Centrality in Networks: Finding the Most Important Nodes

Group centralization of network indices

Importance evaluation of power network nodes based on community division and characteristics of coupled network

Accurate measurement of important nodes in complex networks has great practical and theoretical significance. Mining important nodes should not only consider the core nodes, but also take into account the locations of the nodes in the network. Despite much research on assessing important nodes, the importance of nodes in the structural holes is still easily ignored. Therefore, a local measuring method is proposed, which evaluates the nodes importance by the total constraints caused by the lack of primary structural holes and secondary structural holes around the nodes. This method simultaneously considers both the centrality and the bridging property of the nodes' first-order and second-order neighbours. Further to prove the accuracy of the method TCM, we carry out deliberate attack simulations through selective deletion in a certain proportion of network nodes. Then, we calculate the decreased ratio of network efficiency in the before-and-after attacks. Experiment results show that the average effect of the TCM in four real networks are improved by 50.64% and 14.92% compared to the clustering coefficient index and the k-shell decomposition method, respectively. Obviously, the TCM is more accurate to mine important nodes than other two methods, and it is suitable for quantitative analysis in large-scale networks.

The Middle Route of South-to-North Water Diversion Project (MRSNWD) is the main skeleton of China's National Water Network, its construction has changed the structure of the original water network, and analyzing the topological change of the water network in context with MRSNWD is significant for water network planning and management. In this study, the overall network characteristics of the water network in 2010 and 2020 were analyzed based on the small-world and scale-free characteristics of complex network theory. The topological changes of the water network from a node perspective were examined using three network centrality indexes: degree centrality (DC), closeness centrality (CC), and betweenness centrality (BC), while assessing the important nodes of the water network and recognizing functional areas of cold-hot spots. The results show that the water network's centrality in the study area improved after the project construction, with the average degree of the water network increasing from 2.39 to 2.42 and the average path length decreasing from 111.81 to 97.08. The propagation efficiency and network stability also increased, with a rise in important node proportion from 9.8 to 14.4%. The nodes in the DC hotspot zone along the project route have increased by 1.5%, implying an increase in the connectivity of the water network, while MRSNWD optimizes its north-south hub propagation path. "Small-worldness" indicates that most nodes of a network can be accessed and connected over shorter paths. The water network has a significant "small-worldness" and has been enhanced by the MRSNWD's construction. Approximating the water network as a scale-free network can impact its security by identifying critical nodes. The results of this research can provide the necessary technical support and reference significance for China's National Water Network.

Influential nodes refer to the ability of a node to spread information in complex networks. Identifying influential nodes is an important problem in complex networks which plays a key role in many applications such as rumor controlling, virus spreading, viral market advertising, research paper views, and citations. Basic measures like degree centrality, betweenness centrality, closeness centrality are identifying influential nodes but they are incapable of largescale networks due to time complexity issues. Chen et al. [1] proposed semi-local centrality, which is reducing computation complexity and finding influential nodes in the network. Recently Yang et al. 2020 [2] proposed a novel centrality measure based on degree and clustering coefficient for identifying the influential nodes. Sanjay et al. 2020 [3] gave voterank and neighborhood coreness-based algorithms for finding the influenced nodes in the network. Zhiwei et al. 2019 [4] considered the average shortest path to discover the influenced node in the network. These are the few recent local,global and mixed centralities. In this paper, we show a broad view of recent methods for finding influential nodes in complex networks. It also analyzes the new challenges and limitations for a better understanding of each method in detail. The experimental results based on these methods show better performance compared with existing basic centrality measures.

The identification of critical nodes in the network is of great significance for the research of the invulnerability and structural stability of complex networks. Most of the existing studies evaluate the importance of nodes in networks from a single structural perspective. Considering the local and global characteristics of the network, an identification algorithm of critical nodes in complex networks based on comprehensive topological importance (CTI) is proposed. The algorithm uses the theory of multiple attribute decision making (MADM) to quantitatively analyze the influence of different attributes on nodes and determine the comprehensive topology importance of nodes, which represents their structural importance in complex networks. Based on two simulation networks and five classical real networks, the proposed algorithm is proved to be superior to DC, LLS, P and WL indexes based on local information and BC indexes based on global information through the change of network efficiency under critical nodes attacking.

