Measuring influence in a social network

We study a model of influence in a social network. The point of departure for this model is the framework introduced in [7], and later investigated e.g. in [11, 12]. In [6] we continue research on players ’ interactions in a social network, and introduce tools to mea-sure influence between players. We consider the following situation. In a social network, players (also referred to as agents or voters) are to make an acceptance-rejection decision. Let N: = {1,..., n} be the set of all players. It is assumed that each player has an inclina-tion either to say either YES (denoted by +1) or NO (denoted by −1). Let i = (i1,..., in) denote an inclination vector and I: = {−1,+1}n be the set of all inclination vectors. In particular, (1, 1,..., 1) ∈ I and (−1,−1,...,−1) ∈ I will be denoted by 1N and −1N, respectively. Players may influence each other, and as a consequence of the influence, the final decision of a player may be different from his original inclination. Each inclination vector i ∈ I is therefore transformed into a decision vector Bi = ((Bi)1,..., (Bi)n), where B: I → I is the influence function. Let B denote the set of all influence functions. We are interested in measuring influence a coalition (of players with the same incli-nation) has on an agent outside that coalition. By the influence we mean a situation i