# 统计代写|网络分析代写Network Analysis代考|Graph community and discovery

## 统计代写|网络分析代写Network Analysis代考|Graph community and discovery

Given a real graph, the distribution of edges, as noted in $[18,21]$, is quite inhomogeneous, both globally and locally. This causes the insurgence of regions with a high concentration of edges, i.e., subsets of nodes with a high number of edges among them, and a low number of edges outside. These regions are often referred to as communities, (or clusters or modules), and the property of graphs that have these regions is called a community structure. The relevance of finding communities in a graph is high with applications in many fields. There exist many algorithms for community detection in graphs, and the complete enumeration of them is beyond the scope of this book. Here, we define the main characteristics of a community, then we present some approaches, whereas the interested reader may find more information in [18, $21]$.

As outlined by Fortunato in [18], there is no universal definition of community, and the existing ones are often related to the specific application or context. Trivially, a community in a graph should have two main properties: the number of edges connecting its nodes should be high, whereas the number of the edges connecting its edges to the remaining ones should be low.

More specifically, a community is popularly defined as a cluster having significant intra-community connectivity in comparison to inter-community connectivity in the network. Formally, we can define a network community as follows:

Definition 3.6.1 (Network community). Given a network $\mathcal{G}=$ $(\mathcal{V}, \mathcal{E})$, a network community $\mathcal{C}_i=\left(\mathcal{V}^{\prime}, \mathcal{E}^{\prime}\right)$ is a densely connected subgraph of $\mathcal{G}\left(\mathcal{C}_i \subseteq \mathcal{G}\right)$, where interconnectivity of $\mathcal{V}^{\prime}$ with respect to $\mathcal{E}^{\prime} \subseteq \mathcal{E}$ is higher in comparison to the rest of $\mathcal{V}$, i.e., $\mathcal{V}-\mathcal{V}^{\prime}$.
Consequently, a community should have three desired properties:

• the intra-community density is significantly larger than the average edge density of the graph;
• the inter-community density is significantly lower than the average edge density of the graph;
• $\mathcal{C}_i$ should be connected, i.e., there must be a path between each pair of its vertices, running only through vertices of $\mathcal{C}_i$.
Mostly, communities are treated as exclusive, also called disjoint community. An exclusive community is a subgraph, such that none of its vertices belongs to any other communities. In other words, a node belonging to a community cannot be a member of any other communities simultaneously.

## 统计代写|网络分析代写Network Analysis代考|Few community detection methods

Infomap [32] is a popular community detection method for social networks that employ random walks in the connected network to maximize the amount of information flow on the decomposed subnetworks for community detection. iDBLINK [26] is an incremental density-based community detection algorithm [33] for dynamic network, where the structure of the communities changes over time. Based on the changes in the connections between nodes over time, it immediately updates the corresponding community structures. Inspired from fuzzy granular social networks (FGSN) [20] and Louvain algorithm [5], Nicole et. al. [13] proposed a method for detecting communities in a social network. The molecular complex detection algorithm (MCODE), ${ }^1$ described in the early work of Bader et al., [30], takes in input and interaction networks and tries to find complexes by building clusters. The rationale of MCODE is the separation of dense regions based on an ad hoc defined local density. MCODE has three main stages: (i) node weighting, (ii) complexes prediction, and (iii) postprocessing.

Overlapping cluster generator (OCG) [2] creates a hierarchy of overlapping clusters according to an extension of Newman’s modularity function. Preferential learning and label propagation algorithm, called PLPA [35] is proposed to detect overlapping communities based on learning behavior and information interaction in social networks. COPRA [19] optimizes and expands seed community iteratively, based on the clustering coefficient of each node by taking the average clustering coefficient of neighboring nodes. LPANNI [23], is a label propagation-based method to detect overlapping communities. It quantifies the influence of neighbor nodes by adopting node importance and affinity between node pairs to improve the label update strategy. SLPAD [1] follows label propagation model, such as SLPA [37], for detecting overlapping communities in dynamic networks. AFOCS [28] is an adaptive framework for the detection of overlapping communities as well as tracking the evolution of overlapping communities in incremental networks. TILES [31] extracts overlapping communities in dynamic social networks using peripheral membership and core membership and reevaluate membership of nodes to communities for each new interaction. OSLOM [22] uses local optimization of a fitness function to detect overlapping communities in incremental networks. An extended adaptive density peaks clustering (EADP) [38] is proposed for overlapping community detection based on a novel distance function. EADP adaptively choose cluster centres by employing a linear fitting-based strategy. MCL (Markov clustering algorithm) [14] is an iterative algorithm that simulates random walks using Markov chains. A possible way to define a module within a network is as a collection of nodes that are more connected with each other than to the others. It follows that a random walk starting in any of these nodes is more likely to stay within the cluster rather than to travel between clusters. The simulation is performed by iteratively applying two main operations, usually referred to as expansion and inflation.

Next, we demonstrate the use of R for the implementation of various graph analysis methods.

# 网络分析代考

## 统计代写|网络分析代写Network Analysis代考|Graph community and discovery

• 社区内密度明显大于图的平均边密度；
• 社区间密度明显低于图的平均边缘密度；
• $\mathcal{C}_i$ 应该是连通的，即它的每对顶点之间必须有一条路径，只通过 $\mathcal{C}_i$. 大多数情况下，社区被视为排他性的，也称为不相交的社区。排他社区是一个子 图，因此它的顶点都不属于任何其他社区。换句话说，属于一个社区的节点不能同 时是任何其他社区的成员。

## 统计代写|网络分析代写Network Analysis代考|Few community detection methods

Infomap [32] 是一种流行的社交网络社区检测方法，它在连接的网络中采用随机游走，以最大化用于社区检测的分解子网络上的信息流量。iDBLINK [26] 是一种用于动态网络的基于增量密度的社区检测算法 [33]，其中社区的结构随时间变化。根据节点间连接随时间的变化，立即更新相应的社区结构。受模糊粒度社交网络 (FGSN) [20] 和 Louvain 算法 [5] 的启发，Nicole 等人。阿尔。[13] 提出了一种在社交网络中检测社区的方法。分子复合物检测算法（MCODE），1在 Bader 等人 [30] 的早期工作中有所描述，它接受输入和交互网络并尝试通过构建集群来寻找复合体。MCODE 的基本原理是基于临时定义的局部密度来分离密集区域。MCODE 具有三个主要阶段：(i) 节点加权，(ii) 复合物预测，以及 (iii) 后处理。

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