Modeling centrality measures in social network analysis using bi-criteria network flow optimization problems

Centrality measures play an important role in the field of network analysis. In the particular case of social networks, the flow represents the way in which information passes through the network nodes. Freeman et al. (1991) were the first authors to relate centrality measures to network flow optimization problems in terms of betweenness, closeness, and the influence of one node over another one. Such measures are single dimensional and, in general, they amalgamate several heterogeneous dimensions into a single one, which is not suitable for dealing with most real-world problems. In this paper we extend the betweenness centrality measure (or concept) to take into account explicitly several dimensions (criteria). A new closeness centrality measure is defined to deal not only with the maximum flow between every ordered pair of nodes, but also with the cost associated with communications. We shall show how the classical measures can be enhanced when the problem is modeled as a bi-criteria network flow optimization problem.     

Leverage centrality analysis of infrastructure networks

Leverage centrality is a novel centrality measure proposed to identify the critical nodes that are highly influential within the network. Leverage centrality considers the extent of connectivity of a node relative to the connectivity of its neighbors. The leverage centrality of a node in a network is determined by the extent to which its direct neighbors rely on that node for information. In this paper, leverage centralities of the nodes of infrastructure networks are computed and critical nodes within the network are identified.     

Identifying sets of key players in a social network

A procedure is described for finding sets of key players in a social network. A key assumption is that the optimal selection of key players depends on what they are needed for. Accordingly, two generic goals are articulated, called KPP-POS and KPP-NEG. KPP-POS is defined as the identification of key players for the purpose of optimally diffusing something through the network by using the key players as seeds. KPP-NEG is defined as the identification of key players for the purpose of disrupting or fragmenting the network by removing the key nodes. It is found that off-the-shelf centrality measures are not optimal for solving either generic problem, and therefore new measures are presented. Stephen P. Borgatti is Professor of Organization Studies at the Carroll School of Management, Boston College. His research is focused on social networks, social cognition and knowledge management. He is also interested in the application of social network analysis to the solution of managerial problems.     

超网络中心性度量的υ-Position值方法

利用合作博弈理论的分配规则如Shapley值、Banzhaf值等来度量政治、经济和社会网络中节点的中心性或者重要性是识别网络中关键节点的一类重要方法。考虑到在超网络中代表各类组织的超边在网络中发挥的作用不同,本文研究了超网络博弈上一类广义Position值的分配规则,被称为υ-position值。它可以作为网络中度值测度的一类推广,以此来度量网络中参与者的中心性和相对重要性。其次,证明了超网络结构上类Shapley-position值可由分支超边指数和局部平衡超边贡献两个性质所唯一刻画。最后, 举例分析了υ-position值在超网络中心性测度中的应用。     

Quasi-stationary distributions as centrality measures for the giant strongly connected component of a reducible graph

A random walk can be used as a centrality measure of a directed graph. However, if the graph is reducible the random walk will be absorbed in some subset of nodes and will never visit the rest of the graph. In Google PageRank the problem was solved by the introduction of uniform random jumps with some probability. Up to the present, there is no final answer to the question about the choice of this probability. We propose to use a parameter-free centrality measure which is based on the notion of a quasi-stationary distribution. Specifically, we suggest four quasi-stationary based centrality measures, analyze them and conclude that they produce approximately the same ranking.     

基于Erdos1合著网络和论文引用网络的复杂网络节点特征研究

首先,研究了Erdos1合著网络的特征属性,一方面使用节点的度、介数、接近中心性来描述Erdos1合著网络节点重要性,另一方面使用特征向量中心性和本文提出的高阶度参数来描述Erdos1合著网络节点影响力;然后,分别用逼近理想解的排序算法（TOPSIS算法）和主成份分析（PCA）对节点重要性和影响力排序;最后,利用修改的网页排名算法（PageRank算法）讨论了网络科学原创性论文中最具影响力的论文。     

Local and consistent centrality measures in parameterized networks

We propose an axiomatic approach to characterize centrality measures for which the centrality of an agent is recursively related to the centralities of the agents she is connected to. This includes the Katz–Bonacich and the eigenvector centrality. The core of our argument hinges on the power of the consistency axiom, which relates the properties of the measure for a given network to its properties for a reduced problem. In our case, the reduced problem only keeps track of local and parsimonious information. Our axiomatic characterization highlights the conceptual similarities among those measures.     

On a matrix-based measure of the degree of coreness of a node in a network

The paper focuses on the right and left eigenvectors of a network matrix that belong to the largest eigenvalue. It is shown that each of vector entries measures the walk centrality of the corresponding node’s position in the network’s link structure and of the positions of the node’s adjacent nodes; as a result, it indicates to which degree the node can be associated with the structure’s core, i.e., the structural coreness of the node. The relationship between the vectors’ coordinates and the position of the nodes, as well as the actual computation of the coordinates, is based on an iterative computational scheme known as the power method. The paper studies the method’s convergence for networks of different structure. Some possible applications are discussed. The paper also includes a numerical example dealing with a real network of 197 nodes and 780 links.     

