ranking analysis by using complex networks: structure and...

Introduction and basic definitionsStructural properties of Competitivity Graphs

(Weighted) Competitivity Graphs of rankings (without ties)(Weighted) Competitivity Graphs of rankings with ties

Conclusions

Ranking analysis by using complex networks:Structure and some applications

Miguel Romance(joint work with R.Criado, E.Garcıa, F.Pedroche & V.E.Sanchez)

Applied Mathematics Dept. Appl. and Biol. Mathematics Lab.ESCET Center for Biomedical Technology (CTB)

Universidad Rey Juan Carlos Universidad Politecnica de Madrid

IX EITA Research Meeting inApproximation Theory,

Alquezar, October 19th 2014

Miguel Romance Ranking analysis by using complex networks

Conclusions

Contents

1 Introduction and basic definitionsData, information and rankings(Un-weighted) Competitivity Graphs

2 Structural properties of Competitivity GraphsPermutation, Comparability and Competitivity GraphsMain result

3 (Weighted) Competitivity Graphs of rankings (without ties)Definition and propertiesAn application to the analysis of Major European Soccer Leagues

4 (Weighted) Competitivity Graphs of rankings with tiesDefinition and propertiesAn application to the analysis of the IBEX35 Index

5 Conclusions

Conclusions

Data, information and rankings(Un-weighted) Competitivity Graphs

Data, information and rankings

We are surrounded by tons of data.

How can we extract the most valuable information containedin these data?

We can construct rankings that give the data ordered by theirrelevance.

Conclusions

An example: Web data

There more than 107 web pages about the conceptApproximation Theory.

The PageRank algorithm of Google produce a ranking of allthese web pages....but there are other web-rankers...

Conclusions

There more than 107 web pages about the conceptApproximation Theory.The PageRank algorithm of Google produce a ranking of allthese web pages.

...but there are other web-rankers...

Conclusions

There more than 107 web pages about the conceptApproximation Theory.The PageRank algorithm of Google produce a ranking of allthese web pages....but there are other web-rankers...

Conclusions

The PageRank algorithm of Google produce a ranking of allthese web pages.

Conclusions

The PageRank algorithm of Google produce a ranking of allthese web pages.

Conclusions

Given a Data Set, we can associate it several rankings.

A similar situation occurs in many other contexts...

Conclusions

Other example: Sport rankings

A slightly different example can be found in Sport Rankings, wherewe have a sequence of temporal rankings (one ranking after eachmatch day).

Conclusions

Given a Data Set with several rankings associated, whatinformation can we obtain?

Segal’s Law

A man with one watch knows what time it is, a man with two isnever sure.

Segal’s Law for Rankings

A man with one ranking knows what it is relevant, a man with twois never sure.

Question

Can mathematics help solving this problem/limitation?

Conclusions

Segal’s Law

Question

Conclusions

Segal’s Law

Question

Conclusions

Segal’s Law

Question

Conclusions

Definition 1.1.

Given a set N = {1, · · · , n} with n ∈ N, a ranking (without ties) isa bijection c : N −→ N . We will denote c by

c : (c(1), c(2), · · · , c(n)),

where c(i) represents the position of element i in the ranking c .

There is a huge scientific literature analyzing different rankings,including mathematician, physicist, economist, social scientists,and other, such as

M.G.Kendall and C.Spearman (in the 1930’s),

P.Diacoris, R.Fagin, E.J.Emond, L.Groot, ...

Conclusions

Definition 1.1.

Given a set N = {1, · · · , n} with n ∈ N, a ranking (without ties) isa bijection c : N −→ N . We will denote c by

c : (c(1), c(2), · · · , c(n)),

where c(i) represents the position of element i in the ranking c .

There is a huge scientific literature analyzing different rankings,including mathematician, physicist, economist, social scientists,and other, such as

M.G.Kendall and C.Spearman (in the 1930’s),

P.Diacoris, R.Fagin, E.J.Emond, L.Groot, ...

Conclusions

A good reference is the book of A.N.Langville and C.D.Meyer.

Conclusions

(Un-weighted) Competitivity Graphs

Main Goal

Given a family of rankings R = {c1, · · · , cr} of a setN = {1, · · · , n}, associate to R a graph that gives valuableinformation about R.