Evaluating the importance of nodes in complex networks is an important topic in the research of network characteristics. Its relevant research has a wide range of applications, such as network supervision and rumor control. At present, many methods have been proposed to evaluate the importance of nodes in complex networks, but most of them have the deficiency of one-sided evaluation or too high time complexity. In order to break through the limitations of existing methods, in this paper a novel method of evaluating the importance of complex network nodes is proposed based on Tsallis entropy. This method takes into account both the local and global topological information of the node. It considers the structural hole characteristics and K-shell centrality of the node and fully takes into account the influence of the node itself and its neighboring nodes. To illustrate the effectiveness and applicability of this method, eight real networks are selected from different fields and five existing methods of evaluating node importance are used as comparison methods. On this basis, the monotonicity index, SIR (susceptible-infectious-recovered) model, and Kendall correlation coefficient are used to illustrate the superiority of this method and the relationship among different methods. Experimental results show that this method can effectively and accurately evaluate the importance of nodes in complex networks, distinguish the importance of different nodes significantly, and can show good accuracy of evaluating the node importance under different proportions of nodes. In addition, the time complexity of this method is <inline-formula><tex-math id="M2">\begin{document}$ O({n^2}) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20210979_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20210979_M2.png"/></alternatives></inline-formula>, which is suitable for large-scale complex networks.

SummaryDetermining the centrality of nodes in complex networks provides practical benefits in many areas such as detecting influencer nodes, viral marketing, and preventing the spread of rumors. On the other hand, there is no consensus for the definition of centrality. Therefore, different centrality measures such as degree, closeness, and betweenness have been developed to measure the centrality of a node. However, each centrality measure highlights the various characteristics of the nodes in the network from its own point of view. This causes each centrality measure to rank the nodes in a different order. In recent years, researchers have focused on approaches that combine multiple centrality measures. Thus, the perspectives of different centrality measures can be considered simultaneously. In this study, we have proposed a fast and efficient method using the analytic hierarchy process and entropy weighting to combine multiple centrality measures. We tested the proposed method with synthetic and real datasets and compared the results with those of state‐of‐the‐art methods. The experimental results showed the proposed method to be competitive with these advanced methods, whereas it performed much better than the other methods in terms of computational speed. This indicated that our proposed method could be applied to large and dynamic complex networks.

It is well known that heterogeneous networks are vulnerable to the intentional removal of a small fraction of highly connected or loaded nodes, implying that to protect the network effectively, the important nodes should be allocated more defense resource than the others. However, if too much resource is allocated to the few important nodes, the numerous less-important nodes will be less protected, which if attacked together can still lead to devastating damage. A natural question is therefore how to efficiently distribute the limited defense resource among the network nodes such that the network damage is minimized against any attack strategy. In this paper, taking into account the factor of attack cost, the problem of network security is reconsidered in terms of efficient network defense against cost-based attacks. The results show that, for a general complex network, there exists an optimal distribution of the defense resource with which the network is best protected from cost-based attacks. Furthermore, it is found that the configuration of the optimal defense is dependent on the network parameters. Specifically, networks of larger size, sparser connection and more heterogeneous structure will more likely benefit from the defense optimization.

There is currently much interest in the problem of measuring the centrality of nodes in networks/graphs; such measures have a range of applications, from social network analysis, to chemistry and biology. In this paper we propose the first measure of node centrality that takes into account the community structure of the underlying network. Our measure builds upon the recent literature on game-theoretic centralities, where solution concepts from cooperative game theory are used to reason about importance of nodes in the network. To allow for flexible modelling of community structures, we propose a generalization of the Owen value—a well-known solution concept from cooperative game theory to study games with a priori-given unions of players. As a result we obtain the first measure of centrality that accounts for both the value of an individual node's relationships within the network and the quality of the community this node belongs to.

As an important data type, the demand of network analysis and learning is increasingly prominent. A key problem of network analysis is to study how to reasonably represent the feature information in the network, that is network embedding. However, in the study of network embedding, only the first-order proximity relationship of nodes is characterized, while the second-order proximity relationship hidden in the network nodes is ignored. Therefore, we propose an algorithm, called SINE, to realize the representation learning of nodes in the network which fuses the first-order and second-order proximity of nodes in the original network. Through applying SINE to three real networks, we obtain the feature representation of the nodes in the networks and cluster the nodes based on these features. Comparing with the existing network embedding learning methods-Node2vec, Large-Scale Information Network Embedding (LINE) and Structural Deep Network Embedding (SDNE), the SINE algorithm showed better performance in clustering tasks.