The economic effect of interlocking directorates in Italy: new evidence using centrality measures

We use measures of vertex centrality to examine interlocking directorates and their economic effects in Italy. We employ centrality measures like degree, eigenvector centrality, betweenness, and flow betweenness, along with the clustering coefficient. We document the existence of a negative relationship between both degree and eigenvector centrality and firm value. Betweenness and flow betweenness, on the other hand, are not associated with lower firm valuations. We argue that these differences derive from the different properties of these measures: while degree and eigenvector centrality measures the influence and the power of the connections, betweenness and flow betweenness are proxies for the volume of information that passes between the nodes. This result is robust with respect to the use of both stock market and operating performance measures, as well as several controlling variables.     

Application of modal analysis in assessing attack vulnerability of complex networks

In this paper we propose an alternative way to study robustness and vulnerability of complex networks, applying a modal analysis. The modal weights of the network nodes are considered as a measure for their busyness, which is further used for preferential removal of nodes and attack simulation. Analyses of the attack vulnerability are carried out for several generic graphs, generated according to ER and BA algorithms, as well as for some examples of manmade networks. It was found that a modal weight based attack causes significant disintegration of manmade networks by removing a small fraction of the busiest nodes, comparable to the one based on the node degree and betweenness centrality.     

Key potential-oriented criticality analysis for complex military organization based on FINC-E model

The complex social organizations, which can self-organize into the region “at the edge of chaos”, neither too ordered nor too random, now have become an interdisciplinary research topic. As a kind of special social organization, the complex military organization usually has its key entities and relations, which should be well protected in case of attacks. In order to do the criticality analysis for the military organization, finding the key entities or relations which can disrupt the functions of the organization, two problems should be seriously considered. First, the military organization should be well modeled, which can work well in the specialized military context; secondly it is critical to define and identify the key entities or relations, which should incorporate the topological centrality and weighted nodes or edges. Different from the traditional military organizations which are usually task-oriented, this paper proposes the Force, Intelligence, Networking, and C2 Extended (FINC-E) Model for complex military organization, with which a more detailed and quantitative analysis for the military organization is available. This model provides the formal representation for the nodes and edges in the military organization, which provides a highly efficient and concise network topology. In order to identify the critical nodes and edges, a method based on key potential is proposed, which acts as the measurement of criticality for the heterogeneous nodes and edges in the complex military organization. The key potential is well defined on the basis of topology structure and of the node’s or edge’s capability, which helps to transform the organization from the heterogeneity to the homogeneity. In the end, the criticality analysis case study is made for both small-world networked military organization and scale-free networked military organization, showing that the measure of key potential has the advantage over other classical measures in locating the key entities or relations for complex military organization.     

A new centrality measure of nodes in multilayer networks under the framework of tensor computation

One challenging issue in information science, biological systems and many other field is to determine the most central agents in multilayer networked systems characterized by different types of interrelationships. In this paper, using a fourth-order tensor to represent multilayer networks, we propose a new centrality measure, referred to as the Singular Vector of Tensor (SVT) centrality, which is used to quantitatively evaluate the importance of nodes connected by different types of links in multilayer networks. First, we present a novel iterative method to obtain four alternative metrics that can quantify the hub and authority scores of nodes and layers in multilayer networked systems. Moreover, we use the theory of multilinear algebra to prove that the four metrics converge to four singular vectors of the adjacency tensor of the multilayer network under reasonable conditions. Furthermore, a novel SVT centrality measure is obtained by integrating these four metrics. The experimental results demonstrate that the proposed method is a new centrality measure that significantly outperforms six other published centrality methods on two real-world multilayer networks related to complex diseases, i.e., gastric and colon cancers.     

国内外黄金现货价格联动波动复杂网络拓扑性质

为了研究国内外黄金现货价格联动性波动特征,选取2003年1月1日至2013年9月6日上海黄金交易所Au9999黄金现货收盘价与伦敦标准黄金现货的下午定盘价分别作为国内与国外黄金现货价格的样本数据.依据粗粒化方法,将Au9999与伦敦标准金价格的联动波动转化为由5个{P,O,M}三个字符构成的字符串,每一个字符串代表每5天的黄金现货价格联动波动模态,共产生106个模态.将模态作为节点,模态之间的转化为边,构建国内外黄金现货价格联动性波动复杂网络.对联动性复杂网络的点强度与强度分布、聚集系数、平均路径长度、中介中心性及凝集子群特征进行研究分析.结果表明:点强度值较大的前32个节点累积强度分布达到了92.05%,点强度与度分布、点强度与其等级均呈幂律分布;加权集聚系数与点强度之间并没有表现出良好的相关性,网络中存在14个小群簇;网络平均最短路径长度为7.668;节点中介性差异不太明显,前40个节点对整个网络的中介中心性贡献率为62.29%;8个节点的凝集子群有2个,8个节点以上的子群不存在.从网络结构拓扑性质角度验证了国内外黄金现货价格变化的复杂特征,这些对于掌握国内外黄金价格波动的内在规律和了解价格变化信息有一定指导意义,能够为我国黄金价格制定、风险投资和规避经济风险提供决策参考.     