Definition 1.2. [CGPR2]

Given a family of rankings R = {c1, · · · , cr} of a setN = {1, · · · , n} with n ∈ N, we define the (un-weighted)Competitivity Graph Gc(R) = (N ,ER), where {i , j} ∈ ER if thereare cs , ct ∈ R such that cs(i) < cs(j) but ct(i) > ct(j). in thiscase we will say that there is a crossing between i and j .

Conclusions

Main Goal

Given a family of rankings R = {c1, · · · , cr} of a setN = {1, · · · , n}, associate to R a graph that gives valuableinformation about R.

Given a family of rankings R = {c1, · · · , cr} of a setN = {1, · · · , n} with n ∈ N, we define the (un-weighted)Competitivity Graph Gc(R) = (N ,ER), where {i , j} ∈ ER if thereare cs , ct ∈ R such that cs(i) < cs(j) but ct(i) > ct(j). in thiscase we will say that there is a crossing between i and j .

Conclusions

Consider the following rankings of the set N = {1, · · · , 6}

N c1 c2 c3 c4

1 1 1 1 42 2 4 2 23 3 2 4 14 4 3 5 65 5 5 3 56 6 6 6 3

Conclusions

N c1 c2 c3 c4

1 1 1 1 42 2 4 2 23 3 2 4 14 4 3 5 65 5 5 3 56 6 6 6 3

Conclusions

N c1 c2 c3 c4

1 1 1 1 42 2 4 2 23 3 2 4 14 4 3 5 65 5 5 3 56 6 6 6 3

There is a link between 1 and 2 since c1(1) < c1(2) butc4(1) > c4(2).

Conclusions

N c1 c2 c3 c4

1 1 1 1 42 2 4 2 23 3 2 4 14 4 3 5 65 5 5 3 56 6 6 6 3

There is no link between 3 and 4 since ck(3) < ck(4) for all1 ≤ k ≤ 4.

Conclusions

Main Idea

Given a family of rankings R = {c1, · · · , cr} of a setN = {1, · · · , n}, then structural properties of the (un-weighted)Competitivity Graph Gc(R) = (N ,ER) correspond with propertiesof the rankings.

The more dense Gc(R) is, the more different the rankings{c1, · · · , cr} are.

The neighbors of each i ∈ N are the elements j ∈ N thatcross with i , i.e. they are the competitors of i according tothe rankings {c1, · · · , cr}.

Conclusions

Main Idea

Conclusions

Main Idea

Conclusions

Permutation, Comparability and Competitivity GraphsMain result

Structural properties of Competitivity Graphs

If we take a family of rankings R = {c1, · · · , cr} of a setN = {1, · · · , n}, we construct its Competitivity GraphGc(R) = (N ,ER).

Question 2.1.

Given an undirected and un-weighted graph G = (N ,E ) withN = {1, · · · , n},, can we find a family of rankingsR = {c1, · · · , cr} such that G = Gc(R)?

NO, in general!

Question 2.2.

Can we relate the family of Competitivity Graphs with other classicfamilies of graphs?

Conclusions

Question 2.1.

NO, in general!

Question 2.2.

Conclusions

Question 2.1.

NO, in general!

Question 2.2.

Conclusions

Question 2.1.

NO, in general!

Question 2.2.

Conclusions

Permutation Graphs

Definition 2.3.

A graph G = (N ,E ) is a permutation graph if its verticesrepresent the elements of a permutation and each one of its edgescorrespond to a pair of elements that are reversed by thepermutation.

Conclusions

Permutation Graphs

Definition 2.3.

A graph G = (N ,E ) is a permutation graph if its verticesrepresent the elements of a permutation and each one of its edgescorrespond to a pair of elements that are reversed by thepermutation.

Conclusions

Permutation graphs

Every Permutation graph is a Competitivity graph given by tworankingsG is given by the permutation

(4 3 5 1 2)

Then if we take R = {c1, c2}N c1 c2

1 1 42 2 53 3 24 4 15 5 3

Then G = Gc(R).

Conclusions

Permutation graphs

(4 3 5 1 2)

1 1 42 2 53 3 24 4 15 5 3

Then G = Gc(R).

Conclusions

Permutation graphs

(4 3 5 1 2)

1 1 42 2 53 3 24 4 15 5 3

Then G = Gc(R).

