Comparing rankings by means of competitivity graphs: structural properties and computation (original) (raw)
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Competitivity graphs analysis and structural comparison of rankings
A complex networks based method is introduced for comparing different complete rankings of a finite family of elements. The concepts of competitivity graph and evolutive competitivity graph are introduced as the main tools for analyzing an (ordered) family of rankings of a fixed set of elements. It is shown how the structural properties of these competitivity graphs give deep information about the competitiveness of the elements according to the rankings considered. The relationships between competitivity graphs and some other wellknown families of graphs, such as permutation graphs, comparability graphs and chordal graphs are also presented. Finally some applications are presented, including the analysis of sports rankings and, more precisely, the study of European soccer leagues.
SIAM Journal on Discrete Mathematics, 1998
A vertex (edge) coloring c : V ! f1;2;:::;tg (c 0 : E ! f1;2;:::; tg) of a graph G = (V; E) is a vertex (edge) t-ranking if for any two vertices (edges) of the same color every path between them contains a vertex (edge) of larger color. The vertex ranking number r (G) (edge ranking number 0 r (G)) is the smallest value of t such that G has a vertex (edge) t-ranking. In this paper we study the algorithmic complexity of the vertex ranking and edge ranking problems. Among others it is shown that r (G) can be computed in polynomial time when restricted to graphs with treewidth at most k for any xed k. We characterize those graphs where the vertex ranking number r and the chromatic number coincide on all induced subgraphs, show that r (G) = (G) implies (G) = !(G) (largest clique size) and give a formula for 0 r (K n ).
A constructive solution to a problem of ranking tournaments
Discrete Mathematics, 2019
A tournament is an oriented complete graph. The problem of ranking tournaments was firstly investigated by P. Erdős and J. W. Moon. By probabilistic methods, the existence of "unrankable" tournaments was proved. On the other hand, they also mentioned the problem of explicit constructions. However, there seems to be only a few of explicit constructions of such tournaments. In this note, we give a construction of many such tournaments by using skew Hadamard difference sets which have been investigated in combinatorial design theory. 2010 Mathematics Subject Classification. 05C20.
2007
We provide a comprehensive picture of how to compare partial rankings, that is, rankings that allow ties. We propose several metrics to compare partial rankings and prove that they are within constant multiples of each other.
Ranking the vertices of a complete multipartite paired comparison digraph
Discrete Applied Mathematics, 1996
A paired comparison digraph (abbreviated to PCD) D = (V, A) is a weighted digraph in which the sum of the weights of arcs, if any, joining two distinct vertices equals one. A one-to-one mapping α from V onto {1, 2, ..., |V |} is called a ranking of D. For every ranking α, an arc vu ∈ A is said to be forward if α(v) < α(u), and backward, otherwise. The length of an arc vu is (vu) = (vw)|α(v) − α(u)|, where (vw) is the weight of vu. The forward (backward) length f D (α) (b D (α)) of α is the sum of the lengths of all forward (backward) arcs of D. A ranking α is forward (backward) optimal if f (α) is maximum (b(α) is minimum). M. Kano (Disc. Appl. Math., 17 (1987) 245-253) characterized all backward optimal rankings of a complete multipartite PCD D and raised the problem to characterize all forward optimal rankings of a complete multipartite PCD L. We show how to transform the last problem into the single machine job sequencing problem of minimizing total weighted completion time subject to precedence "parallel chains" constraints. This provides an algorithm for generating all forward optimal rankings of L as well as a polynomial algorithm for finding * Corresponding author. This work was supported by the Danish Research Council under grant no. 11-0534-1. The support is gratefully acknowledged. the average rank of every vertex in L over all forward optimal rankings of L.
Greedy rankings and arank numbers
Information Processing Letters, 2009
A ranking on a graph is an assignment of positive integers to its vertices such that any path between two vertices of the same rank contains a vertex of strictly larger rank. A ranking is locally minimal if reducing the rank of any single vertex produces a non ranking. A ranking is globally minimal if reducing the ranks of any set of vertices produces a non ranking. A ranking is greedy if, for some ordering of the vertices, it is the ranking produced by assigning ranks in that order, always selecting the smallest possible rank. We will show that these three notions are equivalent. If a ranking satisfies one property it satisfies all three. As a consequence of this and known results on arank numbers of paths we improve known upper bounds for on-line ranking.
Computing distances between partial rankings
Information Processing Letters, 2009
We give two efficient algorithms for computing distances between partial rankings (i.e. rankings with ties). Given two partial rankings over n elements, and with b and c equivalence classes, respectively, our first algorithm runs in O (n log n/ log log n) time, and the second in O (n log min{b, c}) time.
Discrete Mathematics, 2000
A (vertex) k-ranking of a graph G =(V; E) is a proper vertex coloring ' : V → {1; : : : ; k} such that each path with endvertices of the same color i contains an internal vertex of color ¿i +1. In the on-line coloring algorithms, the vertices v 1; : : : ; vn arrive one by one in an unrestricted order, and only the edges inside the set {v1; : : : ; vi} are known when the color of vi has to be chosen. We characterize those graphs for which a 3-ranking can be found on-line. We also prove that the greedy (First-Fit) on-line algorithm, assigning the smallest feasible color to the next vertex at each step, generates a (3 log 2 n)-ranking for the path with n¿2 vertices, independently of the order in which the vertices are received.