One way of characterizing the topological and structural properties of vertices and edges in a graph is by using structural similarity measures. Measures like Cosine, Jaccard and Dice compute the similarities restricted to the immediate neighborhood of the vertices, bypassing important structural properties beyond the locality. Others measures, such as the generalized edge clustering coefficient, go beyond the locality but with high computational complexity, making them impractical in large-scale scenarios. In this paper we propose a novel similarity measure that determines the structural similarity by dynamically diffusing and capturing information beyond the locality. This new similarity is modeled as an iterated function that can be solved by fixed point iteration in super-linear time and memory complexity, so it is able to analyze large-scale graphs. In order to show the advantages of the proposed similarity in the community detection task, we replace the local structural similarity used in the SCAN algorithm with the proposed similarity measure, improving the quality of the detected community structure and also reducing the sensitivity to the parameter @math of the SCAN algorithm.
The degree centrality is based on the idea that vertices with high degree are involved more frequently in communications than vertices with low degree. For a vertex @math , the degree centrality is basically its degree. This is a local function, because it only depends on the neighborhood of @math . Several definitions based on the degree centrality have been proposed, like k-path centrality and edge-disjoint k-path centrality. To compute these centrality measures several randomnized algorithms are proposed @cite_1 .
@cite_1
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Betweenness centrality is a fundamental measure in social network analysis, expressing the importance or influence of individual vertices (or edges) in a network in terms of the fraction of shortest paths that pass through them. Since exact computation in large networks is prohibitively expensive, we present two efficient randomized algorithms for betweenness estimation. The algorithms are based on random sampling of shortest paths and offer probabilistic guarantees on the quality of the approximation. The first algorithm estimates the betweenness of all vertices (or edges): all approximate values are within an additive factor @math ??(0,1) from the real values, with probability at least @math 1-?. The second algorithm focuses on the top-K vertices (or edges) with highest betweenness and estimate their betweenness value to within a multiplicative factor @math ?, with probability at least @math 1-?. This is the first algorithm that can compute such approximation for the top-K vertices (or edges). By proving upper and lower bounds to the VC-dimension of a range set associated with the problem at hand, we can bound the sample size needed to achieve the desired approximations. We obtain sample sizes that are independent from the number of vertices in the network and only depend on a characteristic quantity that we call the vertex-diameter, that is the maximum number of vertices in a shortest path. In some cases, the sample size is completely independent from any quantitative property of the graph. An extensive experimental evaluation on real and artificial networks shows that our algorithms are significantly faster and much more scalable as the number of vertices grows than other algorithms with similar approximation guarantees. 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. Abstract Vertex betweenness centrality is a metric that seeks to quantify a sense of the importance of a vertex in a network in terms of its ‘control’ on the flow of information along geodesic paths throughout the network. Two natural ways to extend vertex betweenness centrality to sets of vertices are (i) in terms of geodesic paths that pass through at least one of the vertices in the set, and (ii) in terms of geodesic paths that pass through all vertices in the set. The former was introduced by Everett and Borgatti [Everett, M., Borgatti, S., 1999. The centrality of groups and classes. Journal of Mathematical Sociology 23 (3), 181–201], and called group betweenness centrality. The latter, which we call co-betweenness centrality here, has not been considered formally in the literature until now, to the best of our knowledge. In this paper, we show that these two notions of centrality are in fact intimately related and, furthermore, that this relationship may be exploited to obtain deeper insight into both. In particular, we provide an expansion for group betweenness in terms of increasingly higher orders of co-betweenness, in a manner analogous to the Taylor series expansion of a mathematical function in calculus. We then demonstrate the utility of this expansion by using it to construct analytic lower and upper bounds for group betweenness that involve only simple combinations of (i) the betweenness of individual vertices in the group, and (ii) the co-betweenness of pairs of these vertices. Accordingly, we argue that the latter quantity, i.e., pairwise co-betweenness, is itself a fundamental quantity of some independent interest, and we present a computationally efficient algorithm for its calculation, which extends the algorithm of Brandes [Brandes, U., 2001. A faster algorithm for betweenness centrality. Journal of Mathematical Sociology 25, 163] in a natural manner. Applications are provided throughout, using a handful of different communication networks, which serve to illustrate the way in which our mathematical contributions allow for insight to be gained into the interaction of network structure, coalitions, and information flow in social networks. The betweenness centrality of a vertex in a graph is a measure for the participation of the vertex in the shortest paths in the graph. The Betweenness centrality is widely used in network analyses. Especially in a social network, the recursive computation of the betweenness centralities of vertices is performed for the community detection and finding the influential user in the network. Since a social network graph is frequently updated, it is necessary to update the betweenness centrality efficiently. When a graph is changed, the betweenness centralities of all the vertices should be recomputed from scratch using all the vertices in the graph. To the best of our knowledge, this is the first work that proposes an efficient algorithm which handles the update of the betweenness centralities of vertices in a graph. In this paper, we propose a method that efficiently reduces the search space by finding a candidate set of vertices whose betweenness centralities can be updated and computes their betweenness centeralities using candidate vertices only. As the cost of calculating the betweenness centrality mainly depends on the number of vertices to be considered, the proposed algorithm significantly reduces the cost of calculation. The proposed algorithm allows the transformation of an existing algorithm which does not consider the graph update. Experimental results on large real datasets show that the proposed algorithm speeds up the existing algorithm 2 to 2418 times depending on the dataset.