# A model of inï¬uence with an ordered set of possible actions

# Abstract

In the paper, a yes-no model of inﬂuence is generalized to a multi-choice framework. We introduce and study weighted inﬂuence indices of a coalition on a player in a social network, where players have an ordered set of possible actions. Each player has an inclination to choose one of the actions. Due to mutual inﬂuence among players, the ﬁnal decision of each player may be different from his original inclination. In a particular case, the decision of the player is closer to the inclination of the inﬂuencing coalition than his inclination was, i.e., the distance between the inclinations of the player and of the coalition is greater than the distance between the decision of the player and the inclination of the coalition in question. The weighted inﬂuence index which captures such a case is called the weighted positive inﬂuence index. We also consider the weighted negative inﬂuence index, where the ﬁnal decision of the player goes farther away from the inclination of the coalition. We consider several inﬂuence functions deﬁned in the generalized model of inﬂuence and study their properties. The concept of a follower of a given coalition, and its particular case, a perfect follower, are deﬁned. The properties of the set of followers are analyzed.

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