On Fractional Approach to Analysis of Linked Networks (original) (raw)
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We introduce the link-space formalism for analyzing network models with degree-degree correlations. The formalism is based on a statistical description of the fraction of links li,j connecting nodes of degrees i and j. To demonstrate its use, we apply the framework to some pedagogical network models, namely, random-attachment, Barabási-Albert preferential attachment and the classical Erdős and Rényi random graph. For these three models the link-space matrix can be solved analytically. We apply the formalism to a simple one-parameter growing network model whose numerical solution exemplifies the effect of degree-degree correlations for the resulting degree distribution. We also employ the formalism to derive the degree distributions of two very simple network decay models, more specifically, that of random link deletion and random node deletion. The formalism allows detailed analysis of the correlations within networks and we also employ it to derive the form of a perfectly non-assortative network for arbitrary degree distribution.
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In some networks nodes belong to predefined groups (e.g., authors belong to institutions). Common network centrality measures do not take this structure into account. Gefura measures are designed as indicators of a node's brokerage role between such groups. They are defined as variants of betweenness centrality and consider to what extent a node belongs to shortest paths between nodes from different groups. In this article we make the following new contributions to their study: (1) We systematically study unnormalized gefura measures and show that, next to the 'structural' normalization that has hitherto been applied, a 'basic' normalization procedure is possible. While the former normalizes at the level of groups, the latter normalizes at the level of nodes. (2) Treating undirected networks as equivalent to symmetric directed networks, we expand the definition of gefura measures to the directed case. (3) It is shown how Brandes' algorithm for betweenness centrality can be adjusted to cover these cases.
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The centrality of vertices has been a key issue in network analysis. For unweighted networks where edges are just present or absent and have no weight attached, many centrality measures have been presented, such as degree, betweenness, closeness, eigenvector and subgraph centrality. There has been a growing need to design centrality measures for weighted networks, because weighted networks where edges are attached weights would contain rich information. Some network measures have been proposed for weighted networks, including three common measures of vertex centrality: degree, closeness, and betweenness. In this paper, we propose a new centrality measure called the Laplacian centrality measure for weighted networks. The Laplacian energy is defined as E L ðGÞ ¼ P i k 2 i , where k i 's are eigenvalues of the Laplacian matrix of weighted network G. The importance (centrality) of a vertex v is reflected by the drop of the Laplacian energy of the network to respond to the deactivation (deletion) of the vertex from the network. We also prove an algebraic graph theory result that provides a structural description of the Laplacian centrality measure which is in terms of the number of all kinds of 2-walks. Laplacian centrality unveils more structural information about connectivity and density around v (further than its immediate neighborhood). That is, comparing with other standard centrality measures proposed for weighted networks (e.g. degree, closeness or betweenness centrality), Laplacian centrality is an intermediate measuring between global and local characterization of the importance (centrality) of a vertex. We further investigate the validness and robustness of this new centrality measure by illustrating this method to some classical weighted social network data sets and obtain reliable results, which provide strong evidences of the new measure's utility.