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Some Unique Properties of Eigenvector Centrality

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Abstract

Eigenvectors, and the related centrality measure Bonacich's c(β), have advantages over graph-theoretic measures like degree, betweenness, and closeness centrality: they can be used in signed and valued graphs and the beta parameter in c(β) permits the calculation of power measures for a wider variety of types of exchange. Degree, betweenness, and closeness centralities are defined only for classically simple graphs—those with strictly binary relations between vertices. Looking only at these classical graphs, where eigenvectors and graph–theoretic measures are competitors, eigenvector centrality is designed to be distinctively different from mere degree centrality when there are some high degree positions connected to many low degree others or some low degree positions are connected to a few high degree others. Therefore, it will not be distinctively different from degree when positions are all equal in degree (regular graphs) or in core-periphery structures in which high degree positions tend to be connected to each other.

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... Nodes with high eigenvector centrality are connected to other well-connected nodes, creating a network of interconnected critical points. According to Bonacich (2007) eigenvector centrality of node is given by [77]: ...
... Nodes with high eigenvector centrality are connected to other well-connected nodes, creating a network of interconnected critical points. According to Bonacich (2007) eigenvector centrality of node is given by [77]: ...
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... Eigenvector centrality [15] evaluates a node's importance based on the importance of its neighbors, in contrast to degree centrality, which only considers the number of direct connections. As a result, eigenvector centrality provides a more comprehensive assessment of node significance in a network [16], incorporating the influence of well-connected nodes with high centrality [17]. ...
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... In general, there are two main methods for cyberspace data sparsification. The first method involves ranking and selecting nodes based on their importance or attributes, such as degree centrality [45], betweenness centrality [46], closeness centrality [47], or eigenvector centrality [48]. The other method involves ranking and selecting edges based on their weights or attributes. ...
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... We gathered attachment network data following a similar approach to that in Berán, Pléh, Soltész, Rácz, Kardos, Czobor, and Unoka [41], which asked participants to name 5-8 members of their core support network and to rate each of these alters along several dimensions, such as con ict, trust, and shared values. We then also asked them to rate how close each alter was to the others, generating an alter-alter network that was used to calculate eigenvector centrality, which estimates centrality both on the nodes' own place in the network in addition to how well-connected their neighbors are [42]. After review, three participants were excluded from the analyses involving this support network due to illogical responses in the name generator (listing plurals like "my friends," etc.). ...
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... Degree is computed as the row sum of the binary matrix, B, with entries b ij = 1 is node i and node j are connected and b ij = 0 otherwise. Similarly, eigenvector centrality is a measure of influence of a node on the network and provides a value for each node based on the idea that a highly central node is connected to other central nodes even if the node does not have many direct connections [16]. Finally, closeness centrality is a metric that provides information of how close a node is to the other nodes in the network [17]. ...
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... Entities with high betweenness centrality act as bridges and control information flow across the network. Eigenvector centrality measures an entity's influence not just based on the number of connections but also on their centrality scores [50][51][52]. This means that if an entity is connected to other highly central entities, its own influence is correspondingly enhanced. ...
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... A network can be represented as a matrix. Laplacian and adjacency matrixes are important tools capable of showing the topology of a complex network; therefore, eigenvalues and eigenvectors are proposed to describe the network dynamics [8,9]. For example, Zheng et al. found that the threshold value of time delay is proportional to the minimum eigenvalue of the network matrix when studying the impact of both network and time delays on Turing instability [10]. ...
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Network science is a powerful tool for understanding the complex interactions between individuals and is widely used to study the spread of infectious diseases. Men who have sex with men (MSM) have a high risk of HIV transmission, and sex-role preference is an essential element of HIV spread. Considering the preferences of MSM groups and the effective connections with actual transmission rates, this study established a random network (symmetric degree distribution) and a scale-free network (asymmetric degree distribution), respectively. The matrix centrality theory and computer numerical simulation are combined to analyze HIV transmission patterns in MSM groups. The results indicate that the stochasticity in the asymmetric degree distribution network is higher than in the symmetric degree distribution network. Degree and eigenvector centrality are similar in asymmetric or symmetric degree distribution networks. The centrality eigenvector can reflect more information because it includes both the node’s degree and its connections’ degrees. However, when many individuals are infected, the degree of centrality may directly come into play.
