Zachary Karate Club Network Research Articles

TÌM KIẾM CỘNG ĐỒNG MẠNG DỰA TRÊN TÍNH GÓC GIỮA HAI VÉC TƠ

Recently, the problem of community detection has attracted the attention of many scientists. Most types of networks such as computer networks, biological networks and social networks, have a community structure. Community detection helps to understand the structure and properties of that real network. There have been many algorithms with different approaches, including coordinating vertices and building appropriate distances between them. In this paper, a random walk has been used to coordinate the vertices of the graph and use the cosine of the angle between two vectors to detect network communities. The article also presents the Modularity function to evaluate graph clustering. Some experimental results on randomly generated graphs and graphs generated from the real data set Zachary's karate club network have been presented and compared with the K-means++ algorithm.

Distributed Network Embedding: A Distributed Control Approach for Undirected Networks

In this paper, a new network embedding method is proposed for an effective network representation of vertices. While the existing embedding methods have embedded a set of vertices via network representation learning in a fully centralized way, we present a novel distributed network embedding (DNE) method without representation learning. By coupling first-order proximity and second-order proximity with distance embedding constraints, a distributed and explicit embedding protocol steers a set of vertices in a vector space to preserve local and global structural information of networks. The proposed method does not require representation learning process, and is distributed, explicit, and straightforward. Therefore, we expect our approach to be fast, cost-effective, especially suitable for dynamic networks, and robust for sparse networks. Lastly, numerical experiments will briefly validate the proposed embedding method using a well-known Zachary's karate club network.

Generalized fuzzy variable precision rough sets based on bisimulations and the corresponding decision-making.

Recently, the classical rough set has been extended in many ways. However, some of them are based on binary relations which only excavate “one step” information to distinguish objects. The “one step” in the binary relation means that the ordered pair of the starting and end points of the step belongs to the relation. Faced with some complex data sets, the “one step” information may be not feasible. Motivated by the notion of bisimulation in computer science, three types of bisimulation-based generalized fuzzy variable precision rough set (BGFVPRS) models are constructed. Different from many existed rough set models which are based on binary relations, the BGFVPRS models can distinguish objects by excavating the “multi-step” information of underlying relations. The related properties and relationships of BGFVPRS models are investigated. The uncertainty measure of BGFVPRS models and the reduction of fuzzy bisimulations are also discussed. Furthermore, learning from the PROMETHEE II method and combining it with our presented BGFVPRS models, a novel multiple-attribute decision-making method is provided. This method can effectively deal with complex problems including attribute data and relational data. The flexibility and effectiveness of our decision-making method are illustrated by comparative analysis and sensitivity analysis in the Zachary karate club network.

Community Detection Research Based on Line Graph

Community structure is an important property of complex networks, in which network nodes are tightly split into different groups. In this paper, a new overlapping community detection algorithm based on line graph is proposed. Spectral analysis based on Normal Laplace Matrix of line graph is introduced. We not only make the spectral clustering to be an overlapping algorithm but also define a new similarity between edges by spectral analysis. We test our method on three-community network and Zachary karate club network. The experimental result shows that our method can detect community structure and overlapping nodes effectively. It is worth to note that our method is against undirected and unweighted graphs.

An efficient counting method for the colored triad census

The triad census is an important approach to understand local structure in network science, providing comprehensive assessments of the observed relational configurations between triples of actors in a network. However, researchers are often interested in combinations of relational and categorical nodal attributes. In this case, it is desirable to account for the label, or color, of the nodes in the triad census. In this paper, we describe an efficient algorithm for constructing the colored triad census, based, in part, on existing methods for the classic triad census. We evaluate the performance of the algorithm using empirical and simulated data for both undirected and directed graphs. The results of the simulation demonstrate that the proposed algorithm reduces computational time by approximately many-fold over the naïve approach. We also apply the colored triad census to the Zachary karate club network dataset. We simultaneously show the efficiency of the algorithm, and a way to conduct a statistical test on the census by forming a null distribution from 1000 realizations of a mixing-matrix conditioned graph and comparing the observed colored triad counts to the expected. From this, we demonstrate the method's utility in our discussion of results about homophily, heterophily, and bridging, simultaneously gained via the colored triad census. In sum, the proposed algorithm for the colored triad census brings novel utility to social network analysis in an efficient package.