A network-based approach for increasing discrimination in data envelopment analysis

Data envelopment analysis (DEA) is known to produce more than one efficient decision-making unit (DMU). This paper proposes a network-based approach for further increasing discrimination among these efficient DMUs. The approach treats the system under study as a directed and weighted network in which nodes represent DMUs and the direction and strength of the links represent the relative relationship among DMUs. In constructing the network, the observed node is set to point to its referent DMUs as suggested by DEA. The corresponding lambda values for these referent DMUs are taken as the strength of the network link. The network is weaved by not only the full input/output model, but also by models of all possible input/output combinations. Incorporating these models into the system basically introduces the merits of each DMU under various situations into the system and thus provides the key information for further discrimination. Once the network is constructed, the centrality concept commonly used in social network analysis—specifically, eigenvector centrality—is employed to rank the efficient DMUs. The network-based approach tends to rank high the DMUs that are not specialized and have balanced strengths.     

Combining evolutionary game theory and network theory to analyze human cooperation patterns

As natural systems continuously evolve, the human cooperation dilemma represents an increasingly more challenging question. Humans cooperate in natural and social systems, but how it happens and what are the mechanisms which rule the emergence of cooperation, represent an open and fascinating issue. In this work, we investigate the evolution of cooperation through the analysis of the evolutionary dynamics of behaviours within the social network, where nodes can choose to cooperate or defect following the classical social dilemmas represented by Prisoner’s Dilemma and Snowdrift games. To this aim, we introduce a sociological concept and statistical estimator, “Critical Mass”, to detect the minimum initial seed of cooperators able to trigger the diffusion process, and the centrality measure to select within the social network. Selecting different spatial configurations of the Critical Mass nodes, we highlight how the emergence of cooperation can be influenced by this spatial choice of the initial core in the network. Moreover, we target to shed light how the concept of homophily, a social shaping factor for which “birds of a feather flock together”, can affect the evolutionary process. Our findings show that homophily allows speeding up the diffusion process and make quicker the convergence towards human cooperation, while centrality measure and thus the Critical Mass selection, play a key role in the evolution showing how the spatial configurations can create some hidden patterns, partially counterbalancing the impact of homophily.     

基于复杂网络的中国股市房地产板块股票强相关性研究

为分析中国股市房地产板块股票的强相关特性,以101只股票为结点,以近17年股票对数回报的相关系数为加权边,建立复杂网络模型,通过对网络拓扑参数计算,发现该网络为无尺度网络,结点度分布P(s)~s-|s|δ,在不同相关系数阈值下,δ值介于0.8~1.6之间.网络平均集聚系数为0.53.文章也对网络中心性进行测量和子群划分,发现代码为000592和601588的结点在网络中具有很高的中介性,网络中大部分结点都受其影响较大.     

Ranking hubs and authorities using matrix functions

The notions of subgraph centrality and communicability, based on the exponential of the adjacency matrix of the underlying graph, have been effectively used in the analysis of undirected networks. In this paper we propose an extension of these measures to directed networks, and we apply them to the problem of ranking hubs and authorities. The extension is achieved by bipartization, i.e., the directed network is mapped onto a bipartite undirected network with twice as many nodes in order to obtain a network with a symmetric adjacency matrix. We explicitly determine the exponential of this adjacency matrix in terms of the adjacency matrix of the original, directed network, and we give an interpretation of centrality and communicability in this new context, leading to a technique for ranking hubs and authorities. The matrix exponential method for computing hubs and authorities is compared to the well known HITS algorithm, both on small artificial examples and on more realistic real-world networks. A few other ranking algorithms are also discussed and compared with our technique. The use of Gaussian quadrature rules for calculating hub and authority scores is discussed.     

An investigation of synchrony in transport networks

The cumulative degree distributions of transport networks, such as air transportation networks and respiratory neuronal networks, follow power laws. The significance of power laws with respect to other network performance measures, such as throughput and synchronization, remains an open question. Evolving methods for the analysis and design of air transportation networks must be able to address network performance in the face of increasing demands and the need to contain and control local network disturbances, such as congestion. Toward this end, we investigate functional relationships that govern the performance of transport networks; for example, the links between the first nontrivial eigenvalue, λ₂, of a network's Laplacian matrix—a quantitative measure of network synchronizability—and other global network parameters. In particular, among networks with a fixed degree distribution and fixed network assortativity (a measure of a network's preference to attach nodes based on a similarity or difference), those with small λ₂ are shown to be poor synchronizers, to have much longer shortest paths and to have greater clustering in comparison to those with large λ₂. A simulation of a respiratory network adds data to our investigation. This study is a beginning step in developing metrics and design variables for the analysis and active design of air transport networks. © 2008 Wiley Periodicals, Inc. Complexity, 2009