Conclusions

Comparability Graphs

Definition 2.4.

Given any partial ordered set (N ,�) we can associate a directedgraph G� to (N ,�) as follows: the vertex set is N and there is alink from i to j , i 6= j , if i � j .A graph G = (N ,E ) is a Comparability graph if it is thenon-directed graph obtained after removing orientation in G� forsome partial order � of N .

Conclusions

Definition 2.4.

Conclusions

Definition 2.4.

Conclusions

Chordal Graphs

Definition 2.5.

A graph G = (N ,E ) is Chordal if each of its cycles of four or morevertices has a chord, which is an edge joining two nodes that arenot adjacent in the cycle.

Conclusions

Chordal Graphs

Definition 2.5.

Conclusions

Chordal Graphs

Definition 2.5.

Conclusions

Chordal Graphs

Definition 2.5.

Conclusions

Main result

Theorem 2.6. [CGPR2]

Let 6 ≤ n ∈ N. If we denote

1 G the set of all undirected graphs of n nodes.

2 CG the set of all Competitivity graphs of n nodes.

3 CPG the set of all Comparability graphs of n nodes.

4 CHG the set of all Chordal graphs of n nodes.

then each region in the figure is non-empty.

Conclusions

Ingredients of the proof

The main tools used in the proof of this result are:

1 A lemma that shows that every Competitivity graph is thesum of several Permutation graphs [CGPR2].

2 A result of M.C.Golumbic et al. (1983) that relatescomparability graphs with sums of permutation graphs.

3 A lemma that characterizes the Competitivity graphs in termsof the existence of some partial orders with some extraproperties (called semi-cohesive orders) that extends someresults of S.V.Gervacio et al. (2013) about Permutationgraphs [CGPR2].

Conclusions

Definition and propertiesAn application to the analysis of Major European Soccer Leagues

Definition and properties

Given an (ordered) sequence of (temporal) rankingsR = {c1, · · · , cr} of a set N = {1, · · · , n}, there is a evolutive ortemporal crossing between i ∈ N and j ∈ N if there is1 ≤ k ≤ r − 1 such that rk(i) < rk(j) and rk+1(i) > rk+1(j) orrk(i) > rk(j) and rk+1(i) < rk+1(j).

Given an (ordered) sequence of rankings complex networkR = {c1, · · · , cr} of a set N = {1, · · · , n}, the Evolutive(weighted) Competitivity Graph G e

c (R) = (N ,ER) is a undirectedand weighted graph, such that the weight of each link {i , j} ∈ ERis the number of temporal crossing between i and j .

Conclusions

Given an (ordered) sequence of (temporal) rankingsR = {c1, · · · , cr} of a set N = {1, · · · , n}, there is a evolutive ortemporal crossing between i ∈ N and j ∈ N if there is1 ≤ k ≤ r − 1 such that rk(i) < rk(j) and rk+1(i) > rk+1(j) orrk(i) > rk(j) and rk+1(i) < rk+1(j).

Given an (ordered) sequence of rankings complex networkR = {c1, · · · , cr} of a set N = {1, · · · , n}, the Evolutive(weighted) Competitivity Graph G e

c (R) = (N ,ER) is a undirectedand weighted graph, such that the weight of each link {i , j} ∈ ERis the number of temporal crossing between i and j .

Conclusions

An example

N c1 c2 c3 c4

1 1 1 1 42 2 4 2 23 3 2 4 14 4 3 5 65 5 5 3 56 6 6 6 3

Conclusions

The relationship between two rankings c1 and c2 was studied byM.G.Kendall in the 1930’s

Definition 3.3.

Given an (pair) of rankings c1 and c2 of a set N = {1, · · · , n}, theKendall’s coefficient τ(c1, c2) is defined by

τ(c1, c2) =K (c1, c2)− K (c1, c2)(n

) ∈ [−1, 1],

K (c1, c2) is the number of pairs of nodes that do not have a(temporal) crossing,

K (c1, c2) is the number of pairs of nodes that have a(temporal) crossing.

