Adjustable reach in a network centrality based on current flows

Centrality, which quantifies the "importance" of individual nodes, is among the most essential concepts in modern network theory. Most prominent centrality measures can be expressed as an aggregation of influence flows between pairs of nodes. As there are many ways in which influence can be defined, many different centrality measures are in use. Parametrized centralities allow further flexibility and utility by tuning the centrality calculation to the regime most appropriate for a given network. Here, we identify two categories of centrality parameters. Reach parameters control the attenuation of influence flows between distant nodes. Grasp parameters control the centrality's potential to send influence flows along multiple, often nongeodesic paths. Combining these categories with Borgatti's centrality types [S. P. Borgatti, Social Networks 27, 55-71 (2005)], we arrive at a novel classification system for parametrized centralities. Using this classification, we identify the notable absence of any centrality measures that are radial, reach parametrized, and based on acyclic, conservative flows of influence. We therefore introduce the ground-current centrality, which is a measure of precisely this type. Because of its unique position in the taxonomy, the ground-current centrality has significant advantages over similar centralities. We demonstrate that, compared to other conserved-flow centralities, it has a simpler mathematical description. Compared to other reach centralities, it robustly preserves an intuitive rank ordering across a wide range of network architectures. We also show that it produces a consistent distribution of centrality values among the nodes, neither trivially equally spread (delocalization), nor overly focused on a few nodes (localization). Other reach centralities exhibit both of these behaviors on regular networks and hub networks, respectively

Absorbing Random Walks Interpolating Between Centrality Measures on Complex Networks

Centrality, which quantifies the "importance" of individual nodes, is among the most essential concepts in modern network theory. As there are many ways in which a node can be important, many different centrality measures are in use. Here, we concentrate on versions of the common betweenness and it closeness centralities. The former measures the fraction of paths between pairs of nodes that go through a given node, while the latter measures an average inverse distance between a particular node and all other nodes. Both centralities only consider shortest paths (i.e., geodesics) between pairs of nodes. Here we develop a method, based on absorbing Markov chains, that enables us to continuously interpolate both of these centrality measures away from the geodesic limit and toward a limit where no restriction is placed on the length of the paths the walkers can explore. At this second limit, the interpolated betweenness and closeness centralities reduce, respectively, to the well-known it current betweenness and resistance closeness (information) centralities. The method is tested numerically on four real networks, revealing complex changes in node centrality rankings with respect to the value of the interpolation parameter. Non-monotonic betweenness behaviors are found to characterize nodes that lie close to inter-community boundaries in the studied networks