A symmetric network consists of a set of positions and a set of bilateral links between these positions. For every symmetric network we define a cooperative transferable utility game that measures the “power” of each coalition of positions in the network. Applying the Shapley value to this game yields a network power measure, the β-measure, which reflects the power of the individual positions in the network. Applying this power distribution method iteratively yields a limit distribution, which turns out to be (...) proportional to the well-known degree measure. We compare the β-measure and degree measure by providing characterizations, which differ only in the normalization that is used. (shrink)

We deal with the ranking problem of the nodes in a directed graph. The bilateral relationships specified by a directed graph may reflect the outcomes of a sport competition, the mutual reference structure between websites, or a group preference structure over alternatives. We introduce a class of scoring methods for directed graphs, indexed by a single nonnegative parameter α. This parameter reflects the internal slackening of a node within an underlying iterative process. The class of so-called internal slackening scoring methods, (...) denoted by λα, consists of the limits of these processes. It is seen that λ0 extends the invariant scoring method, while λ∞ extends the fair bets scoring method. Method λ1 corresponds with the existing λ-scoring method of Borm et al. (Ann Oper Res 109(1):61–75, 2002) and can be seen as a compromise between λ0 and λ∞. In particular, an explicit proportionality relation between λα and λ1 is derived. Moreover, the internal slackening scoring methods are applied to the setting of social choice situations where they give rise to a class of social choice correspondences that refine both the Top cycle correspondence and the Uncovered set correspondence. (shrink)

In the paper, we study a model of influence in a social network. It is assumed that each player has an inclination to say YES or NO which, due to influence of other players, may be different from the decision of the player. The point of departure here is the concept of the Hoede-Bakker index - the notion which computes the overall decisional 'power' of a player in a social network. The main drawback of the Hoede-Bakker index is that it (...) hides the actual role of the influence function, analyzing only the final decision in terms of success and failure. In this paper, we separate the influence part from the group decision part, and focus on the description and analysis of the influence part. We propose among other descriptive tools a definition of a (weighted) influence index of a coalition upon an individual. Moreover, we consider different influence functions representative of commonly encountered situations. Finally, we propose a suitable definition of a modified decisional power. (shrink)