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A Bellman-Ford algorithm for the path-length-weighted distance in graphs

R. Arnau, J. M. Calabuig, L. M. García Raffi, E. A. Sánchez Pérez, S. Sanjuan

TL;DR

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, and is based on arguments similar to those at work for the Bellman–Ford and Dijkstra methods.

Abstract

Consider a finite directed graph without cycles in which the arrows are weighted. 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. 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. We lay out the appropriate framework for its computation, showing the constraints and requirements for its use, along with some illustrative examples.

A Bellman-Ford algorithm for the path-length-weighted distance in graphs

TL;DR

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, and is based on arguments similar to those at work for the Bellman–Ford and Dijkstra methods.

Abstract

Consider a finite directed graph without cycles in which the arrows are weighted. 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. 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. We lay out the appropriate framework for its computation, showing the constraints and requirements for its use, along with some illustrative examples.

Paper Structure

This paper contains 6 sections, 3 theorems, 20 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main definitions and context
  3. Previous to the algorithm
  4. The algorithm
  5. Examples of graphs
  6. Special cases: constraints on weights and proximity matrix to improve the efficiency of the algorithm

Key Result

Proposition 3.2

Consider a graph $G$ and the corresponding elements considered above. Fix $a, b \in V$ and let $P, P' \in \mathcal{P}(a,b)$ such that $P \preceq P'$. Then, for any $c \in V$ and $R \in \mathcal{P}(b,c)$, we have that the paths $Q = P \sqcup R$ and $Q' = P' \sqcup R$ in $\mathcal{P}(a,c)$ satisfies $

Figures (3)

  • Figure 1: A graph in which a Bellman-Ford type algorithm to calculate the smallest distance between two nodes would not work. The weight of each edge is written on the corresponding arrow.
  • Figure 2: Example of a Pareto front associated to the possible paths in a graph. The framed dots represent all possible paths from one node to another. The red dots represent the set of paths needed to calculate distances, with $\mathcal{P}^{*}$ being the set of all of them.
  • Figure 10: Example of a graph satisfying \ref{['eq:condition2_2']}.

Theorems & Definitions (6)

  • Example 3.1
  • Proposition 3.2
  • proof
  • Proposition 6.1
  • proof
  • Proposition 6.2