In many social networks, certain nodes play more important roles than others. Consequently, the concept of centrality index has been extensively studied in the literature. More recently, a number of new centrality indices have been proposed in an attempt to reflect the following fundamental property of nodes: the role they each play in connecting the network. Typically, such measures are built upon the well-known coalitional-game model of Myerson, where cooperation is restricted to connected coalitions. Unfortunately, although these recent indices offered new insights, they were only evaluated empirically.
To address this issue, we propose an axiomatic approach built around five basic requirements. Building upon these axioms, we prove that Degree Centrality is the unique index satisfying Normalization, Locality, Fairness and Monotonicity. Furthermore, we prove that replacing Monotonicity with Gain-loss leads to yet another unique index, which we call: Attachment Centrality; this is the first axiomatized centrality index focusing on connectivity in the literature.
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