A Bellman–Ford Algorithm for the Path-Length-Weighted Distance in Graphs

Abstract

Consider a finite directed graph without cycles in which the arrows are weighted by positive weights. We present an algorithm for the computation of a new distance, called path-length-weighted distance, which has proven useful for graph analysis in the context of fraud detection. The idea is that the new distance explicitly takes into account the size of the paths in the calculations. It has the distinct characteristic that, when calculated along the same path, it may result in a shorter distance between far-apart vertices than between adjacent ones. This property can be particularly useful for modeling scenarios where the connections between vertices are obscured by numerous intermediate vertices, such as in cases of financial fraud. For example, to hide dirty money from financial authorities, fraudsters often use multiple institutions, banks, and intermediaries between the source of the money and its final recipient. Our distance would serve to make such situations explicit. Thus, although our algorithm is based on arguments similar to those at work for the Bellman–Ford and Dijkstra methods, it is in fact essentially different, since the calculation formula contains a weight that explicitly depends on the number of intermediate vertices. This fact totally conditions the algorithm, because longer paths could provide shorter distances—contrary to the classical algorithms mentioned above. We lay out the appropriate framework for its computation, showing the constraints and requirements for its use, along with some illustrative examples.

Keywords

• graph
• distance
• Bellman–Ford
• algorithm
• path-length-weighted