page_title: Centrality
section_title: Dissimilarity based centrality measures
hierarchical_section_title: Centrality / Dissimilarity based centrality measures
caption_attribution_description: English: In the illustrated network, green and red nodes are the most dissimilar because they do not share neighbors between them. So the green node is more relevant than gray nodes for the red one, because only through green node, the red one can access to the blues and it can access directly to any gray node.
context_page_description: In graph theory and network analysis, indicators of centrality identify the most important vertices within a graph. Applications include identifying the most influential person in a social network, key infrastructure nodes in the Internet or urban networks, and super-spreaders of disease. Centrality concepts were first developed in social network analysis, and many of the terms used to measure centrality reflect their sociological origin.
  They should not be confused with node influence metrics, which seek to quantify the influence of every node in the network.
context_section_description: In order to obtain better results in the ranking of the nodes of a given network,  in  are used dissimilarity measures (specific to the theory of classification and data mining) to enrich the centrality measures in complex networks. This is illustrated with eigenvector centrality, calculating the centrality of each node through the solution of the eigenvalue problem
  where  (coordinate-to-coordinate product) and  is an arbitrary dissimilarity matrix, defined through a dissimilitary measure, e.g., Jaccard dissimilarity given by
  Where this measure permits us to quantify the topological contribution (which is why is called contribution centrality) of each node to the centrality of a given node, having more weight/relevance those nodes with greater dissimilarity, since these allow to the given node access to nodes that which themselves can not access directly.
  Is noteworthy that  is non-negative because  and  are non-negative matrices, so we can use the Perron–Frobenius theorem to ensure that the above problem has a unique solution for λ = λₘₐₓ with c non-negative, allowing us to infer the centrality of each node in the network. Therefore, the centrality of the i-th node is
  where  is the number of the nodes in the network. Several dissimilarity measures and networks were tested in  obtaining improved results in the studied cases.
caption: In the illustrated network, green and red nodes are the most dissimilar because they do not share neighbors between them. So, the green one contributes more to the centrality of the red one than the gray ones, because the red one can access to the blue ones only through the green, and the gray nodes are redundant for the red one, because it can access directly to each gray node without any intermediary.
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