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The flow network method

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Abstract

In this paper we propose an in-depth analysis of a method, called the flow network method, which associates with any network a complete and quasi-transitive binary relation on its vertices. Such a method, originally proposed by Gvozdik (Abstracts of the VI-th Moscow conference of young scientists on cybernetics and computing. Scientific Council on Cybernetics of RAS, Moscow, p 56, 1987), is based on the concept of maximum flow. Given a competition involving two or more teams, the flow network method can be used to build a relation on the set of teams which establishes, for every ordered pair of teams, if the first one did at least as good as the second one in the competition. Such a relation naturally induces procedures for ranking teams and selecting the best k teams of a competition. Those procedures are proved to satisfy many desirable properties.

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Notes

  1. The National Football League (NFL) and the NCAA Division I Football Bowl Subdivision are examples of competitions where there are teams which never play each other on the field.

  2. We describe a competition by a table in which every row has the shape

    meaning that the matches involving the teams x and y were \(m+n\) and that x won m times and y won n times.

  3. We stress that Belkin and Gvozdik (1989) mainly focus on the problem of building rankings.

  4. A similar interpretation appears in Patel (2015).

  5. Due to the identification between competitions and networks, we are going to freely use the terminology of competitions for networks too.

  6. Note that in many applications networks capacities are allowed to be nonnegative real numbers and networks are identified with their adjacency matrix.

  7. The Copeland method is sometimes called net flow method (Bouyssou 1992).

  8. We emphasize that some of the properties considered by González-Díaz et al. (2014) are satisfied by the flow network method as described in course of the paper.

  9. The flow network method fulfils the three main properties stated in that paper. Property I (opponent strength) has been already discussed; Property II (incentive to win) is the content of Proposition 11; Property III (sequence of matches) follows from the very definition of the method.

  10. Note that the (normalized) invariant and the fair-bets methods are also known as the (normalized) long path and the Markov methods, respectively.

  11. A tournaments is a complete and asymmetric digraph. Tournaments can be identified with networks whose capacities are 0 or 1 and such that the sum of the capacities of any pair of opposite arcs is 1. They are used to represent round robin-competitions.

  12. A balanced network is a network whose sum of the capacities of any pair of opposite arcs is constant. Balanced networks represent competitions where any pair of teams confront each other the same number of time.

  13. The equality between the flow and the Borda network methods on balanced networks is stated, without proof, in Belkin and Gvozdik (1989).

  14. Complete and quasi-acyclic relations always admit linear refinements (Proposition 27).

  15. That fact can be easily checked considering the network in Example 56.

  16. Given a relation R on a nonempty finite set V, a k-maximum set of R is a subset W of size k of V having the property that, for every \(x\in W\) and \(y\not \in W\), \((x,y)\in R\). Complete and quasi-acyclic relations always admit k-maximum sets (Proposition 30).

  17. Weak tournaments and partial tournaments can be naturally identified with networks whose arcs always have capacities that are 0 or 1 and such that the sum of the capacities of each pair of opposite arcs is at least 1 and at most 1 respectively.

  18. Those authors use the term anonymity instead of neutrality.

  19. For a more general approach, see also the main result in Hartmann and Schneider (1993).

  20. We thank László Lovász for useful advices about this proposition.

  21. That fact seems to be known in the literature even though we could not find a precise reference. Thus, for completeness, we provide a proof.

  22. Note that, on the network \(N_{\textsf {D}}\), the Kemeny network rule and the ranked pair network rule are equal to \(\{L, \,\textsc {c}\succ \textsc {b}\succ \textsc {a},\, \textsc {a}\succ \textsc {c}\succ \textsc {b}\}\), while \(\mathfrak {F}_{\diamond }(N_{\textsf {D}})=\{L\}\).

  23. Actually, this definition is less general than the one in the paper of Dutta and Laslier. Nevertheless, it is sufficient for our purposes.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica U. Dini, Università degli Studi di Firenze, viale Morgagni 67/a, 50134, Florence, Italy

    Daniela Bubboloni

  2. Dipartimento di Scienze per l’Economia e l’Impresa, Università degli Studi di Firenze, via delle Pandette 9, 50127, Florence, Italy

    Michele Gori

Authors
  1. Daniela Bubboloni
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  2. Michele Gori
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Correspondence to Michele Gori.

Additional information

We wish to thank an anonymous referee for letting us know the existence of the papers by Gvozdik (1987) and Belkin and Gvozdik (1989) (in Russian), where the flow network method was first formulated. We also thank Andrey Sarychev for translating the mentioned papers. Daniela Bubboloni was partially supported by GNSAGA of INdAM.

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Bubboloni, D., Gori, M. The flow network method. Soc Choice Welf 51, 621–656 (2018). https://doi.org/10.1007/s00355-018-1131-7

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