On the Euclidian Metric for Undirected Graphs and Exact Calculations

Abstract

The Euclidian metric gives better results when organizing a multi-agent interaction environment. The analytical basis for this metric is the matrix inverse to a simple matrix description of an undirected graph. In many applied tasks, the usual matrix inversion and real numbers in the framework of finite bit-depth calculations are quite sufficient. However, the unweighted undirected graph is a discrete object, and traditional metrics are able to support processing in the field of rational numbers. Here we show that the Euclidian metric has the same property. Moreover, the space with the dot product is much richer in possibilities in comparison to the spaces where only norms are introduced. Here these possibilities are at the heart of a simple algorithm for calculating the rational entries of the required inverse matrix. Also in the Euclidian space one can use the most important relationship between its elements - orthogonality. The results of numerical experiments are presented.