Comparing series of rankings with ties by using complex networks: An analysis of the Spanish stock market (IBEX-35 index)

Abstract

In this paper we extend the concept of Competitivity Graph to compare series of rankings with ties (partial rankings). We extend the usual method used to compute Kendall's coefficient for two partial rankings to the concept of evolutive Kendall's coefficient for a series of partial rankings. The theoretical framework consists of a four-layer multiplex network. Regarding the treatment of ties, our approach allows to define a tie between two values when they are close enough, depending on a threshold. We show an application using data from the Spanish Stock Market; we analyse the series of rankings defined by 25 companies that have contributed to the IBEX-35 return and volatility values over the period 2003 to 2013.

Ranking dynamics and volatility

“…In this new framework measures must still satisfy all requirements discussed in section 5. Differences and similarities between our work and that of Pedroche et al (2015) will be discussed in section 7.…”

“…2) Whether the ranking is complete or if ties are allowed. Criado et al (2013) studied rankings without ties, but they removed this restriction in Pedroche et al (2015). Indeed, ties form an essential aspect of real-world rankings.…”

“…Moreover, it may easily happen that ties occur. This last aspect was taken into account by Pedroche et al (2015). Hence, we will adapt the previous framework so that one can take new entrants, leavers and ties into account.…”

“…As we will apply their idea to any ranking, not just football competitions, and as the term competitiveness has a specific meaning in economics (Ambec et al, 2013;Buser et al, 2014;World Economic Forum, 2016), we will not use their term competitiveness but replace it by the term ranking dynamics, referring to the phenomenon of changes in rankings, mainly over time. In a follow-up paper Pedroche, Criado, García, Romance and Sánchez (2015), extended their approach to partial rankings, i.e., rankings in which ties may occur. In this generalization the authors apply the theory of multiplex networks (Boccaletti et al, 2014) and introduce a generalization of Kendall's distance with penalty p (Fagin et al, 2006).…”

“…In this generalization the authors apply the theory of multiplex networks (Boccaletti et al, 2014) and introduce a generalization of Kendall's distance with penalty p (Fagin et al, 2006). Pedroche et al (2015) apply their generalization to a study of the Spanish stock market.…”