Transmission and closeness centrality in networks

Abstract

In the study of graphs, distances play a central role throughout. Major newlineinvestigation of distance concepts in graphs was given by their wide newlineapplicability. Their applications range from facility location problems and newlinenetwork designs in Operations Research to prediction of properties of chemical newlinecompounds in chemistry, from measuring closeness of groups of individuals in newlinesociology to identifying important role of players, for example, in the internet. newlineWith Wiener s discovery of a close correlation between the boiling points of newlinecertain alkanes and the sum of the distances between vertices in the graphs newlinerepresenting their molecular structures, it became apparent that graph newlineparameters and topological indices, can potentially be used to predict properties newlineof chemical compounds. newlineIn the classical study of distances in graph theory, the main focus has newlinebeen on the study of the two main graph parameters concerned with distance, newlinenamely the diameter and the radius. Methods developed to determine the newlinediameter of a graph became the foundation for most research in other distance newlinerelated graph parameters. The transmission of a vertex (in some literature, also newlinecalled farness) is defined as the sum of the lengths of all shortest paths between newlinethe chosen vertex and all other vertices in G. The transmission of a graph G is newlinethe sum of the transmissions of all its vertices which is nothing but twice its newlineWiener index. In location theory, sets of vertices with the minimum (or newlinemaximum) distance in a graph, play a special role because they form target sets newlinefor locations of facilities. In this thesis, we adapt different strategies to compute newlinethe transmission of a vertex without using the distance matrix. From the newlinetransmission value, we have obtained closeness centrality newline