Absorbing random walks interpolating between centrality measures on complex networks

Fractional centralities on networks: Consolidating the local and the global

We propose a new centrality incorporating two classical node-level centralities, the degree centrality and the information centrality, which are considered as local and global centralities, respectively. These two centralities have expressions in terms of the graph Laplacian L, which motivates us to exploit its fractional analog L^{γ} with a fractional parameter γ. As γ varies from 0 to 1, the proposed fractional version of the information centrality makes intriguing changes in the node centrality rankings. These changes could not be generated by the fractional degree centrality since it is mostly influenced by the local aspect. We prove that these two fractional centralities behave similarly when γ is close to 0. This result provides its complete understanding of the boundary of the interval in which γ lies since the fractional information centrality with γ=1 is the usual information centrality. Moreover, our computation for the correlation coefficients between the fractional information centrality and the degree centrality reveals that the fractional information centrality is transformed from a local centrality into being a global one as γ changes from 0 to 1.

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 purpose and 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 tendency to send influence flows along multiple, often nongeodesic paths. Combining these categories with Borgatti's centrality types [Borgatti, Soc. Networks 27, 55 (2005)0378-873310.1016/j.socnet.2004.11.008], we arrive at a 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, conserved 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 differs significantly from similar centralities. We demonstrate that, compared to other conserved-flow centralities, it has a simpler mathematical description. Compared to other reach-parametrized centralities, it robustly preserves an intuitive rank ordering across a wide range of network architectures, capturing aspects of both the closeness and betweenness centralities. 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-parametrized centralities exhibit both of these behaviors on regular networks and hub networks, respectively. We compare the properties of the ground-current centrality with several other reach-parametrized centralities on four artificial networks and seven real-world networks.