Large Scale Graph Evaluation for Find Communities in Big Data

Most recently, the large scale graphs with billions of edges and nodes have become quite prevalent and significant. These graphs are intensely being used in complex networks associated with the package of big data like social networks etc. some of common challenges associated with these graphs that hinder the performance of complex systems and needs to be addressed are the huge number of nodes and edges present in the data, diverse types of edges and nodes, significant amount of important information is not available and many clustering problems are NP-hard. As a matter of demand, analyzing and processing graphs has came up as a challenging issue faced by the big data researchers across the globe. In the light of the similar context, this research paper addresses and investigates validation techniques and methodology for various existing algorithms like edge betweenness, walktrap, leading eigen vector, and infomap for the datasets like Zachary karate club network and Immunograph. This research paper also proposed an algorithm namely random walk fast greedy and compared it with existing algorithms. The implementation has been carried out on R version 3.3.2. The authors have reported and analyzed the comparative performance characteristics of these algorithms promising to contribute towards the large scale graph community for complex systems and higher analytical inferences. The results achieved while implementation depicts markable improvement as compared with the current state of art techniques.

An Efficient Counting Method for the Colored Triad Census

The triad census is an important approach to understand local structure in network science, providing comprehensive assessments of the observed relational configurations between triples of actors in a network. However, researchers are often interested in combinations of relational and categorical nodal attributes. In this case, it is desirable to account for the label, or color, of the nodes in the triad census. In this paper, we describe an efficient algorithm for constructing the colored triad census, based, in part, on existing methods for the classic triad census. We evaluate the performance of the algorithm using empirical and simulated data for both undirected and directed graphs. The results of the simulation demonstrate that the proposed algorithm reduces computational time by approximately 17,400% over the naive approach. We also apply the colored triad census to the Zachary karate club network dataset. We simultaneously show the efficiency of the algorithm, and a way to conduct a statistical test on the census by forming a null distribution from 1,000 realizations of a mixing-matrix conditioned graph and comparing the observed colored triad counts to the expected. From this, we demonstrate the method's utility in our discussion of results about homophily, heterophily, and bridging, simultaneously gained via the colored triad census. In sum, the proposed algorithm for the colored triad census brings novel utility to social network analysis in an efficient package.

Cross-disciplinary detection and analysis of network motifs.

The detection of network motifs has recently become an important part of network analysis across all disciplines. In this work, we detected and analyzed network motifs from undirected and directed networks of several different disciplines, including biological network, social network, ecological network, as well as other networks such as airlines, power grid, and co-purchase of political books networks. Our analysis revealed that undirected networks are similar at the basic three and four nodes, while the analysis of directed networks revealed the distinction between networks of different disciplines. The study showed that larger motifs contained the three-node motif as a subgraph. Topological analysis revealed that similar networks have similar small motifs, but as the motif size increases, differences arise. Pearson correlation coefficient showed strong positive relationship between some undirected networks but inverse relationship between some directed networks. The study suggests that the three-node motif is a building block of larger motifs. It also suggests that undirected networks share similar low-level structures. Moreover, similar networks share similar small motifs, but larger motifs define the unique structure of individuals. Pearson correlation coefficient suggests that protein structure networks, dolphin social network, and co-authorships in network science belong to a superfamily. In addition, yeast protein–protein interaction network, primary school contact network, Zachary’s karate club network, and co-purchase of political books network can be classified into a superfamily.

