Ranking dynamics and volatility. Ronald Rousseau KU Leuven & Antwerp University, Belgium

Transcription

1 Ranking dynamis and volatility Ronald Rousseau KU Leuven & Antwerp University, Belgium

2 Joint work with Carlos Garía-Zorita, Sergio Marugan Lazaro and Elias Sanz-Casado Department of Library and Information Siene. Laboratory of Metri Studies on Information (LEMI). Carlos III University of Madrid. C/Madrid 126, Getafe, Madrid, Spain

3 Football ompetitions Reently Criado, Garia, Pedrohe and Romane (2013) studied rankings in European football ompetitions, trying to answer the question Whih ompetition is the most exiting, in the sense that there are many position swithes in the rankings. They answered this question using ompetitivity graphs and derived measures of ompetitiveness. As we will apply their idea to any ranking, not just football ompetitions, and as the term ompetitiveness has a speifi meaning in eonomis we will not use their term ompetitiveness but replae it by ranking dynamis, referring to the phenomenon of hanges in rankings, mainly over time.

4 Our work In this ontribution we will disuss the notion of ranking dynamis, onsider how to measure it, look in more detail to the approah proposed by Criado et al. (2013), introdue a generalization and suggest appliations.

5 The Criado et al. (2013) framework Consider a set E of n elements or nodes (when desribed in a network ontext), denoted as {e 1, e 2,, e n }. Next we onsider an ordered set R of rankings of these n elements. Rankings denoted as 1,, r are ordered (usually in time, referred to as instanes), where eah j is a omplete ranking (no ties) of the n elements at instane j. We say that element e i hanges position with element e j if they exhange their relative positions between two onseutive rankings. Roughly speaking the more position shifts the more dynami a ranking system, e.g., a football ompetition.

6 An example We present a simple example: let n 6 and let E {e 1, e 2,, e 6 }. ( e, e, e, e e, e ) ( ee,, e, ee, e) ( ee,, e, ee, e) ( e, e, e, e e, e ) , , , , 5 6

7 Different aspets when studying the dynamis of rankings 1) The underlying soring method leading to a ranking. 2) Whether the ranking is omplete or if ties are allowed. 3) The timing of the r rankings. 4) One may study rankings of different entities as by Criado et al. who studied four football ompetitions; or rankings based on different riteria for the same entity (journals ranked by IF, immediay index, total number of reeived itations). 5) Dynamis, see next slide.

8 Dynamis a) In the football ase hanges between onseutive rankings (weeks) are small as the maximum hange in the underlying sore is 3, but if one onsiders the final ranking at the end of the season then anything is (theoretially) possible. b) Another aspet is whether numbers on whih rankings are based are ompletely independent between events (as for yearly final rankings in a national football ompetition) or are umulative (as in the football ompetition data based on weekly or h-indies for researhers). ) Finally, another dynami aspet is the fat that one must take into aount that some teams/journals enter or leave the rankings.

9 Representing ompetitiveness in the sense of Criado et al. Criado et al. (2013) only studied the ase of a fixed number of elements without ties in the ranking. We first desribe their framework. They represent a set of rankings R, as a weighted network, with nodes (e j ) j1,, n. Two nodes are linked with weight k if these nodes perform k position shifts. If there are r instanes, then there are at most r-1 position shifts (reall that position shifts are always onsidered between onseutive rankings or instanes). We present a simple example: let n 6 and let E {e 1, e 2,, e 6 }. ( e, e, e, e e, e ) ( ee,, e, ee, e) ( ee,, e, ee, e) ( e, e, e, e e, e ) , , , , 5 6

10 Matrix and network representations ( e, e, e, e e, e ) ( ee,, e, ee, e) ( ee,, e, ee, e) ( e, e, e, e e, e ) , , , , 5 6 e e e e e e The number of position shifts between two elements an be expressed in a (symmetri) matrix M or, equivalently, in a network. We see that the four possible ases: no shifts, 1, 2 and 3 shifts our in this example. As a weighted network, alled the ranking dynamis graph this leads to the network shown on the left.

11 Dynamis and volatility (Criado) Ranking dynamis (ompetitiveness in the ase of (Criado et al., 2013)) is a property of an ordered set of rankings. The sum of all position shifts of an element is alled its volatility. Using the matrix representation M (m ij ) ij, the volatility of element n e j is defined as: vvv e j i1 m ii.

12 Measuring ranking dynamis (Criado) Absolute measure This measure is defined as the sum of the node strengths of all nodes in the ranking dynamis graph. This is twie the sum of all weights, or the sum of all elements in the matrix M. Seond measure: normalized form The normalized mean strength, denoted as NS is the normalized sum of all node strengths in the ranking dynamis graph: NS(R) n j 1 d S ( e ) nn ( 1)( r 1) j

13 Properties of measures of ranking dynamis (a) Axiom 1: Relabelling. Relabelling the elements (tehniality: applying a permutation) must not hange the resulting measure for dynamis. The dynamis is only determined by a hange in overall onfiguration. This anonymity axiom must always be satisfied by a bona fide measure of ranking dynamis. Axiom 2: Repliation. If the ordered set of r rankings, R ( 1, 2,, r ) is replaed by the ordered set of r+1 rankings R ( 1, 1, 2,, r ), where one of the j s (not neessarily the first) is dupliated, then this has no influene on absolute ranking dynamis. Of ourse, if more than one dupliation ours this has no influene either. A repliation operation has no influene on the network and its weights. Note that dupliation must our between onseutive rankings. Repliation must derease relative ranking dynamis.

