A graph interpretation of the least squares ranking method

The paper aims at analyzing the least squares ranking method for generalized tournaments with possible missing and multiple paired comparisons. The bilateral relationships may reflect the outcomes of a sport competition, product comparisons, or evaluation of political candidates and policies. It is shown that the rating vector can be obtained as a limit point of an iterative process based on the scores in almost all cases. The calculation is interpreted on an undirected graph with loops attached to some nodes, revealing that the procedure takes into account not only the given object's results but also the strength of objects compared with it. We explore the connection between this method and another procedure defined for ranking the nodes in a digraph, the positional power measure. The decomposition of the least squares solution offers a number of ways to modify the method.