Topic
Katz centrality
About: Katz centrality is a research topic. Over the lifetime, 601 publications have been published within this topic receiving 77858 citations.
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TL;DR: Li et al. as discussed by the authors established the co-inventing network and examined the moderate role of network connectivity, measured by classifying the individuals into two cohorts: inventors in the largest connected component and inventors from other isolated components.
23 citations
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TL;DR: In this article, the authors propose an axiomatic approach to characterize a class of centrality measures for which the centrality of an agent is recursively related to the centralities of the agents she is connected to.
Abstract: The centrality of an agent in a network has been shown to be crucial in explaining different behaviors and outcomes. In this paper, we propose an axiomatic approach to characterize a class of 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. This is possible because all the centralities study here are local in the sense that the centrality measure of an agent only depends on her set of neighbors and their centralities. Our axiomatic characterization highlights the conceptual similarities among this class of measures.
22 citations
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TL;DR: This estimate gives an analytical comparison of the eigenvector centrality of G with thecentrality of L(G) in terms of some irregularity measure of G.
Abstract: Given a network G, it is known that the Bonacich centrality of the bipartite graph B(G) associated with G can be obtained in terms of the centralities of the line graph L(G) associated with G and the centrality of the network G+gr (whose adjacency matrix is obtained by adding to the adjacency matrix A(G) the diagonal matrix D=bij, where bii is the degree of node i in G) and conversely. In this contribution, we use the centrality of G to estimate the centrality of G+gr and show that the error committed is bounded by some measure of the irregularity of G. This estimate gives an analytical comparison of the eigenvector centrality of G with the centrality of L(G) in terms of some irregularity measure of G.
22 citations
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01 Jan 2017
TL;DR: This work answers the question of which pairwise rankings are reliable given an approximate solution to the linear system and obtains bounds on the accuracy of the approximation compared to the exact solution with respect to the highly ranked nodes.
Abstract: Graphs and networks are prevalent in modeling relational datasets from many fields of research. By using iterative solvers to approximate graph measures (specifically Katz Centrality), we can obtain a ranking vector consisting of a number for each vertex in the graph identifying its relative importance. We use the residual to accurately estimate how much of the ranking from an approximate solution matches the ranking given by the exact solution. Using probabilistic matrix norms and applying numerical analysis to the computation of Katz Centrality, we obtain bounds on the accuracy of the approximation compared to the exact solution with respect to the highly ranked nodes. This relates the numerical accuracy of the linear solver to the data analysis accuracy of finding the correct ranking. In particular, we answer the question of which pairwise rankings are reliable given an approximate solution to the linear system. Experiments on many real-world networks up to several million vertices and several hundred million edges validate our theory and show that we are able to accurately estimate large portions of the approximation. By analyzing convergence error, we develop confidence in the ranking schemes of data mining.
22 citations
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01 Dec 2011TL;DR: This paper calculates and normalizes the three centrality measures values for each node in the Fuzzy Cognitive Map and transforms these values into linguistic terms using 2-tuple fuzzy linguistic representation model, and provides new important measures to overcome the above drawbacks.
Abstract: The Fuzzy Cognitive Map (FCM) provides a robust model for knowledge representation. FCM is a fuzzy signed weighted directed graph that depicts the knowledge of the domain as nodes representing the factors of the domain and arcs representing the connections among these factors. The centrality of a node in FCM, also called the importance of a node in this paper, is considered the most important index of all the graph theory indices applying to FCM which helps decision makers in analysing their FCM models. By finding the centrality values of the nodes in FCM, the important (central) nodes, which are the focal point for decision makers, are determined. The highest centrality value of a node in FCM is the most important one. Little research has addressed the centrality of the nodes in an FCM using only the degree centrality measure. The degree centrality measure only accounts for the direct connections of the node. Although the degree centrality index is considered an important measure in determining the centrality of a node in an FCM, it is not sufficient and has significant shortcomings; it ignores the importance of the indirect connections, the role of the node's position and flow of information through that node, i.e., how a node is close to other nodes and how the node contributes to the flow of information (communication control) through that node. In the literature, there are other centrality measures that can handle direct and indirect connections to determine the central nodes in a directed graph. This paper presents a new method for identifying the central nodes in an FCM. In order to achieve that, we provide, in addition to the degree measure, new important measures to overcome the above drawbacks. These new centrality measures are: betweenness and closeness measures. In this paper, we calculate and normalize the three centrality measures values for each node in the FCM. These values are then transformed into linguistic terms using 2-tuple fuzzy linguistic representation model. We use the 2-tuple model because it describes the granularity of uncertainty of the fuzzy sets and avoids the loss of information resulted from the imprecision and normalization of the measures. The calculated centrality measures values for each node in the FCM are then aggregated using a 2-tuple fuzzy fusion approach to obtain consensus centrality measure. The resulting aggregated values are then ranked in descending order to identify the most central nodes in the FCM, and this would improve the decision-making and help in simplify the FCM by removing the least important nodes from it. Finally, a list of future works related to this paper is suggested.
22 citations