... Therefore, it considers the entire connectivity pattern of the network. It has been shown to be a more adequate index to measure the entire connectivity of a network than other measurements like degree centrality or betweenness centrality (Bienenstock & Bonacich, 2021;Bonacich, 2007;Lohmann et al., 2010). ...
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Will our brains get to know a new face better if we look at its external features first? Here we offer neurophysiological evidence of the relevance of external versus internal facial features for constructing new face representations, by contrasting successful face processing with a prototypical case of face agnosia. A woman with acquired prosopagnosia (E.C.) and 14 age-matched typical participants (7 women) were exposed to a face-feature matching task. External (E), internal (I) features, and whole target faces of unknown individuals (from an IdentiKit gallery) were displayed according to two different sequences: E → I → whole faces, or I → E → whole faces. Then, we studied the induced EEG activity using ‘isolated effective coherence’ to analyse the intracortical causal information flow among face-sensitive nodes. Initial presentation of external features (E before I), when compared to internal ones, triggered connections encompassing extensively the right-hemisphere face processing pathway [from posterior visual cortices for initial structural analysis, towards both intermediate (occipitotemporal) and high-level (prefrontal) relay stations], in which face-identity is thought to emerge progressively. Also, whereas exposure to internal features as second stimulus seemed to demand some sort of basic visual processing, external features triggered again more widespread and integrative connections. Connections for whole faces closing the E-I sequence resembled those for external features initiating the same sequence. Meanwhile, the predominant connections for whole faces completing the I-E sequence were more restricted to specific brain areas, with relevant prefrontal activity and a few connected nodes in right posterior regions, suggesting high attentional load plus initial and intermediate processing of face identity. Interestingly, the pattern of connections that distinguished typical participants from E.C. in the I-E sequence was the recruitment of left posterior visual regions, presumably underlying analytical subroutines for structural encoding of facial stimuli. These findings support that initial exposure to external features, followed by internal ones, provides the best visual cue to acquire new face configurations. Nevertheless, in case of face agnosia after right posterior damage, relying preferentially on internal features and left hemisphere specialized subroutines might be an alternative for cognitive training.
... Eigenvector centrality values take the entire network into account, such that nodes may obtain a high centrality by being connected to many low-centrality nodes or by being connected to a smaller number of high-centrality nodes (Bonacich, 2007). We extracted undirected eigenvector centrality, which is a combined measure of the individual's direct and indirect associations. ...
... While BioGRID and iRefIndex displayed clear linear correlations between these two measures, there are some genes in ReactomeFI and STRING that have low degrees but high eigenvector centrality. By the definition of eigenvector centrality, these low-degree genes are typically connected to some high-degree genes [40]. This connection suggests that, although indirect, their positions in a PPI network are significantly inf luenced by the degree bias. ...
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Network-based methods utilize protein–protein interaction information to identify significantly perturbed subnetworks in cancer and to propose key molecular pathways. Numerous methods have been developed, but to date, a rigorous benchmark analysis to compare the performance of existing approaches is lacking. In this paper, we proposed a novel benchmarking framework using synthetic data and conducted a comprehensive analysis to investigate the ability of existing methods to detect target genes and subnetworks and to control false positives, and how they perform in the presence of topological biases at both gene and subnetwork levels. Our analysis revealed insights into algorithmic performance that were previously unattainable. Based on the results of the benchmark study, we presented a practical guide for users on how to select appropriate detection methods and protein–protein interaction networks for cancer pathway identification, and provided suggestions for future algorithm development.
... The performance of ISCG-IM was benchmarked against a wide range of adaptive centrality-based methods, including degree(k), k-core [41], betweenness [48], closeness [49], eigenvector [50], Katz [51], subgraph [52], collective influence [47], and non-backtracking centrality [53]. These methods iteratively recalculate node importance after each selection, dynamically adapting to structural changes in the network. ...
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... Closeness centrality indicates the proximity of a node to all other nodes, thereby giving insights into its potential to inf luence them efficiently [57,58]. Eigenvector centrality takes the connectivity of the associated nodes into account, indicating that a taxon plays a significant role in the overall community by being part of an important subnetwork [59,60]. Transitivity, also known as clustering coefficient, quantifies the clustering of nodes in a network by measuring the probability that the neighbors of a node are connected and may give indications about niche specialization [57]. ...
