### 统计代写|复杂网络代写complex networks代考| A First Principles Approach to Block

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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|Mapping the Problem

Common to all of the before-mentioned approaches is their attempt to discover patterns in the link structure of networks. Patterns were either block structures in the adjacency matrix or-more specifically-cohesive subgroups. We will try to define a quality function for block structure in networks and optimize the ordering of rows and columns of the matrix as to maximize the quality of the blocking. The search for cohesive subgroups will prove to be a special case of this treatment. It makes sense to require that our quality function will be independent of the order of rows and columns within one block. It will depend only on the assignment of nodes, i.e., rows and columns, into blocks. Finding a good assignment into blocks is hence a combinatorial optimization problem. In many cases, it is possible to map such a combinatorial optimization problem onto minimizing the energy of a spin system [1]. This approach had been suggested for the first time by Fu and Anderson in 1986 [2] in the context of bi-partitioning of graphs and it has been applied successfully to other problems such as vertex cover [3], k-sat [4], the traveling salesmen [5] and many others as well.

Before introducing such a quality function, it is instructive to leave the field of networks for a moment and take a detour into the dimensionality reduction of multivariate data.

## 统计代写|复杂网络代写complex networks代考|Dimensionality Reduction with Minimal Squared Error

Suppose we are given a set of real valued measurements of some objects. As an example, for all boats in a marina, we measure length over all, width, height of the mast, the area of the sail, power of the engine, length of the waterline, and so forth. Let $N$ be the number of measurements, i.e., the number of boats in the marina, and let the measurements be vectors of dimension $d$, i.e., the number of things we have measured. We compile our measurements into a

data matrix $\mathbf{A} \in \mathbb{R}^{N \times d}$, i.e., we write the individual measurement vectors as the rows of matrix A. Let us further assume that we have already subtracted the mean across all measurements from each individual sample such that the columns of A sum to zero, i.e., we have centered our data.

Now we see that $\mathbf{A}^{T} \mathbf{A}$ is a $d \times d$ matrix describing the covariance of the individual dimensions in which we measured our data.

We now ask if we can drop some of the $d$ dimensions and still describe our data well. Naturally, we want to drop those dimensions in which our data do not vary much or we would like to replace two dimensions which are correlated by a single dimension. We can discard the unnecessary dimensions by projecting our data from the $d$-dimensional original space in a lower dimensional space of dimension $q<d$. Such a projection can be achieved by a matrix $\mathbf{V} \in \mathbb{R}^{d \times q}$. Taking measurement $\mathbf{a}{\mathbf{i}} \in \mathbb{R}^{d}$ from row $i$ of $\mathbf{A}$. we find the coordinates in the new space to be $\mathbf{b}{\mathrm{i}}=\mathbf{a}{\mathrm{i}} \mathbf{V}$ with $\mathbf{b}{\mathrm{i}} \in \mathbb{R}^{q}$. We can also use the transpose of $\mathbf{V}$ to project back into the original space of dimension $d$ via $\mathbf{a}{\mathrm{i}}^{\prime}=\mathbf{b}{\mathrm{i}} \mathbf{V}^{T}$. Since in the two projections we have visited a lower dimensional space, we find that generally the reconstructed data point does not coincide with the original datum $\mathbf{a}{\mathbf{i}} \mathbf{V V}^{T}=\mathbf{a}{\mathbf{i}}^{\prime} \neq \mathbf{a}_{\mathbf{i}}$.

However, if we would have first started in the $q$-dimensional space with $\mathbf{b}{\mathrm{i}}$ and projected it into the $d$-dimensional space via $\mathbf{V}^{T}$ and then back again via $V$ we require that our projection does not lose any information and hence $\mathbf{b}{\mathbf{i}} \mathbf{V}^{T} \mathbf{V}=\mathbf{b}_{\mathbf{i}}$. This means that we require $\mathbf{V}^{T} \mathbf{V}=\mathbb{1}$ or in other words we require that our projection matrix $V$ be unitary.

