### 统计代写|复杂网络代写complex networks代考| A New Error Function

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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|A New Error Function

We already said that we would like to use a statistical mechanics approach. The problem of finding a block structure which reflects the network as good as possible is then mapped onto finding the solution of a combinatorial optimization problem. Trying to approximate the adjacency matrix A of rank $r$ by a matrix B of rank $q<r$ means approximating $\mathbf{A}$ with a block model of only full and zero blocks. Formally, we can write this as $\mathbf{B}{i j}=B\left(\sigma{i}, \sigma_{j}\right)$ where $B(r, s)$ is a ${0,1}^{q \times q}$ matrix and $\sigma_{i} \in{1, \ldots, q}$ is the assignment of node $i$ from A into one of the $q$ blocks. We can view $B(r, s)$ as the adjacency matrix of the blocks in the network or as the image graph discussed in the previous chapter and its nodes represent the different equivalence classes into which the vertices of A may be grouped. From Table 3.1, we see that our error function can have only four different contributions. They should

1. reward the matching of edges in $\mathbf{A}$ to edges in $\mathbf{B}$,
2. penalize the matching of missing edges (non-links) in $\mathbf{A}$ to edges in $\mathbf{B}$,
3. penalize the matching of edges in $\mathbf{A}$ to missing edges in $\mathbf{B}$ and
4. reward the matching of missing edges in $\mathbf{A}$ to edges in $\mathbf{B}$

These four principles can be expressed via the following function:
\begin{aligned} Q({\sigma}, \mathbf{B})=& \sum_{i j} a_{i j} \underbrace{A_{i j} B\left(\sigma_{i}, \sigma_{j}\right)}{\text {links to links }}-\sum{i j} b_{i j} \underbrace{\left(1-A_{i j}\right) B\left(\sigma_{i}, \sigma_{j}\right)}{\text {non-links to links }} \ &-\sum{i j} c_{i j} \underbrace{A_{i j}\left(1-B\left(\sigma_{i}, \sigma_{j}\right)\right)}{\text {links to non-links }}+\sum{i j} d_{i j} \underbrace{\left(1-A_{i j}\right)\left(1-B\left(\sigma_{i}, \sigma_{j}\right)\right)}{\text {non-links to non-links }} \end{aligned} in which $A{i j}$ denotes the adjacency matrix of the graph with $A_{i j}=1$, if an edge is present and zero otherwise, $\sigma_{i} \in{1,2, \ldots, q}$ denotes the role or group index of node $i$ in the graph and $a_{i j}, b_{i j}, c_{i j}, d_{i j}$ denote the weights of the individual contributions, respectively. The number $q$ determines the maximum number of groups allowed and can, in principle, be as large as $N$, the number of nodes in the network. Note that in an optimal assignment of nodes into groups it is not necessary to use all group indices as some indices may remain unpopulated in the optimal assignment.

## 统计代写|复杂网络代写complex networks代考|Fitting Networks to Image Graphs

The above-defined quality and error functions in principle consist of two parts. On one hand, there is the image graph $\mathbf{B}$ and on the other hand, there is the mapping of nodes of the network to nodes in the image graph, i.e., the assignment of nodes into blocks, which both determine the fit. Given a network $\mathbf{A}$ and an image graph $\mathbf{B}$, we could now proceed to optimize the assignment of nodes into groups ${\sigma}$ as to optimize (3.6) or any of the derived forms. This would correspond to “fitting” the network to the given image graph. This allows us to compare how well a particular network may be represented by a given image graph. We will see later that the search for cohesive subgroups is exactly of this type of analysis: If our image graph is made of isolated vertices which only connect to themselves, then we are searching for an assignment of nodes into groups such that nodes in the same group are as densely connected as possible and nodes in different groups as sparsely as possible. However, ultimately, we are interested also in the image graph which best fits to the network among all possible image graphs B. In principle, we could try out every possible image graph, optimize the assignment of nodes into blocks ${\sigma}$ and compare these fit scores. This quickly becomes impractical for even moderately large image graphs. In order to solve this problem, it is useful to consider the properties of the optimally fitting image graph $\mathbf{B}$ if we are given the networks plus the assignment of nodes into groups ${\sigma}$.