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... For data analysis, SNA was used to create a network that related all sequences of game complexes by set, considering undirected nodes, allowing observation of how game complexes connect based on their effects. Eigenvector centrality was utilised, as it reflects the relationship among game complexes, based on the understanding that a node has higher centrality when it is connected to more central nodes, with a centrality value ranging from 0 to 1 [27,28]. therefore, node centrality depends on adjacent nodes and their interaction characteristics [19]. ...
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Purpose The objective of the present manuscript was to analyse the flow of play in high-level women’s volleyball by sequencing game complexes per set. Methods The sample analysed 135 matches played in the 2021/2022 Brazilian Women’s Superliga, with the number of matches analysed ranging from 14 and 25 matches for all teams participating in the championship. Eigenvector Centrality and Social Network Analysis conducted the connectivity and specificity of relationships as, and inferential analysis was performed using the chi-square test with a Monte Carlo correction. Results The results showed that the highest eigenvalues were for the continuity effect, except for complex 0. Regarding the continuity effect, the eigenvalues in the first four sets were higher for complexes III, IV, and V. In the fifth set, the highest eigenvalues were close between complexes II, III, IV, and V for the continuity effect. Furthermore, the results indicated that there was no association between the sequencing of game complexes and the set played (f? = 2470.01 and φ = 0.34, p = 0.272), nor between the number of complexes and the set played = 17.63 and φ = 0.03, p = 0.346). Conclusions In conclusion, women’s volleyball presents gameplay strategies to sustain play, promoting game continuity, regardless of the set played. The number of game complexes required to secure a point varies between two and three game complexes. Coaches should consider the dynamics in women’s volleyball matches, preparing teams for less risky and more continuous play.
... The same idea can be extended to centrality properties other than the degrees of the nodes. For illustration, we now consider the so-called eigenvector centrality, i.e. we rank the nodes' importance based on the eigenvector associated with the largest eigenvalue of the adjacency matrix [49,50]. ...
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... Over 30 different centrality measures (e.g., degree centrality, betweenness centrality, closeness centrality, eigenvector centrality, and control centrality) have been examined in the literature [6][7][8][9] . Among these, eigenvector centrality, defined as the leading eigenvector of the adjacency matrix of a graph, has received increasing attention 10,11 . It is worth noting that PageRank, a variant of eigenvector centrality, is the primary algorithm used in Google's search engine 12,13 . ...
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Centrality is widely recognized as one of the most critical measures to provide insight in the structure and function of complex networks. While various centrality measures have been proposed for single-layer networks, a general framework for studying centrality in multilayer networks (i.e., multicentrality) is still lacking. In this study, a tensor-based framework is introduced to study eigenvector multicentrality, which enables the quantification of the impact of interlayer influence on multicentrality, providing a systematic way to describe how multicentrality propagates across different layers. This framework can leverage prior knowledge about the interplay among layers to better characterize multicentrality for varying scenarios. Two interesting cases are presented to illustrate how to model multilayer influence by choosing appropriate functions of interlayer influence and design algorithms to calculate eigenvector multicentrality. This framework is applied to analyze several empirical multilayer networks, and the results corroborate that it can quantify the influence among layers and multicentrality of nodes effectively.
... This model predicts possibility of total eradication of an epidemics through preventive behaviors [34]. In a subsequent study [40], authors considered an information-dissemination network as an alternative alerting mechanism, and proposed the optimal design solution for an information-dissemination network based on eigenvector centralities [41] in the contact network graph. The SAIS model has been further explored in [42]- [44]. ...
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People change their physical contacts as a preventive response to infectious disease propagations. Yet, only a few mathematical models consider the coupled dynamics of the disease propagation and the contact adaptation process. This paper presents a model where each agent has a default contact neighborhood set, and switches to a different contact set once she becomes alert about infection among her default contacts. Since each agent can adopt either of two possible neighborhood sets, the overall contact network switches among 2^N possible configurations. Notably, a two-layer network representation can fully model the underlying adaptive, state-dependent contact network. Contact adaptation influences the size of the disease prevalence and the epidemic threshold---a characteristic measure of a contact network robustness against epidemics---in a nonlinear fashion. Particularly, the epidemic threshold for the presented adaptive contact network belongs to the solution of a nonlinear Perron-Frobenius (NPF) problem, which does not depend on the contact adaptation rate monotonically. Furthermore, the network adaptation model predicts a counter-intuitive scenario where adaptively changing contacts may adversely lead to lower network robustness against epidemic spreading if the contact adaptation is not fast enough. An original result for a class of NPF problems facilitate the analytical developments in this paper.