. Miguel Romance Ranking analysis by using complex networks

Conclusions

Theorem 3.4.[CGPR1]

Given an (ordered) sequence of (temporal) rankingsR = {c1, · · · , cr} of a set N = {1, · · · , n} and its EvolutiveCompetitivity Graph G e

c (R) = (N ,ER), if

NS(G ec (R)) =

)(r − 1)

∑i ,j∈N

w({i , j})

is the normalized mean strength of G ec (R), then

2NS(G ec (R)) = 1− τ(R), where τ(R) is the Kendall’s coefficient

Conclusions

Analysis of Major European Soccer Leagues

In [CGPR1] we consider the day match rankings of the EuropeanSoccer Leagues

Bundesliga (Germany),

Lega Serie A (Italy),

Liga BBVA (Spain),

Premier League (United Kingdom),

along 2011-12 and 2012-13.

Conclusions

Analysis of Major European Soccer Leagues

The Competitivity Graphs for the season 2012-13 is

Conclusions

An application to the analysis of Major European SoccerLeagues

The Normalized Node Strength for the seasons 2011-12 and2012-13 is

Conclusions

We performed many other structural analysis such as:

Other structural global parameters, including the node degree,clustering, diameter,...

In the mesoscale, the community structure was considered.

A sensibility studied was done in term of the number ofcrossings.

All these analysis produced similar conclusions about the leagues.

Conclusions

Definition and propertiesAn application to the analysis of the IBEX35 Index

Definition 4.1.

Given a set N = {1, · · · , n} with n ∈ N, a ranking with ties is anapplication (probably non injective) c : N −→ N . If c(i), c(j),then i and j are tied.

Example 4.2.

σ1 σ2 σ3 σ4 σ5

2 5 5, 6 2, 5 13, 5 6 2 3, 6 61, 6 2, 3 3 4 44 1 1, 4 1 2, 3, 5

Conclusions

Definition 4.1.

Given a set N = {1, · · · , n} with n ∈ N, a ranking with ties is anapplication (probably non injective) c : N −→ N . If c(i), c(j),then i and j are tied.

Example 4.2.

σ1 σ2 σ3 σ4 σ5

2 5 5, 6 2, 5 13, 5 6 2 3, 6 61, 6 2, 3 3 4 44 1 1, 4 1 2, 3, 5

Conclusions

R.Fagin et al. extended in 2006 the Kendall’s coefficient for tworankings with ties c1 and c2, but in this case they must considerthree events for a pair of nodes i , j :

1 There is a crossing between i and j ,

c1(i) < c1(j) c2(i) > c2(j)

c1(i) > c1(j) c2(i) < c2(j),

Conclusions

2 i and j are tied in c1 and c2,

c1(i) = c1(j) c2(i) = c2(j),

Conclusions

2 i and j are tied in c1 and c2,

3 They are tied in c1 and not tied in c2 or viceversa,

c1(i) = c1(j) c2(i) 6= c2(j)

c1(i) 6= c1(j) c2(i) = c2(j).

Conclusions

In order to extend this idea to a sequence of (temporal) rankingswith ties R = {c1, · · · , cr} we must consider four events for a pairof nodes i , j :

1 There is a temporal crossing between i and j ,

ct(i) < ct(j) ct+1(i) > ct+1(j)

ct(i) > ct(j) ct+1(i) < ct+1(j),

Conclusions

2 i and j are tied along R,

c1(i) = c1(j), c2(i) = c2(j) · · · cr (i) = cr (j),

Conclusions

3 There is a long term semi-crossing,

ct(i) < ct(j), ct+1(i) = ct+1(j) · · · ct+s(i) < ct+s(j),

ct(i) > ct(j), ct+1(i) = ct+1(j) · · · ct+s(i) > ct+s(j),

Conclusions

3 There is a long term semi-crossing,

4 There is a long term crossing,

ct(i) < ct(j), ct+1(i) = ct+1(j) · · · ct+s(i) > ct+s(j),

ct(i) > ct(j), ct+1(i) = ct+1(j) · · · ct+s(i) < ct+s(j).

Conclusions

Competitivity Multiplex Graph

Conclusions

An application to the analysis of the IBEX35 Index

We studied the fluctuations in the rankings related with theSpanish Stock Market from 2003 to 2013.

What rankings are useful?

What metrics are useful?

We will consider two metrics:

The (annualized) return,

The (annualized) volatility

Conclusions

Definition 4.3.

