Graph colorings and power in experimental exchange networks

The principal goal of studying experimental exchange networks is to understand the relationship between power and network position. In this paper we provide a formal definition of the appropriate notion of position, and explore some of the consequences of assuming that power is a function of position. It is shown that, in highly structured graphs, the space of possible power outcomes is significantly reduced if power is entirely structural. Drawing on the notion of role colorings (Everett and Borgatti 1991), we formalize the frequently expressed intuitive idea that a node's power is a function of the powers of its neighbors, just as their power is determined by the powers of their neighbors, and so on. We use a combination of two role colorings to express this idea. One, called ecological coloring, states that if two nodes have the same power neighborhoods (i.e. distinct levels of power exhibited by their neighbors), then they must have equal power. The other, called regular coloring, states that if two nodes have equal power, then we can infer that they have the same power neighborhoods. Together, these colorings imply a one-to-one relationship between the power of a node, and the power(s) of its neighbors. It is found that applying these colorings in addition to assuming power is a function of position, radically reduces the sample space of possible power outcomes, leaving only a few possibilities. With two revealing exceptions, the reduced space of possible power outcomes always contains the experimentally observed result.