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      On Rich Clubs of Path-Based Centralities in Networks

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          Abstract

          Many scale-free networks exhibit a rich club structure, where high degree vertices form tightly interconnected subgraphs. In this paper, we explore the emergence of rich clubs in the context of shortest path based centrality metrics. We term these subgraphs of connected high closeness or high betweeness vertices as rich centrality clubs (RCC). Our experiments on real world and synthetic networks highlight the inter-relations between RCCs, expander graphs, and the core-periphery structure of the network. We show empirically and theoretically that RCCs exist, if the core-periphery structure of the network is such that each shell is an expander graph, and their density decreases from inner to outer shells. The main contributions of our paper are: (i) we demonstrate that the formation of RCC is related to the core-periphery structure and particularly the expander like properties of each shell, (ii) we show that the RCC property can be used to find effective seed nodes for spreading information and for improving the resilience of the network under perturbation and, finally, (iii) we present a modification algorithm that can insert RCC within networks, while not affecting their other structural properties. Taken together, these contributions present one of the first comprehensive studies of the properties and applications of rich clubs for path based centralities.

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          Most cited references 25

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          Network structure and minimum degree

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            Identification of influential spreaders in complex networks

            Networks portray a multitude of interactions through which people meet, ideas are spread, and infectious diseases propagate within a society. Identifying the most efficient "spreaders" in a network is an important step to optimize the use of available resources and ensure the more efficient spread of information. Here we show that, in contrast to common belief, the most influential spreaders in a social network do not correspond to the best connected people or to the most central people (high betweenness centrality). Instead, we find: (i) The most efficient spreaders are those located within the core of the network as identified by the k-shell decomposition analysis. (ii) When multiple spreaders are considered simultaneously, the distance between them becomes the crucial parameter that determines the extend of the spreading. Furthermore, we find that-- in the case of infections that do not confer immunity on recovered individuals-- the infection persists in the high k-shell layers of the network under conditions where hubs may not be able to preserve the infection. Our analysis provides a plausible route for an optimal design of efficient dissemination strategies.
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              The average distances in random graphs with given expected degrees.

              Random graph theory is used to examine the "small-world phenomenon"; any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log nlog d, where d is the weighted average of the sum of squares of the expected degrees. Of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1kbeta for some fixed exponent beta. For the case of beta > 3, we prove that the average distance of the power law graphs is almost surely of order log nlog d. However, many Internet, social, and citation networks are power law graphs with exponents in the range 2 < beta < 3 for which the power law random graphs have average distance almost surely of order log log n, but have diameter of order log n (provided having some mild constraints for the average distance and maximum degree). In particular, these graphs contain a dense subgraph, which we call the core, having n(clog log n) vertices. Almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core.
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                Author and article information

                Journal
                08 August 2018
                Article
                1808.02903

                http://creativecommons.org/licenses/by-sa/4.0/

                Custom metadata
                Accepted at ACM CIKM 2018, 10 Pages
                cs.SI physics.soc-ph

                Social & Information networks, General physics

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