The (annualized) return Ri of a stock is defined by

Ri = Ny

(SPi − SPi−1

SPi−1

where Ny is the number of trading days of the previous year andSPi and SPi−1 are the daily Stock Price of that stock in,respectively, the considered day and the day before.The (annualized) volatility Vi of a stock is the standard deviationof the (annualized) return Ri of the stock along the last year.

Conclusions

Dataset and parameters

We consider two sequences of rankings (one for returns andother for volatility) of the 25 stocks of IBEX35 that appear inall the sessions of the Spanish Stock Market from January2003 until December 2013 (2529 trading days), i.e. twosequences of 2529 rankings with ties of 25 stocks each.

For each sequence of rankings, we compute the CompetitivityMultiplex Graph. For the ties, we consider a precision of∆x = 0.05 in the round-off.

10^(-7) 10^(-4) 10^(-3) 0.03 0.07 0.8 2 3 4 5 6 70

Delta x

Return , dx=variable

20032004200520062007200820092010201120122013

Conclusions

10^(-7) 10^(-4) 10^(-3) 0.03 0.07 0.8 2 3 4 5 6 70

Delta x

Return , dx=variable

20032004200520062007200820092010201120122013

Conclusions

The parameters consider are α = 1, p = 1/2, q = 1 andγ = 0.

We compute the evolution of the normalized mean strength.

Conclusions

50 100 150 200 2500.02

Return , Delta x=0.05

20032004200520062007200820092010201120122013

50 100 150 200 2500

Volatility , Delta x=0.05

20032004200520062007200820092010201120122013

Conclusions

50 100 150 200 2500

Return 2004, dx=0.05

50 100 150 200 2500

50 100 150 200 250

Volatility 2004, dx=0.05

50 100 150 200 250

Conclusions

Summary

Structural properties of the Competitivity graphs correspondto properties of the rankings.

There is a structural characterization of the Competitivitygraphs in term of known families of graphs.

These techniques can be applied to real cases, such as sportrankings or economic rankings.

Talking about Real Zaragoza, Competitivity graphs say that amiracle in needed...

Many thanks for your attention!

Conclusions

Summary

Conclusions

Summary

Conclusions

Summary

Conclusions

Summary

Conclusions

S.Boccaletti, G.Bianconi, R.Criado, C.I. del Genio,J.Gomez Gardenes, M.Romance, I.Sendina and M.Zanin,“Structure and dynamics of multilayer networks”, to appear inPhysics Reports (2014), 1–155.

R.Criado, E.Garcia, F. Pedroche and M.Romance, “Anew method for comparing rankings through complex networks:Model and analysis of competitiveness of major European soccerleagues”, Chaos 23 (2013), 043114.

R.Criado, E.Garcia, F. Pedroche and M.Romance,“Comparing rankings by means of competitivity graphs: structuralproperties and computation”, Submitted (2014), 1–20.

R.Criado, E.Garcia, F. Pedroche, M.Romance andV.E. Sanchez, “Comparing series of rankings with ties by usingcomplex networks: an analysis of the Spanish Stock Market (IBEX35index)”, Submitted (2014), 1–24.

Conclusions

Ranking analysis by using complex networks:Structure and some applications

Miguel Romance(joint work with R.Criado, E.Garcıa, F.Pedroche & V.E.Sanchez)

Applied Mathematics Dept. Appl. and Biol. Mathematics Lab.ESCET Center for Biomedical Technology (CTB)

Universidad Rey Juan Carlos Universidad Politecnica de Madrid

IX EITA Research Meeting inApproximation Theory,

Alquezar, October 19th 2014

Conclusions

Contents

1 Introduction and basic definitionsData, information and rankings(Un-weighted) Competitivity Graphs

2 Structural properties of Competitivity GraphsPermutation, Comparability and Competitivity GraphsMain result

3 (Weighted) Competitivity Graphs of rankings (without ties)Definition and propertiesAn application to the analysis of Major European Soccer Leagues

4 (Weighted) Competitivity Graphs of rankings with tiesDefinition and propertiesAn application to the analysis of the IBEX35 Index

5 Conclusions

Conclusions

European Soccer Leagues 2013-14

ranking analysis by using complex networks: structure and...

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