On graphs associated to sets of rankings

Abstract

a b s t r a c tIn this paper we analyze families of rankings by studying structural properties of graphs. Given a finite number of elements and a set of rankings of those elements, two elements compete when they exchange their relative positions in at least two rankings, and we can associate an undirected graph to a set of rankings by connecting elements that compete. We call this graph a competitivity graph. Competitivity graphs have already appeared in the literature as co-comparability graphs, f -graphs or intersection graphs associated to a concatenation of permutation diagrams. We introduce certain important sets of nodes in a competitivity graph. For example, nodes that compete among them form a competitivity set and nodes connected by chains of competitors form a set of eventual competitors. These sets are analyzed and a method to obtain sets of eventual competitors directly from a set of rankings is shown.

Network analysis: An indispensable tool for curricula design. A real case-study of the degree on mathematics at the URJC in Spain

“…In network analysis, a key point is to devise measures suitable for quantifying the strategic importance of a node, an edge, a set of nodes, or a set of edges, with the goal of identifying the optimal functioning of the system represented by the network [2][3][4][5]. Topological measures, such as different versions of vulnerability, efficiency, centrality, and clustering, can be used to quantify, compare, and rank different configurations for a specific system [2,[6][7][8][9][10][11][12][13]. A task specially difficult to solve for the university management authorities is the design of the different curriculum plans.…”

Mathematical modeling and computational methods

“…B. Royo et al provide in [41] a solution of the long distance routing problem to help pallet and package delivery companies to the decision making, considering a mixed delivery system to improve the use of resources. Finally, R. Criado et al [42] analyze families of rankings by studying structural properties of graphs. They introduce certain important sets of nodes in a competitivity graph.…”

On PageRank versatility for multiplex networks: properties and some useful bounds

“…Now, note that by (7) we have that ( ) = 0, for = 1, … , since each (for = 1, … ) is a Laplacian matrix. Note also that for Corollary 1 we have that ( ) = 0 for = 1, … , , and ( ) = .…”

“…This problem is directly related to understanding the relevance of each element within the structure of a system, which is a first step in understanding its behavior. This problem appears in multiple fields ranging from biological and technological systems to social systems 5,7,11,15,21,36 . A great help in determining the relevance of the nodes of a specific network is provided by centrality measures that allow us to detect the most important nodes by associating a numerical value to each vertex of the system.…”