Many of the most important complex systems take the form of networks, including biological systems like neural connections and food webs, technological systems like the Internet and power grids, and social networks like political affiliations of social media users. In recent decades, the analysis of complex networks using the methods of statistical physics has become increasingly popular. Centrality, which quantifies the “importance” of individual network 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. We identify two categories of centrality parameters. Grasp parameters control the centrality’s potential to send influence flows along nongeodesic paths. Reach parameters control the attenuation of influence flows between distant nodes. We develop two grasp-parametrized conditional walker-flow centralities to interpolate between prominent variations of the closeness and betweenness. The first of these is the con- ditional resistance-closeness centrality, which interpolates between the standard closeness at low grasp and a form of the information centrality at high grasp. The second is the condi- tional current-betweenness centrality, which interpolates between the standard betweenness at low grasp and Newman’s random-walk betweenness at high grasp. The conditional walker- flow centralities are based on absorbing Markov chains and employ a grasp parameter that controls the absorption of walkers as they cross the network’s edges. Combining the reach and grasp categories with Borgatti’s centrality types [S. P. Borgatti, Social Networks 27, 55-71 (2005)], we arrive at a novel classification scheme for parametrized centralities. The classification scheme includes many of the most prominent parametrized centrality measures: the communicability, Katz, and PageRank centralities. Furthermore, two of the most prominent non-parametrized centralities are recovered as limits of the reach- parametrized centralities within the classification: the degree centrality is the low-reach limit of the communicability, while the eigenvector centrality is obtained in the high-reach limit. Using the centrality-classification scheme, we identify the notable absence of any measures that are radial, reach parametrized, and based on acyclic, conservative flows of influence. We therefore introduce the reach-parametrized 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 measures. Finally, we study methods of quantifying the extent of a centrality’s reach and grasp. To this end, we develop two statistical methods: the influence-reach distribution and the influence-grasp distribution. These tools provide additional insights into the ways that cen- trality parameters affect the flow of influence in networks. They can also be used to extract the length scales inherent in a network’s structure.
