In this paper, we present a framework for studying the following fundamental question in network analysis: How should one assess the centralities of nodes in an information/influence propagation process over a social network? Our framework systematically extends a family of classical graph-theoretical centrality formulations, including degree centrality, harmonic centrality, and their sphere-of-influence generalizations, to influence-based network centralities. We further extend natural group centralities from graph models to influence models, since group cooperation is essential in social influences. This in turn enables us to assess individuals centralities in group influence settings by applying the concept of Shapley value from cooperative game theory. Mathematically, using the property that these centrality formulations are Bayesian, we prove the following characterization theorem: Every influence-based centrality formulation in this family is the unique Bayesian centrality that conforms with its corresponding graph-theoretical centrality formulation. Moreover, the uniqueness is fully determined by the centrality formulation on the class of layered graphs, which is derived from a beautiful algebraic structure of influence instances modeled by cascading sequences. Our main mathematical result that layered graphs in fact form a basis for the space of influence-cascading-sequence profiles could also be useful in other studies of network influences. We further provide an algorithmic framework for efficient approximation of these influence-based centrality measures. Our study provides a systematic road map for comparative analyses of different influence-based centrality formulations, as well as for transferring graph-theoretical concepts to influence models.
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