... In some methods, instead of considering the single term quantities, it is more appropriate to take into account these quantities only when the corresponding terms appear along with other highly frequent terms (see survey [3]). We propose to rank the importance of a keyword in a static semantic network using the eigenvector centrality [7], and then introduce the dynamic eigenvector centrality to capture emergent keywords and summarize the trends. We also generalize the frequency and degree centralities with similar dynamic versions. ...
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Methods for detecting and summarizing emergent keywords have been extensively studied since social media and microblogging activities have started to play an important role in data analysis and decision making. We present a system for monitoring emergent keywords and summarizing a document stream based on the dynamic semantic graphs of streaming documents. We introduce the notion of dynamic eigenvector centrality for ranking emergent keywords, and present an algorithm for summarizing emergent events that is based on the minimum weight set cover. We demonstrate our system with an analysis of streaming Twitter data related to public security events.
... Eigenvector centrality measures the connectivity of a user as well as the connectivity of its network neighbours; a user can acquire a high connectivity from the number and weight of his interactions or from being connected to highly connected users [13]. Eigenvector centrality can be easily calculated for large weighted network [14]. The average eigenvector centrality values of the 2836 Twitter users who wrote about the strike both before and after the passage of Bill-78 Twitter users decreased after Bill-78 (paired permutation test with 9999 permutations, p=0.001). ...
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Social media have been widely used to organize citizen movements. In 2012, 75% university and college students in Quebec, Canada, participated in mass protests against an increase in tuition fees, mainly organized using social media. To reduce public disruption, the government introduced special legislation designed to impede protest organization. Here, we show that the legislation changed the behaviour of social media users but not the overall structure of their social network on Twitter. Thus, users were still able to spread information to efficiently organize demonstrations using their social network. This natural experiment shows the power of social media in political mobilization, as well as behavioural flexibility in information flow over a large number of individuals.
... Towards this direction, there are many problems to be solved. The influential degree in the proposed discount strategy may be replaced with other influence indexes [42][43][44][45][46][47], resulting in many similar discount strategies. A comparative study can be conducted so as to find out the best one. ...
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This paper addresses the discount pricing in word-of-mouth (WOM) marketing. A new discount strategy known as the Infection-Based Discount (IBD) strategy is proposed. The basic idea of the IBD strategy lies in that each customer enjoys a discount that is linearly proportional to his/her influence in the WOM network. To evaluate the performance of the IBD strategy, the WOM spreading process is modeled as a dynamic model known as the DPA model, and the performance of the IBD strategy is modeled as a function of the basic discount. Next, the influence of different factors, including the basic discount and the WOM network, on the dynamics of the DPA model is revealed experimentally. Finally, the influence of different factors on the performance of the IBD strategy is uncovered experimentally. On this basis, some promotional measures are recommended.
... Specifically, for an undirected simple network, ( , ) G V E with p nodes and q edges, we use the topological features of nodes in the network as the input, including the degree of the node, the average degree of its neighbors, the average clustering coefficient 97 of its neighbors, the number of internal edges of its egonet (the substructure of a node and its first-order neighbors) 98,99 , the number of external edges of its egonet, the eigenvector centrality 100 ...
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Network robustness design is a significant engineering task in complex systems including urban planning, communication programming, and chip designing. With the embedded vulnerability of complex networks, the relationship between network topology and its robustness remains unknown, presenting a significant challenge in designing robust networks. Existing approaches—ranging from empirical manual designs, statistically-driven rules to optimization via Monte Carlo simulations, struggle to meet the design demands of large-scale networks under multidimensional attacks. Here, we introduce a general framework for designing robust networks based on AI reinforcement learning. This framework establishes an interactive environment between network attack strategies and design models, enabling the learning of effective robustness design strategies against specific attacks. Our framework enables efficient design of robust large-scale networks for a given cost, surpassing existing methods. Notably, we find that during the design process, the network may develop suitable multilayer backbones that mitigate its current vulnerability, offering insight into higher-order relations in real-world networks. Our approach can be adopted to various network design scenarios, which provides an integrative intelligent solution for designing robust complex systems.