14 Properties of measures of ranking dynamis (b) Axiom 3: No hange. If R (,,,, ) then there are no hanges in rankings for onseutive instanes, hene there is no dynamis. We require that a measure of ranking dynamis must take the value zero for this ase. The no-hange operation happens if one always applies alphabetial order. Axiom 4: Adding an element whih is always ranked first or last. When this operation is applied there must be no influene on absolute measures but a derease on relative measures of ranking dynamis. Axiom 5. Instane reversion. If the order of the instanes is reversed then this has no influene on ranking dynamis

15 New entrants, leavers and ties In real-world rankings it often happens that new entrants join the set of elements or that some leave. Moreover, it may easily happen that ties our. Hene, we will adapt the previous framework so that one an take new entrants, leavers and ties into aount.

16 Notation (a) As in the Criado framework we onsider a stritly ordered row of r instanes. Eah instane is a ranked set of elements, where ties are allowed. By definition, tied elements have the same rank. The symbol S denotes the set of all (different) elements ourring in the r instanes; let #S n. We add to eah instane those elements in S whih are absent in this partiular instane, ranking them as ties on the last position. The original elements in a given instane are alled the ative elements, the added ones are alled the inative elements. Being ative or inative is referred to as the state of an element in a given instane. Ative and inative elements are separated by the symbol ;.

17 Notation (b) Denoting a tie between elements e and f as..., ef,,... and assuming that elements x, y and z are missing in instane, this means that (s 1, s 2,, s n-3 ) is rewritten as s1, s2,..., sn 3; xyz,, The rank of an element s in instane is denoted as r (s).

18 Priniples to alulate a dynamis indiator We will alulate the value of the ranking dynamis indiator NS. In its absolute form this indiator is the sum of the volatility sores of all elements in S. Besides position shifts, also leaving or entering, i.e. beoming ative or beoming inative are signs of dynamis and are taken into aount.

19 Soring (a) We obtain a volatility sore for eah element, e, by omparing with eah other element. This sore is obtained by a omparison of the positions and states of e and the other element in onseutive instanes, leading to partial volatility sore. The sum of all these volatility sores is the (global) volatility sore of element e. Partial volatility sores are symmetri: the partial volatility sore between elements e and f is the same as the one between f and e. In eah omparison between instanes the sore either stays unhanged or inreases by one. The initial sore is zero.

20 Soring (b) When the state of (at least) one of the two elements hanges from ative (A) to inative (I) or vie versa then the sore inreases; if both elements do not hange state, then we ompare the relative position of e and the other element between two onseutive instanes. If < hanges to > or vie versa, then the volatility sore inreases; if there is no hange in the relative ranking of e and the other element the sore stays the same. If the two elements are ative and beome tied then the sore does not hange, but the previous relative position is kept in memory. If the two elements were tied and are still tied, then nothing hanges; if they were tied and are not tied anymore then the last time they were not tied determines if there is a position hange and hene if the volatility sore inreases or not.

21 An illustration ( stuv,,, ) ( suwx,,, ) (, suwy,, ) (,,,) zyus The set S is here {s,t,u,v,w,x,y,z}, with #S8. Consequently the four instanes are rewritten as: stuvwxyz,,, ;,,, suwxtvyz,,, ;,,, ( suwytvxz,,, ;,,, ) ( zyustvwx,,, ;,,, )

22 Soring: first steps stuvwxyz,,, ;,,, suwxtvyz,,, ;,,, ( suwytvxz,,, ;,,, ) ( zyustvwx,,, ;,,, ) Now we illustrate, step by step, how we ount the partial volatility sore for {t,x}. We start with showing the initial position. Here, this initial position is: (t,a,x,i,1,<,0,*). A first omparison yields: (t,i,x,a,2,>,1,*): the sore beomes 1 beause t beomes inative (and moreover x beomes ative, but this has no influene on the sore anymore). Next we have: (t,i,x,i,3,,2,>): x beomes inative (leading to an inrease in the sore); we note that moreover t and x are tied and the last time they were not tied x was ranked before t (t>x) (t,i,x,i,4,,2,>): this is the end result: the partial volatility sore for the pair {t,x} is 2.

23 Complete results elements total elements total s-t u-w s-u u-x s-v u-y s-w u-z s-x v-w s-y v-x s-z v-y t-u v-z t-v w-x t-w w-y t-x w-z t-y x-y t-z x-z u-v y-z

24 The final result The global volatility of the elements in S are shown below. Clearly leavers and entrants have the highest volatility and hene ontribute most to the overall ranking dynamis (also beause in this simple example there are relatively many of them!). Finally, the total strength is 102 and NS, the normalized mean strength, is 102/(8x7x3) Elements s t u v w x y z Volatility

25 Stability In many appliation not the dynamis but the stability of rankings is of importane. Clearly, 1 NS yields a relative measure of stability. In the previous example the orresponding stability measure, defined as 1 NS, is

26 An example: JIF rankings ( ) JCR ategory # Journals r NS(R) Eonomis Information Siene & Library Siene (LIS) Biology

27 Thank you for your attention