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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Cohesive Subgroups or Communities as Block

The abundance of diagonal block models or modular structures makes modularity a concept so important that it is often studied outside the general framework of block modeling. One explanation may be that in social networks it may even be the dominant blocking structure. The reason may be that homophily $[16]$, i.e., the tendency to form links with agents similar to oneself, is a dominant mechanism in the genesis of social networks. Recall, however, that the concept of functional roles in networks is much wider than mere cohesiveness as it specifically focuses on the inter-dependencies between groups of nodes. Modularity or community structure, emphasizing the absence of dependencies between groups of nodes is only one special case. It may also be that the concept of modularity appeals particularly to physicists because it is reminiscent of the reductionist approach of taking systems apart into smaller subsystems that has been so successful in the natural sciences.

Nevertheless, in the literature, there is no generally accepted definition of what a community or module actually is. A variety of definitions exist that all imply that members of a community are more densely connected among themselves than to the rest of the network. Two approaches exist to tackle the problem. Either, one starts with a definition of what a community is in the first place and then searches for sets of nodes that match this definition. Or one can use a heuristic approach by designing an algorithm and define a community as whatever this algorithm outputs. Both of these approaches differ in one fundamental way: When starting from a definition of community, it often occurs that some nodes in the network will not be placed into any community. The algorithmic approaches on the other hand will generally partition the set of vertices such that all nodes are found in some community. Whether all nodes need to be assigned into a community needs to be decided by the researcher and may determine which definitions and methods are useful in the analysis of actual data. With these considerations in mind we shall briefly review the approaches taken in the literature.

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

The study of community structure has a long tradition in the field of sociology and it comes as no surprise that the example that sparked the interest of physicists in the field was a sociological one $[17,18]$. Alternatively to community, the term cohesive subgroup is often used to subsume a number of definitions

that emphasize different aspects of the problem. These can be grouped into definitions based on reachability, nodal degree or the comparison of within to outside links [11].

Cliques are complete subgraphs, such that every member is connected to every other member in the clique. An $n$-clique is a maximal subgraph, such that the geodesic distance $d(i, j)$ between any two members $i, j$ is smaller or equal to $n$. Naturally, cliques are 1-cliques. Note that the shortest path may also run through nodes not part of the n-clique, such that the diameter of an $\mathrm{n}$-clique may be larger than $n$. An $n$-clan denotes an $\mathrm{n}$-clique with diameter less or equal to $n$. Naturally, all n-clans are also n-cliques. Alternatively, an $n$-club is a maximal subgraph of diameter $n$.

These definitions are problematic in several ways. Cliques can never get larger than the smallest degree among the member nodes which limits these communities to be generally very small in large networks with limited degrees. The other definitions relying on distances are problematic if the network possesses the small world property. The overlap of such communities will generally be as large as a typical group.

Another group of definitions is based on the degree of the members of a community. A $k$-plex is a maximal subgraph of $n$ nodes, such that each member has at least $n-k$ connections to other nodes in the k-plex. This definition is less strict than that of a clique as it allows some links to be missing. At the same time, a k-plex only contains nodes with minimum degree $d \geq(n-k)$. A $k$-core denotes a maximal subgraph, such that each node has at least $k$ connections to other members of the k-core.

Here again, the size of k-plexes is limited by the degrees of the nodes. K-cores are problematic also because they disregard all nodes with degree smaller than $k$ even if they have all their connections to nodes within this core.

While the two former groups of definitions are based primarily on internal connections, a number of definitions of cohesive subgroups exist which compare intra- and inter-group connections. One example are LS sets. A set of $n$ nodes is an LS set, if each of its proper subsets has more ties to its complement than to the rest of the network.

## 统计代写|复杂网络代写complex networks代考|Definitions from Physicists

The diversity of definitions from sociology already indicates the conceptual difficulties involved and demonstrates that the question of what a community is may not have a simple answer. To make things worse, a number of alternative definitions have been and continue to be contributed by physicists as well $[19,20]$.

Radicchi et al. [21] have introduced the notion of community in a strong sense and in a weak sense. For a subgraph $V$ of $\mathcal{S}$ to be a community in the strong sense, they require
$$k_{i}^{i n}>k_{i}^{\text {out }} \quad \forall i \in V,$$
i.e., the number of internal connections $k_{i}^{i n}$ to other members of $V$ shall be larger than the number of external connections $k_{i}^{\text {out }}$ to the rest of the network. Note that $k_{i}^{\text {in }}+k_{i}^{\text {out }}=k_{i}$, the degree of node $i$. Relaxing this condition, for a subgraph $V$ to be a community in a weak sense they require

A paradoxical issue arising from both of these definitions is that communities in the strong or weak sense can be formed of disconnected subgraphs as long as these subgraphs also obey the definition. It should be noted, however, that this definition was initially proposed as a stop criterion for hierarchical agglomerative or divisive clustering algorithms.

Palla et al. $[8,22]$ have given an alternative definition based on reachability, though defined through a clique percolation process and not via paths in the network. Two $k$-cliques are adjacent if they share a (k-1)-clique, i.e., they differ by only one node. Note that the term k-cliques here denotes complete subgraphs with $k$ nodes. As a community or k-clique percolation cluster, they define the set of nodes connected by $(\mathrm{k}-1)$-cliques. An example will clarify these issues. Two vertices connected by an edge form a 2-clique. Two triangles (3-cliques) are adjacent if they share an edge, i.e., a 2-clique. This definition allows nodes to be part of more than one community and hence allows for overlap among communities much like the other definitions based on reachability.

Other approaches given by physicists and computer scientists are algorithmically motivated. The next section will discuss this treatment of the problem.

## 统计代写|复杂网络代写complex networks代考|Definitions from Physicists

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## 广义线性模型代考

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