적합도 함수를 이용한 커뮤니티 통합에 필요한 추가에지수 결정 및 위치 선정 방법

본 논문에서는 네트워크 내에 존재하는 두 개의 커뮤니티 A,B(<TEX>$${\mid}A{\mid}{\geq_-}{\mid}B{\mid}$$</TEX>, <TEX>${\mid}{\cdot}{\mid}$</TEX>는 커뮤니티의 노드 개수)를 통합하는데 필요한 에지 수 및 에지 위치를 결정하는 알고리즘을 제안한다. 제안된 알고리즘은 커뮤니티 내,외부로 향하는 에지들의 개수를 이용하여 커뮤니티의 성질을 나타내는 적합도 함수를 이용하고, 큰 값을 가질수록 커뮤니티로서의 성질이 크다는 것을 의미한다. 제안된 알고리즘은 그리디 방식으로, B의 하나의 노드에 대해 해당 노드를 A로 병합할 때 커뮤니티 A의 적합도 값이 증가할 수 있는 최소에지수를 결정한다. 최소에지수가 결정된 후, 새로 추가될 에지의 위치를 결정하기 위해 노드 중앙성을 이용한 커뮤니티 연결도 지표를 정의한다. 추가 에지의 위치는 통합된 커뮤니티 연결도 지표를 최대로 만들 수 있도록 결정한다. B의 모든 노드에 대해 이러한 과정을 적용하여 두 커뮤니티를 통합한다. Zachary의 가라데클럽 네트워크를 이용하여 제안된 알고리즘의 실효성을 검증하였다. In this paper, we propose a method to decide the additional edges in order to integrate two communitites A,B(<TEX>$${\mid}A{\mid}{\geq_-}{\mid}B{\mid}$$</TEX>, <TEX>${\mid}{\cdot}{\mid}$</TEX> is the size of the set). The proposed algorithm uses a fitness function that shows the property of a community and the fitness function is defined by the number of edges which exist in the community and connect two nodes, one is in the community and the other is out of the community. The community has a strong property when the function has a large value. The proposed algorithm is a kind of greedy method and when a node of B is merged to A, the minimum number of additional edges is decided to increase the fitness function value of A. After determining the number of additional edges, we define the community connectivity measures using the node centrality to determine the edges locations. The connections of the new edges are fixed to maximize the connectivity measure of the combined community. The procedure is applied for all nodes in B to integrate A and B. The effectiveness of the proposed algorithm is shown by solving the Zachary Karate Club network.

Towards a Methodology for Validation of Centrality Measures in Complex Networks

BackgroundLiving systems are associated with Social networks — networks made up of nodes, some of which may be more important in various aspects as compared to others. While different quantitative measures labeled as “centralities” have previously been used in the network analysis community to find out influential nodes in a network, it is debatable how valid the centrality measures actually are. In other words, the research question that remains unanswered is: how exactly do these measures perform in the real world? So, as an example, if a centrality of a particular node identifies it to be important, is the node actually important?PurposeThe goal of this paper is not just to perform a traditional social network analysis but rather to evaluate different centrality measures by conducting an empirical study analyzing exactly how do network centralities correlate with data from published multidisciplinary network data sets.MethodWe take standard published network data sets while using a random network to establish a baseline. These data sets included the Zachary's Karate Club network, dolphin social network and a neural network of nematode Caenorhabditis elegans. Each of the data sets was analyzed in terms of different centrality measures and compared with existing knowledge from associated published articles to review the role of each centrality measure in the determination of influential nodes.ResultsOur empirical analysis demonstrates that in the chosen network data sets, nodes which had a high Closeness Centrality also had a high Eccentricity Centrality. Likewise high Degree Centrality also correlated closely with a high Eigenvector Centrality. Whereas Betweenness Centrality varied according to network topology and did not demonstrate any noticeable pattern. In terms of identification of key nodes, we discovered that as compared with other centrality measures, Eigenvector and Eccentricity Centralities were better able to identify important nodes.

