-\frac{\mathcal{H}}{2 M},$$

$$\mathcal{C}=\frac{1}{2}\left(M+E_g\right)=\frac{M}{2}\left(1-2 Q_2\right),$$
$E_g$表示(5.3)的基态能量，很明显，$Q_2$测量了随机分配到组上的分区的改进。

$$\mathcal{H}=-\sum_{i<j} J_{i j} S_i S_j,$$

$$J_{i j}=\left(A_{i j}-\gamma p_{i j}\right) .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。