As the support network of power grid, the reliability and stability of power communication network structure become more and more important with the continuous expansion of power grid scale. Identifying and protecting important nodes in the power communication network in advance is an effective method to improve the robustness of the network. Therefore, this paper proposes a comprehensive evaluation method for the importance of nodes in the power communication network based on MADM (Multiple Attribute Decision Making). This method considers three indexes: degree centrality, betweenness centrality and PageRank value. By determining the weight of each index objectively and calculating the closeness of each node scheme to the ideal scheme, the comprehensive evaluation result of the importance of the node is obtained. When determining the weight of each index, the contrast and correlation of the indexes are considered at the same time, so that the result is closer to reality. The experimental results show that the proposed algorithm is more reliable and effective than the traditional method in ranking the importance of nodes in power communication network.

In network analysis, the betweenness centrality of a node informally captures\nthe fraction of shortest paths visiting that node. The computation of the\nbetweenness centrality measure is a fundamental task in the analysis of modern\nnetworks, enabling the identification of the most central nodes in such\nnetworks. Additionally to being massive, modern networks also contain\ninformation about the time at which their events occur. Such networks are often\ncalled temporal networks. The temporal information makes the study of the\nbetweenness centrality in temporal networks (i.e., temporal betweenness\ncentrality) much more challenging than in static networks (i.e., networks\nwithout temporal information). Moreover, the exact computation of the temporal\nbetweenness centrality is often impractical on even moderately-sized networks,\ngiven its extremely high computational cost. A natural approach to reduce such\ncomputational cost is to obtain high-quality estimates of the exact values of\nthe temporal betweenness centrality. In this work we present ONBRA, the first\nsampling-based approximation algorithm for estimating the temporal betweenness\ncentrality values of the nodes in a temporal network, providing rigorous\nprobabilistic guarantees on the quality of its output. ONBRA is able to compute\nthe estimates of the temporal betweenness centrality values under two different\noptimality criteria for the shortest paths of the temporal network. In\naddition, ONBRA outputs high-quality estimates with sharp theoretical\nguarantees leveraging on the \\emph{empirical Bernstein bound}, an advanced\nconcentration inequality. Finally, our experimental evaluation shows that ONBRA\nsignificantly reduces the computational resources required by the exact\ncomputation of the temporal betweenness centrality on several real world\nnetworks, while reporting high-quality estimates with rigorous guarantees.\n

Burnout is typically viewed as an individual condition, and no link has been identified between burnout and loneliness. To investigate the association of burnout with loneliness and social network degree and centrality. A survey containing the Maslach Burnout Inventory (MBI), a 3-question loneliness scale, and a social connectivity component was sent to residents in a large urban academic medical center internal medicine residency program. The response rate was 77% (95 of 124 residents). We defined significant burnout as MBI subscores of ≥ 27 for emotional exhaustion (EE), ≥ 10 for depersonalization (DP), or both. This was met by 43 (45%), 47 (49%), and 31 (33%) out of 95 respondents, respectively. Those with significant burnout had higher loneliness scores: 5.6 versus 4.5 for EE (P = .002; OR = 1.50; 95% CI 1.15-1.95); 5.4 versus 4.6 for DP (P = .024; OR = 1.33; 95% CI 1.03-1.71); and 5.8 versus 4.6 for both EE and DP (P = .001; OR = 1.54; 95% CI 1.17-2.02). Rating a larger number of coresidents as closer connections on a 5-point Likert scale was not associated with lower burnout scores. No measures of centrality were associated with burnout scores for EE and/or DP. High personal accomplishment subscores on the MBI did correlate significantly with several measures of centrality. Burnout was associated with loneliness in a dose-dependent fashion. Greater sense of personal accomplishment was associated with greater network centrality.

Identifying influential spreaders is an important issue for capturing the dynamics of information diffusion in temporal networks. Most of the identification of influential spreaders in previous researches were focused on analysing static networks, rarely highlighted on dynamics. However, those measures which are proposed for static topologies only, unable to faithfully capture the effect of temporal variations on the importance of nodes. In this paper, a shortest temporal path algorithm is proposed for calculating the minimum time that information interaction between nodes. This algorithm can effectively find out the shortest temporal path when considering the network integrity. On the basis of this, the temporal efficiency centrality (TEC) algorithm in temporal networks is proposed, which identify influential nodes by removing each node and taking the variation of the whole network into consideration at the same time. To evaluate the effectiveness of this algorithm, we conduct the experiment on four real-world temporal networks for Susceptible-Infected-Recovered (SIR) model. By employing the imprecision and the Kendall’s au coefficient, The results show that this algorithm can effectively evaluate the importance of nodes in temporal networks.