On the localization of the personalized PageRank of complex networks

Garcı́a

Chaos: An Interdisciplinary Journal of Nonlinear Science

et al. 2013

In this paper we show a new technique to analyze families of rankings. In particular we focus on sports rankings and, more precisely, on soccer leagues. We consider that two teams compete when they change their relative positions in consecutive rankings. This allows to define a graph by linking teams that compete. We show how to use some structural properties of this competitivity graph to measure to what extend the teams in a league compete. These structural properties are the mean degree, the mean strength and the clustering coefficient. We give a generalization of the Kendall's correlation coefficient to more than two rankings. We also show how to make a dynamic analysis of a league and how to compare different leagues. We apply this technique to analyze the four major European soccer leagues: Bundesliga, Italian Lega, Spanish Liga, and Premier League. We compare our results with the classical analysis of sport ranking based on measures of competitive balance.An important feature of a sport competition is the uncertainty about the outcome. Sport industry, governments and followers are interested in having some degree of uncertainty about the competition. In the field of sport rankings the term "competitive balance" attends to measure this degree of uncertainty about the result of a competition. A high competitiveness means that there is high uncertainty about the teams ranking. Classical measures of competitiveness are based on the ratio of wins of each team or other related measures. In this paper we show a new perspective by using techniques from complex networks. We show how to use an ad hoc graph, that we called "competitivity graph", to give some measures of the competitiveness of a family of rankings. As an application we make a comparison of the four major European soccer leagues during 2011-12 season and 2012-13 season.

Section: Comparison With Some Results Obtained By Using Standard mentioning

confidence: 99%

“…In [17] we show a theoretical analysis of PPR, that gives insight about the concept of competitiveness, and we introduced the concept of effective competitors. This latter concept motivated the study of the competitors in the frame of complex networks theory.…”

Section: Introductionmentioning

confidence: 99%

A new method for comparing rankings through complex networks: Model and analysis of competitiveness of major European soccer leagues

Garcı́a

Chaos: An Interdisciplinary Journal of Nonlinear Science

et al. 2013

“…Along the paper we will recall some notation from [13]. Let G = (N , E) be a directed graph where N = {1, 2, .…”

Section: Introduction and Notationmentioning

confidence: 99%

A biplex approach to PageRank centrality: From classic to multiplex networks

Romance

Chaos: An Interdisciplinary Journal of Nonlinear Science

2016

In this paper we present a new view of the PageRank algorithm inspired by multiplex networks. This new approach allows to introduce a new centrality measure for classic complex networks and a new proposal to extend the usual PageRank algorithm to multiplex networks. We give some analytical relations between these new approaches and the classic PageRank centrality measure and we illustrate the new parameters presented by computing them on real underground networks.One major issue in science, from biological systems to information networks and many other fields, is determining the most relevant elements in a given complex system. Centrality measures help spotting the most important nodes of a complex networks by giving a value to each vertex in the system. There are many different centrality measures in Networks Science, including local parameter (such as the in-degree), metric parameters (such as the betweenness centrality) and spectral centralities (such as the eigenvector centrality), but the PageRank centrality plays a relevant role, since it has many relevant applications. This measure is the basic ingredient of the (probably) most famous web searcher (Google), but it also has many applications to different reallife problems, ranging from biological systems to cibersecurity (hacking detection). In this paper we discuss about a biplex approach to the classic PageRank algorithm and how this new insight allows to introduce new centrality measures on complex and multiplex networks that can be useful for ranking the elements of a complex system according to their relevance.

“…This main fact makes that in many real-life networks (such as WWW networks or social networks) it is crucial for a node i to spot other nodes that can be overcome by i in a ranking based on Personalized PageRank, since these nodes are the nodes that actually compete with i in the ranking based on Personalized PageRank. In their work [4], the last three authors of this paper introduced the notion of effective competitors related to nodes that exchanged their relative positions in the Personalized PageRank algorithm when the personalization vector was modified. This kind of problems had already been posed in the literature in social networks, and actually in [3] a necessary condition for a couple of nodes i, j ∈ N to compete was given in terms of the so-called competitivity intervals.…”

Section: Introductionmentioning

confidence: 99%

“…This kind of problems had already been posed in the literature in social networks, and actually in [3] a necessary condition for a couple of nodes i, j ∈ N to compete was given in terms of the so-called competitivity intervals. Nevertheless, the techniques developed in [4] completely solved the problem of finding the competing nodes of a fixed vertex i and gave a computationally efficient solution to it.…”

Section: Introductionmentioning

confidence: 99%

On graphs associated to sets of rankings

Garcı́a

Journal of Computational and Applied Mathematics

et al. 2016

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.