The natural question is now how to find a unitary matrix such that it minimizes some kind of reconstruction error. Using the mean square error, we could write
\begin{aligned} E \propto \sum_{i}^{N} \sum_{j}^{d}\left(\mathbf{A}-\mathbf{A}^{\prime}\right){i j}^{2} &=\sum{i}^{N} \sum_{j}^{d}\left(\mathbf{A}-\mathbf{A V V ^ { \mathbf { T } } ) _ { i j } ^ { 2 }}\right.\ &=\operatorname{Tr}\left(\mathbf{A}-\mathbf{A V V ^ { T }}\right)^{T}\left(\mathbf{A}-\mathbf{A V V ^ { T }}\right) \end{aligned}
The new coordinates that we project our data onto are called “principal components” of the data set and the technique of finding them is known as “principal component analysis” or PCA for short. Already at this point, we can mention that the $q$ columns of $V$ must be made of the eigenvectors belonging to the largest $q$ eigenvalues of $\mathbf{A}^{T} \mathbf{A}$. To show this, we discuss a slightly different problem, solve it and then show that it is equivalent to the above.
Consider the singular value decomposition (SVD) of a matrix of $\mathbf{A} \in \mathbb{R}^{N \times d}$ into a unitary matrix $\mathbf{U} \in \mathbb{R}^{N \times N}$, a diagonal matrix $\mathbf{S} \in \mathbb{R}^{N \times d}$ (in case $N \neq d$ there are maximally $\min (N, d)$ non-zero entries, the number of nonzero entries in $\mathbf{S}$ is the rank of $\mathbf{A})$ and another unitary matrix $\mathbf{V} \in \mathbb{R}^{d \times d}$ such that $\mathbf{A}=\mathbf{U S V}^{T}$ and $\mathbf{S}=\mathbf{U}^{T} \mathbf{A V}$. The entries on the diagonal of $\mathbf{S}$ are called singular values. We will assume that they are ordered decreasing in absolute value. It is straightforward to see some of the properties of this $\mathbf{S V D}: \mathbf{U}^{T} \mathbf{A}=\mathbf{S V}^{T}$ and $\mathbf{A V}=\mathbf{U S}$ follow from the $\mathbf{U}$ and $\mathbf{V}$ being unitary.

## 统计代写|复杂网络代写complex networks代考|Squared Error for Multivariate Data and Networks

Let us consider in the following the reconstruction of the adjacency matrix of a network $\mathbf{A} \in{0,1}^{N \times N}$ of rank $r$ by another adjacency matrix $\mathbf{B} \in{0,1}^{N \times N}$ possibly of lower rank $q<r$ as before. For the squared error we have
$$E=\sum_{i j}(\mathbf{A}-\mathbf{B})_{i j}^{2}$$
Then, there are only four different cases we need to consider in Table 3.1. The squared error gives equal value to the mismatch on the edges and missing edges in A. We could say it weighs every error by its own magnitude. While this is a perfectly legitimate approach for multivariate data, it is, however, highly problematic for networks. The first reason is that many networks are sparse. The fraction of non-zero entries in $\mathbf{A}$ is generally very, very small compared to the fraction of zero entries. A low rank approximation under the squared error will retain this sparsity to the point that $\mathrm{B}$ may be completely zero. Furthermore, we have seen that real networks tend to have a very heterogeneous degree distribution, i.e., the distribution of zeros and ones per row and column in $\mathbf{A}$ is also very heterogeneous. Why give every entry the same weight in the error function? Most importantly, for multivariate data, all entries of $\mathbf{A}_{i j}$ are equally important measurements in principle. For networks this is not the case: the edges are in principle more important than the missing edges. There are fewer of them and they should hence be given more importance than missing edges. Taken all of these arguments together, we see that our first goal will have to be the derivation of an error function specifically tailored for networks that does not suffer from these deficiencies.

## 广义线性模型代考

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## MATLAB代写

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