## 统计代写|复杂网络代写complex networks代考|The Optimal Image Graph

We have already seen that the two terms of (3.7) are extremized by the same $B\left(\sigma_{i}, \sigma_{j}\right)$. It is instructive to introduce the abbreviations
\begin{aligned} m_{r s} &=\sum_{i j} w_{i j} A_{i j} \delta\left(\sigma_{i}, r\right) \delta\left(\sigma_{j}, s\right) \text { and } \ {\left[m_{r s}\right]{p{i j}} } &=\sum_{i j} p_{i j} \delta\left(\sigma_{i}, r\right) \delta\left(\sigma_{j}, s\right), \end{aligned}
and write two equivalent formulations for our quality function:
\begin{aligned} &\mathbf{Q}^{1}({\sigma}, \mathbf{B})=\sum_{r, s}\left(m_{r s}-\gamma\left[m_{r s}\right]{p{i j}}\right) B(r, s) \text { and } \ &Q^{0}({\sigma}, \mathbf{B})=-\sum_{r, s}\left(m_{r s}-\gamma\left[m_{r s}\right]{p{i j}}\right)(1-B(r, s)) \end{aligned}

Now the sums run over the group indices instead of nodes and $m_{r s}$ denotes the number of edges between nodes in group $r$ and $s$ and $\left[m_{r s}\right]{p{i j}}$ is the sum of penalties between nodes in group $r$ and $s$. Interpreting $p_{i j}$ indeed as a probability or expected weight, the symbol $[\cdot]{p{i j}}$ denotes an expectation value under the assumption of a link(weight) distribution $p_{i j}$, given the current assignment of nodes into groups. That is, $\left[m_{r s}\right]{p{i j}}$ is the expected number (weight) of edges between groups $r$ and $s$. The equivalence of maximizing (3.13) and minimizing (3.14) shows that our quality function is insensitive to whether we optimize the matching of edges or missing edges between the network and the image graph.

Let us now consider the properties of an image graph with $q$ roles and a corresponding assignment of roles to nodes which would achieve the highest $Q$ across all image graphs with the same number of roles. From (3.13) and (3.14) we find immediately that for a given assignment of nodes into blocks ${\sigma}$ we achieve that $Q$ is maximal only when $B_{r s}=1$ for every $\left(m_{r s}-\left[m_{r s}\right]\right)>0$ and $B_{r s}=0$ for every $\left(m_{r s}-\left[m_{r s}\right]\right)<0$. This means that for the best fitting image graph, we have more links than expected between nodes in roles connected in the image graph. Further, we have less links than expected between nodes in roles disconnected in the image graph.

This suggests a simple way to eliminate the need for a given image graph by considering the following quality function:
$$Q({\sigma})=\frac{1}{2} \sum_{r, s}\left|m_{r s}-\gamma\left[m_{r s}\right]\right|$$
The factor $1 / 2$ enters to make the scores of $Q, Q^{0}$ and $Q^{1}$ comparable. From the assignment of roles that maximizes (3.15), we can read off the image graph simply by setting
\begin{aligned} &B_{r s}=1, \text { if }\left(m_{r s}-\gamma\left[m_{r s}\right]\right)>0 \text { and } \ &B_{r s}=0, \text { if }\left(m_{r s}-\gamma\left[m_{r s}\right]\right) \leq 0 \end{aligned}

## 统计代写|复杂网络代写complex networks代考|A New Error Function

1. 奖励边缘的匹配一种到边缘乙,
2. 惩罚缺失边（非链接）的匹配一种到边缘乙,
3. 惩罚边缘的匹配一种缺少边缘乙和
4. 奖励缺失边的匹配一种到边缘乙

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