### 统计代写|复杂网络代写complex networks代考| Optimizing the Quality 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|Optimizing the Quality Function

After having studied some properties of the configurations and image graphs that optimize (3.13), (3.14) or (3.15), let us now turn to the problem of actually finding these configurations. Though any optimization scheme that can deal with combinatorial optimization problems may be implemented $[8,9]$, the

use of simulated annealing $[10]$ for a Potts model [11] is shown, because it yields high-quality results, is very general in its application and very simple to program. We interpret our quality function $Q$ to be maximized as the negative of a Hamiltonian to be minimized, i.e., we write $\mathcal{H}({\sigma})=-Q$. The single site heat bath update rule at temperature $T=1 / \beta$ then reads as follows:
$$p\left(\sigma_{i}=\alpha\right)=\frac{\exp \left(-\beta \mathcal{H}\left(\left{\sigma_{j \neq i}, \sigma_{i}=\alpha\right}\right)\right)}{\sum_{s=1}^{q} \exp \left(-\beta \mathcal{H}\left(\left{\sigma_{j \neq i}, \sigma_{i}=s\right}\right)\right)}$$
That is, the probability of node $i$ being in group $\alpha$ is proportional to the exponential of the energy (negative quality) of the entire system with all other nodes $j \neq i$ fixed and node $i$ in state $\alpha$. Since this is costly to evaluate, one pretends to know the energy of the system with node $i$ in some arbitrarily chosen group $\phi$, which is denoted by $\mathcal{H}{\phi}$. Then one can calculate the energy of the system with $i$ in group $\alpha$ as $\mathcal{H}{\phi}+\Delta \mathcal{H}\left(\sigma_{i}=\phi \rightarrow \alpha\right)$. The energy $\mathcal{H}{\phi}$ then factors out in (3.25) and one is left with $$p\left(\sigma{i}=\alpha\right)=\frac{\exp \left{-\beta \Delta \mathcal{H}\left(\sigma_{i}=\phi \rightarrow \alpha\right)\right}}{\sum_{s=1}^{q} \exp \left{-\beta \Delta \mathcal{H}\left(\sigma_{i}=\phi \rightarrow s\right)\right}}$$
Suppose we are trying to fit a network to a given image graph, i.e., $\mathbf{B}$ is given. Then the change in energy $\Delta \mathcal{F} C\left(\sigma_{i}=\phi \rightarrow \alpha\right)$ is easily calculated from the change in quality according to (3.13):
\begin{aligned} \Delta \mathcal{H}\left(\sigma_{i}=\phi \rightarrow \alpha\right)=& \sum_{s}\left(B_{\phi s}-B_{\alpha s}\right)\left(k_{i \rightarrow s}^{\text {out }}-\gamma\left[k_{i \rightarrow s}^{\text {out }}\right]\right) \ &+\sum_{r}\left(B_{r \phi}-B_{r \alpha}\right)\left(k_{r \rightarrow i}^{i n}-\gamma\left[k_{r \rightarrow i}^{i n}\right]\right) \ =& \sum_{s}\left(B_{\phi s}-B_{\alpha s}\right) a_{i s}+\sum_{r}\left(B_{r \phi}-B_{r \alpha}\right) a_{r i} \end{aligned}

## 统计代写|复杂网络代写complex networks代考|Properties of the Ground State

From the fact that the ground state is a configuration which is a minimum in the configuration space, one can derive a number of properties of the communities that apply to any local minimum of the Hamiltonian in the configuration space. If one takes these properties as defining properties of what a community is, one then finds valid alternative community structures also in the local minima of the Hamiltonian. The energies of these local minima will then allow us to compare these community structures. It may be that alternative

but almost equally “good” community structures exist. Before proceeding to investigate the properties of spin configurations that represent local minima of the Hamiltonian, a few properties of (4.3) as such shall be discussed:

First, note that for $\gamma=1(4.3)$ evaluates to zero in case of assigning all nodes into the same spin state due to the normalization constraint on $p_{i j}$, i.e., $\sum_{i j} p_{i j}=\sum_{i j} A_{i j}=M$, independent of the graph. Second, for a complete graph, any spin configuration yields the same zero energy at $\gamma=1$. Third, for a graph without edges, e.g., only a set of nodes, any spin configuration gives zero energy independent of $\gamma$. Fourth, the expectation value of (4.3) for a random assignment of spins at $\gamma=1$ is zero. These considerations provide an intuitive feeling for the fact that the lower the energy the better the fit of the diagonal block model to the network and that the choice of $\gamma=1$ will result in what could be called “natural partitioning” of the graph into modules.
Let us consider the case of undirected networks which is most often found in applications. Then, the adjacency matrix of the network is symmetric and we have $k_{i}^{\text {in }}=k_{i}^{\text {out }}$ and thus the coefficients of adhesion are also symmetric, i.e., $a_{r s}=a_{s r}$. According to (3.28) the change in energy to move a group of nodes $n_{1}$ from group $s$ to spin state $r$ is
$$\Delta \mathcal{H}=a_{1, s \backslash 1}-a_{1 r}$$
Here $a_{1, s \backslash 1}$ is the adhesion of $n_{1}$ with its complement in group $s$ and $a_{1 r}$ is the adhesion of $n_{1}$ with $n_{r}$. It is clear that if one moves $n_{1}$ to a previously unpopulated spin state, then $\Delta \mathcal{H}=a_{1, s \backslash 1}$. This move corresponds to dividing group $n_{s}$. Furthermore, if $n_{1}=n_{s}$, one has $\Delta \mathcal{H}=-a_{s r}$, which corresponds to joining groups $n_{s}$ and $n_{r}$. A spin configuration can only be a local minimum of the Hamiltonian if a move of this type does not lead to a lower energy. It is clear that some moves may not change the energy and are hence called neutral moves. In cases of equality $a_{1, s \backslash 1}=a_{1, r}$ and $n_{r}$ being a community itself, communities $n_{s}$ and $n_{r}$ are said to have an overlap of the nodes in $n_{1}$.

## 统计代写|复杂网络代写complex networks代考|Simple Divisive and Agglomerative Approaches

The equivalence of modularity with a spin glass energy shows that the problem of maximizing modularity falls into the class of NP-hard optimization problems [3]. For these problems, it is believed that no algorithm exists that is able to produce an optimal solution in a time that grows only polynomial with the size of the problem instance. However, heuristics such as simulated annealing exist, which are able to find possibly very good solutions. In this section, we will discuss an often used approach to clustering, namely hierarchical agglomerative and divisive algorithms and investigate whether they too are good heuristics for finding partitions of maximum modularity.

A number of community detection algorithms presented in Chap. 2 have followed recursive approaches and lead to hierarchical community structures. Hierarchical clustering techniques can be dichotomized into divisive

and agglomerative approaches [4]. It will be shown how a simple recursive divisive approach and an agglomerative approach may be implemented and where they fail.

In the present framework, a hierarchical divisive algorithm would mean to construct the ground state of the q-state Potts model by recursively partitioning the network into two parts according to the ground state of a 2-state Potts or Ising system. This procedure would be computationally simple and result directly in a hierarchy of clusters due to the recursion of the procedure on the parts until the total energy cannot be lowered anymore. Such a procedure would be justified, if the ground state of the q-state Potts Hamiltonian and the repeated application of the Ising system cut the network along the same edges. Let us derive a condition under which this could be ensured.

In order for this recursive approach to work, one must ensure that the ground state of the 2-state Hamiltonian never cuts though a community as defined by the q-state Hamiltonian. Assume a network made of three communities $n_{1}, n_{2}$ and $n_{3}$ as defined by the ground state of the q-state Hamiltonian. For the bi-partitioning, one now has two possible scenarios. Without loss of generality, the cut is made either between $n_{2}$ and $n_{1}+n_{3}$ or between $n_{1}, n_{2}$ and $n_{3}=n_{a}+n_{b}$, parting the network into $n_{1}+n_{a}$ and $n_{2}+n_{b}$. Since the former situation should be energetically lower for the recursive algorithm to work, one arrives at the condition that
$$m_{a b}-\left[m_{a b}\right]{p{i j}}+m_{1 b}-\left[m_{1 b}\right]{p{i j}}>m_{2 b}-\left[m_{2 b}\right]{p{i j}}$$

## 统计代写|复杂网络代写complex networks代考|Optimizing the Quality Function

p\left(\sigma_{i}=\alpha\right)=\frac{\exp \left(-\beta \mathcal{H}\left(\left{\sigma_{j \neq i}, \sigma_{ i}=\alpha\right}\right)\right)}{\sum_{s=1}^{q} \exp \left(-\beta \mathcal{H}\left(\left{\sigma_{j \neq i}, \sigma_{i}=s\right}\right)\right)}p\left(\sigma_{i}=\alpha\right)=\frac{\exp \left(-\beta \mathcal{H}\left(\left{\sigma_{j \neq i}, \sigma_{ i}=\alpha\right}\right)\right)}{\sum_{s=1}^{q} \exp \left(-\beta \mathcal{H}\left(\left{\sigma_{j \neq i}, \sigma_{i}=s\right}\right)\right)}

ΔH(σ一世=φ→一种)=∑s(乙φs−乙一种s)(ķ一世→s出去 −C[ķ一世→s出去 ]) +∑r(乙rφ−乙r一种)(ķr→一世一世n−C[ķr→一世一世n]) =∑s(乙φs−乙一种s)一种一世s+∑r(乙rφ−乙r一种)一种r一世

ΔH=一种1,s∖1−一种1r

## 广义线性模型代考

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## MATLAB代写

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