### 统计代写|复杂网络代写complex networks代考| Non-hierarchical

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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|Non-hierarchical

The non-hierarchical methods approach the problem from a different perspective. In principle, they intend to calculate a full distance matrix for the nodes of the network. This can then be treated by conventional techniques.

One of the earliest approaches to community detection is due to Eriksen et al. $[41,42]$. They study a diffusion process on a network and analyze the decay of the modes of the following diffusive system with discrete time:

2622 Standard Approaches to Network Structure: Block Modeling
$$\rho_{i}(t+1)-\rho_{i}(t)=\sum_{j}\left(T_{i j}-\delta_{i j}\right) \rho_{j}(t)$$
Here $T_{i j}$ represents the adjacency matrix of the network such that $T_{i j}=1 / k_{j}$ for $A_{i j}=1$ and zero otherwise. Hence $T_{i j}$ represents the probability of a random walker to go from $j$ to $i$. The decay of a random initial configuration $\rho(t=0)$ toward the steady state is characterized by the eigenmodes of the transition matrix $T_{i j}$. The eigenvectors corresponding to the largest eigenvalues can then be used to define a distance between nodes which helps in identifying communities. To do this, the eigenvectors belonging to the largest non-trivial positive eigenvalues are plotted against each other. This diffusion approach is very similar in spirit to other algorithms based on the idea of using flow simulations for community detection as suggested by van Dongen [43] under the name of “Markov clustering” (MCL).

The method presented by Zhou [44-46] first converts the sparse adjacency matrix of the graph into a full distance matrix by calculating the average time a Brownian particle needs to move from node $i$ to $j$. Then this distance matrix is clustered using ordinary hierarchical clustering algorithms. This approach is based on the observation that a random walker has shorter traveling time between two nodes if many (short) alternative paths exist.

Another spectral approach has been taken by Muños and Donetti [47]. They work with the Laplacian matrix of the network. The Laplacian is defined as
$$L_{i j}=k_{i} \delta_{i j}-A_{i j} .$$
Otherwise, the method proposed is similar to Ref. [41]. Plotting the nontrivial eigenvectors against each other gives a low-dimensional representation of a distance measure of the network on top of which a conventional clustering procedure then needs to operate.

Though these methods are able to recover known community structures with good accuracy, they suffer from being less intuitive. Communities found can only be interpreted with respect to the particular system under study, be it a diffusive system or the eigen vectors of the Laplacian matrix. Problematic is also that there is no local variant of these methods, i.e., there is no way to find the community around a given node using spectral methods.

## 统计代写|复杂网络代写complex networks代考|Optimization Based

A different approach which is reminiscent of the parametric clustering procedures known in computer science is the idea of searching for partitions with maximum modularity $Q$ using combinatorial optimization techniques [48]. This approach has been adopted by Guimera et al. in Refs. [2, 49] or Massen et al. $[50]$ using simulated annealing [51] or Duch and Arenas using extremal optimization [52].

Though this approach will be the preferred one for the remainder of this book, a number of issues remain. For the hierarchical algorithms, a community was to be understood as whatever the algorithm outputs. Now, it is not the algorithm that defines what a community is, but the quality function, i.e., the modularity $Q$ in this case. Also, the modularity $Q$ as defined by Newman [23] is parameter free and an understanding for hierarchical and overlapping structures needs to be developed.

## 统计代写|复杂网络代写complex networks代考|Conclusion

Block structure in networks is a very common and well-studied phenomenon. The concepts of structural and regular equivalence as well as the types of blocks defined for generalized block modeling are well defined but appear too rigid to be of practical use for large and noisy data sets. Diagonal block models or modular structures have received particular attention in the literature and have developed into an almost independent concept of cohesive subgroups or communities. The comparison of many different community definitions from various fields has shown that the concept of module or community in a network is only vaguely defined. The diversity of algorithms published is only a consequence of this vague definition. None of the algorithms could be called “ideal” in the sense that it combines the features of computational efficiency, accuracy, flexibility and adaptability with regard to the network and easy interpretation of the results. More importantly, none of the above-cited publications allows an estimation to which degree the community structure found is a reality of the network or a product of the clustering process itself. The following chapters are addressing these issues and present a framework in which community detection is viewed again as a special case of a general procedure for detecting block structure in networks.

## 统计代写|复杂网络代写complex networks代考|Non-hierarchical

2622 网络结构的标准方法：块建模
ρ一世(吨+1)−ρ一世(吨)=∑j(吨一世j−d一世j)ρj(吨)

Zhou[44-46]提出的方法首先通过计算布朗粒子从节点移动所需的平均时间，将图的稀疏邻接矩阵转换为全距离矩阵一世到j. 然后使用普通的层次聚类算法对这个距离矩阵进行聚类。这种方法基于以下观察：如果存在许多（短）替代路径，则随机游走者在两个节点之间的旅行时间更短。

Muños 和 Donetti [47] 采用了另一种光谱方法。他们使用网络的拉普拉斯矩阵。拉普拉斯算子定义为

## 广义线性模型代考

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## MATLAB代写

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