모듈래러티를 이용한 소셜 네트워크의 커뮤니티 통합에 필요한 에지 수 결정 방법

In this paper, a method to decide the number of additional edges to integrate two communities in social network by using modularity is studied. The modularity is a measure to be used to describe the quality of the community. By adding additional edges to the communities, the quality of the communities is enhanced, To integrate two communities, we propose methods to decide the number of additional edges by calculating the modularity. Also, the conditions that the proposed method is valid is investigated in a simple test graph and the efficiency of the proposed method is approved by integrating two communities in Zachary Karate Club network.

A novel framework in complex network analysis: Considering both structure of relations and individual characteristics in closeness centrality computation

A B S T R A C T In this paper, we develop a novel framework for defining radial measures of centrality in complex networks. This framework is based on the combination of two approaches: social network analysis and traditional social science approach by considering both structure of relations and individual characteristics. It is always an important issue to detect communities in complex networks as efficiently as possible to understand both the structure and function of the networks and to interpret radial centrality measures. Therefore, we propose spectral clustering by determining the best number of communities as a prerequisite stage before finding radial measures. Based on the proposed framework, an algorithm to compute the closeness centrality in complex networks is developed. We test the proposed algorithm on Zachary's karate club network, which is considerably used as a benchmark for community detection in a network. The preliminary results indicate that the new method is efficient at detecting both good inter-cluster closeness centrality and the appropriate number of clusters.

Finding Statistically Significant Communities in Networks with Weighted Label Propagation

Various networks exist in the world today including biological, social, information, and communication networks with the Internet as the largest network of all. One salient structural feature of these networks is the formation of groups or communities of vertices that tend to be more connected to each other within the same group than to those outside. Therefore, the detection of these communities is a topic of great interest and importance in many applications and different algorithms including label propagation have been developed for such purpose. Speaker-listener label propagation algorithm (SLPA) enjoys almost linear time complexity, so desirable in dealing with large networks. As an extension of SLPA, this study presented a novel weighted label propagation algorithm (WLPA), which was tested on four real world social networks with known community structures including the famous Zachary's karate club network. Wilcoxon tests on the communities found in the karate club network by WLPA demonstrated an improved statistical significance over SLPA. Withthehelp of Wilcoxon tests again, we were able to determine the best possible formation of two communities in this network relative to the ground truth partition, which could be used as a new benchmark for assessing community detection algorithms. Finally WLPA predicted better communities than SLPA in two of the three additional real social networks, when compared to the ground truth.

Community detection by signaling on complex networks

Based on a signaling process of complex networks, a method for identification of community structure is proposed. For a network with n nodes, every node is assumed to be a system which can send, receive, and record signals. Each node is taken as the initial signal source to excite the whole network one time. Then the source node is associated with an n -dimensional vector which records the effects of the signaling process. By this process, the topological relationship of nodes on the network could be transferred into a geometrical structure of vectors in n -dimensional Euclidean space. Then the best partition of groups is determined by F statistics and the final community structure is given by the K -means clustering method. This method can detect community structure both in unweighted and weighted networks. It has been applied to ad hoc networks and some real networks such as the Zachary karate club network and football team network. The results indicate that the algorithm based on the signaling process works well.

Random field Ising model and community structure in complex networks

We propose a method to determine the community structure of a complex network. In this method the ground state problem of a ferromagnetic random field Ising model is considered on the network with the magnetic field Bs = +∞, Bt = -∞, and Bi≠s,t=0 for a node pair s and t. The ground state problem is equivalent to the so-called maximum flow problem, which can be solved exactly numerically with the help of a combinatorial optimization algorithm. The community structure is then identified from the ground state Ising spin domains for all pairs of s and t. Our method provides a criterion for the existence of the community structure, and is applicable equally well to unweighted and weighted networks. We demonstrate the performance of the method by applying it to the Barabási-Albert network, Zachary karate club network, the scientific collaboration network, and the stock price correlation network. (Ising, Potts, etc.)