Common interaction restrictions Sample Clauses

Common interaction restrictions. Modifying the interaction relation is one way in which there can be substantial differences in different agreement problems. A graph is often used to represent an interaction relation. The vertices of a graph represent agents and an edge between two vertices indicates the possibility of interaction between the two agents. An interaction relation can be converted to a graph and vice-versa. Note that the interaction relation allows pairs of agents to have an infinite cost for an interaction. This represents the fact that these two agents cannot directly interact with each other. We can convert any interaction relation into a graph using the algorithm below. 1. Let V be the vertex set. There will be n vertices, one for each agent. vi will be the vertex for αi. Let Et be the set of edges at time t. 2. For each pair of agents αi and αj at time t where Θ(αi, αj, t) < ∞ do the following: (a) Let c = Θ(αi, αj, t) (b) If a directed graph is required, then add edge (vi, vj). (c) if an undirected graph is required, then add edges (vi, vj) and (vj, vi). (d) If a weighted directed/undirected graph is required, add the edge (vi, vj) with weight c (for undirected, add (vj, vi) with weight c as well). We assume that the weights correspond to cost; if not a suitable mapping from weight to cost must be provided. This process creates an edge between every pair of vertices whose associated agents have a cost less than infinity. If a weighted graph is desired then the cost is used as a weight. Thus we can construct an interaction graph at every time step Gt = (V, Et). In cases where the interaction relation does not change with respect to time we drop the subscripts. In the case of the weighted undirected graph case there is a possibility that Θ(αi, αj, t) /= Θ(αj, αi, t). If this occurs the transformation to a graph does not accurately represent the interaction relation. One should use a directed graph to represent the interaction relation instead. We can convert a graph into an interaction relation by reversing the process above – any two nodes with edges between them will have a finite cost assigned to them; for all other pairs of nodes there will be infinite cost. In the rest of the thesis, unless otherwise mentioned, we will assume that interaction relations are specified as undirected graphs. We will use the terms interaction relation and interaction graph interchangeably, depending upon the context. The interaction relation has significant effects on agreement. We discu...