... The importance of an individual in a network depends not only on its own network position characteristics, but also on the position characteristics of the other network individuals to which it is connected. The eigenvector centrality takes into account both the network structure type and the importance of network nodes (Bonacich, 2007), and describes the relative influence and connection quality of marine cities in the internal circulation network. The larger the value, the greater the influence of the member and the more important it is as a core participant in the interaction. ...
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Introduction China’s marine cities have reached a critical juncture after 40 years of rapid development. In this new stage, where internal circulation is the main focus, there is a need to enhance the internal circulation capabilities of these cities and unleash their full economic potential. This paper aims to explore the positioning and improvement path of marine cities in China's internal circulation network, and fully unleash the development potential of marine cities. Methods Based on data from 284 prefecture-level cities in China, this paper constructs the social network of China's urban internal circulation with the help of the modified gravity model, and explores the conditional configuration of the improvement of the status of marine cities in internal circulation network by using the fuzzy-set qualitative comparative analysis (fsQCA) method. Results and discussion (1) The development level of marine cities' internal circulation can be categorized into three tiers, led by Shanghai. The development gap between the 14 marine cities has gradually widened over recent years. (2) Chinese marine cities can be divided into three groups in the topological structure of China's urban internal circulation network: core, periphery, and edge, with Shanghai being the core "bridge" in the network. The traditional advantages of some northern economically strong cities in the construction of the internal circulation network have gradually been lost, and many marine cities have seen their leadership and control over the internal circulation network significantly weakened. (3) No single factor is a necessary condition for achieving a high-level status of marine cities in the internal circulation network. (4) The four conditional variables of demand side, supply side, industrial linkage and inter-regional integration have two sufficient condition configurations to enhance the status of marine cities in internal circulation network. Among them, the "industry-regional integration"-dominated configuration with the core of unblocking the bottlenecks of the internal circulation is the main path.
... The node degree is the sum of the weighted edges connected to a node. The eigenvector centrality represents the transitive influence of a node 57,58 . Betweenness centrality is the sum of the fraction of all-pairs shortest paths that pass through a node 59,60 . ...
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... Here, C denoted as eigenvector centrality (EC) [50], the isolating eigenvector centrality of node u is computed as ISEC(u): ...
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2In an influential paper, Freeman (1979) identified three aspects of centrality: betweenness, nearness, and degree. Perhaps because they are designed to apply to networks in which relations are binary valued (they exist or they do not), these types of centrality have not been used in interlocking directorate research, which has almost exclusively used formula (2) below to compute centrality. Conceptually, this measure, of which c(ot, 3) is a generalization, is closest to being a nearness measure when 3 is positive. In any case, there is no discrepancy between the measures for the four networks whose analysis forms the heart of this paper. The rank orderings by the
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Methods for ranking World Wide Web resources according to their position in the link structure of the Web are receiving considerable attention, because they provide the first effective means for search engines to cope with the explosive growth and diversification of the Web. We show that layouts for effective visualization of an underlying link structure can be computed in sync with the iterative computation utilized in all popular such rankings. Our visualizations provide valuable insight into the link structure and the ranking mechanism alike. Therefore, they are useful for the analysis of query results, maintenance of search engines, and evaluation of Web graph models.
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We introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network. Smaller subgraphs are given more weight than larger ones, which makes this measure appropriate for characterizing network motifs. We show that the subgraph centrality [C(S)(i)] can be obtained mathematically from the spectra of the adjacency matrix of the network. This measure is better able to discriminate the nodes of a network than alternate measures such as degree, closeness, betweenness, and eigenvector centralities. We study eight real-world networks for which C(S)(i) displays useful and desirable properties, such as clear ranking of nodes and scale-free characteristics. Compared with the number of links per node, the ranking introduced by C(S)(i) (for the nodes in the protein interaction network of S. cereviciae) is more highly correlated with the lethality of individual proteins removed from the proteome.
Eigen analysis of networks://www.heinz.cmu.edu/project/INSNA/joss/index1
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Subgraph centrality in complex networks Eigen analysis of networks
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Estrada, E., Rodr ıguez-Velá, J.A., 2005. Subgraph centrality in complex networks. Physical Review, E7. Golub, G.H., Van Loon, C.F., 1983. Matrix Computations. John Hopkins University Press, Baltimore. Richards, W.D., Seary, A.J., 2000. Eigen analysis of networks, Journal of Social Structure. http://www.heinz.cmu.edu/ project/INSNA/joss/index1.html.
Visual ranking of link structures
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