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Measuring scientific production Ranking
Measuring and ranking scientific production
Nicolas Carayol
Observatoire des Sciences et Techniques, Paris
Dimetic, July 2010, Pecs 2010
Measuring scientific production Ranking
Bibliometrics
A field born with the its object : scientific production.
A distributional approach : with Bradford, Zipf and Lotka1926-> 35
RK Merton influence : measuring production and credit.
D. J. de Solla Price
E. Garfield
A connection with economics through Stigler’s influence
Measuring scientific production Ranking
Bibliometrics
A field born with the its object : scientific production.
A distributional approach : with Bradford, Zipf and Lotka1926-> 35
RK Merton influence : measuring production and credit.
D. J. de Solla Price
E. Garfield
A connection with economics through Stigler’s influence
Measuring scientific production Ranking
Bibliometrics
A field born with the its object : scientific production.
A distributional approach : with Bradford, Zipf and Lotka1926-> 35
RK Merton influence : measuring production and credit.
D. J. de Solla Price
E. Garfield
A connection with economics through Stigler’s influence
Measuring scientific production Ranking
Bibliometrics
A field born with the its object : scientific production.
A distributional approach : with Bradford, Zipf and Lotka1926-> 35
RK Merton influence : measuring production and credit.
D. J. de Solla Price
E. Garfield
A connection with economics through Stigler’s influence
Measuring scientific production Ranking
Bibliometrics
A field born with the its object : scientific production.
A distributional approach : with Bradford, Zipf and Lotka1926-> 35
RK Merton influence : measuring production and credit.
D. J. de Solla Price
E. Garfield
A connection with economics through Stigler’s influence
Measuring scientific production Ranking
Bibliometrics
A field born with the its object : scientific production.
A distributional approach : with Bradford, Zipf and Lotka1926-> 35
RK Merton influence : measuring production and credit.
D. J. de Solla Price
E. Garfield
A connection with economics through Stigler’s influence
Measuring scientific production Ranking
Bibliometrics
Measuring scientific production
How to rank ?
Measuring scientific production Ranking
Bibliometrics
Measuring scientific production
How to rank ?
Measuring scientific production Ranking
Sections :
1 Measuring scientific productionVolumeImpactInfluenceMeasures for both quality and quantity
2 RankingThe axiomatic approachThe extended stochastic dominance approach
Core background theoryIllustrative Results
Measuring scientific production Ranking
Bibliometrics
Define the agent ?
Q : quantity
q : quality
Measuring scientific production Ranking
Bibliometrics
Define the agent ?
Q : quantity
q : quality
Measuring scientific production Ranking
Bibliometrics
Define the agent ?
Q : quantity
q : quality
Measuring scientific production Ranking
Volume
The basic data
The structured set of i ’s publications is given by
Si := (s1, s2, ..., s′a, ..., sa, ..., sni ),
with ni the number of articles to be associated to agent i .
Measuring scientific production Ranking
Volume
Ground zero
Total counts :
Ti ,a = # {i mentioned in a}
Total counts by domain :
T ki ,a = # {i mentioned in a} × 1 {j(a) associated to k}
with j(a) is the journal in which a were published.
Measuring scientific production Ranking
Volume
Ground zero
Total counts :
Ti ,a = # {i mentioned in a}
Total counts by domain :
T ki ,a = # {i mentioned in a} × 1 {j(a) associated to k}
with j(a) is the journal in which a were published.
Measuring scientific production Ranking
Volume
Accounting for coauthorship
An article a, referencing at least one address associated toinstitution i , brings a score of :
pki ,a =
# {i mentioned in a}# { j | j mentioned in a}
× 1 {j(a) associated to k} ,
where 1 {.} is the indicator function and # {.} denotes thecardinal of the set defined into brackets.
Measuring scientific production Ranking
Volume
Allocate over disciplines
An article a, referencing at least one address associated toinstitution i , brings a score of :
pki ,a =
# {i mentioned in a}# { j | j mentioned in a}
× 1 {j(a) associated to k}# {k | j(a) associated to k}
,
where 1 {.} is the indicator function and # {.} denotes thecardinal of the set defined into brackets. j(a) is the journal inwhich a were published.
Measuring scientific production Ranking
Volume
Other basic corrections
# pages
...
Measuring scientific production Ranking
Volume
Other basic corrections
# pages
...
Measuring scientific production Ranking
Volume
Authors’ rank
(Assimakis & Adam, scientometrics forthcoming)The linearcontribution index :
lpia = c − δria,
where ria the rank of author i among the na authors of articlea and with c a normalization parameter c = na+1
2 δ + 1na
sothat authros contribution sum to one.
The geometriccontribution index
gpia = κ · λ−na+ri ,
with 0 < λ < 1, and κ he normalization parameter.
Measuring scientific production Ranking
Volume
Authors’ rank
(Assimakis & Adam, scientometrics forthcoming)The linearcontribution index :
lpia = c − δria,
where ria the rank of author i among the na authors of articlea and with c a normalization parameter c = na+1
2 δ + 1na
sothat authros contribution sum to one.The geometriccontribution index
gpia = κ · λ−na+ri ,
with 0 < λ < 1, and κ he normalization parameter.
Measuring scientific production Ranking
Impact
Quality is critical
Rely on peer review
Rely on citation data
Measuring scientific production Ranking
Impact
Quality is critical
Rely on peer review
Rely on citation data
Measuring scientific production Ranking
Impact
Counting citations
The Number of direct citations received :
sa = # { j | tj ∈ w(a) and j cites a} ,
with tj the year of publication of j , and w(a) the citationwindow of article a.
Measuring scientific production Ranking
Impact
The impact Factor
Impact Factor of the journal :
s ′a =# { j | tj ∈ w(a) and j cites i ∈ j(a)}
# { i | i ∈ j(a)}.
Measuring scientific production Ranking
Impact
Correction across fields
Relative Impact Factor of the journal :
s ′′a =s ′a
〈s ′a〉subfield of a
.
Measuring scientific production Ranking
Influence
Prestige centrality
Narin & Pinski 1976
Leibowitz and Palmer 1986
Page rank (Page & al, 1998)
Katz → prestige
Measuring scientific production Ranking
Influence
Prestige centrality
Narin & Pinski 1976
Leibowitz and Palmer 1986
Page rank (Page & al, 1998)
Katz → prestige
Measuring scientific production Ranking
Influence
Prestige centrality
Narin & Pinski 1976
Leibowitz and Palmer 1986
Page rank (Page & al, 1998)
Katz → prestige
Measuring scientific production Ranking
Influence
Prestige centrality
Narin & Pinski 1976
Leibowitz and Palmer 1986
Page rank (Page & al, 1998)
Katz → prestige
Measuring scientific production Ranking
Influence
A unified writing of centrality
A = (ai ) with ai the number of articles published in/of i .
C = (cij) with cij the number of citations i received from j (orthe number of references made in j to i).
The counting method of Bush, Hamelman & Staaf (1974) :
φi = λ
∑j cij/ai∑
k
∑j ckj/ak
.
value is proportional to the average number of citations madeto its articles
The counting modified method :
φi = λ
∑j cij/ai × φj∑
k
∑j ckj/ak × φj
.
value is proportional to the average number of citations madeto its articles
Measuring scientific production Ranking
Influence
A unified writing of centrality
A = (ai ) with ai the number of articles published in/of i .
C = (cij) with cij the number of citations i received from j (orthe number of references made in j to i).
The counting method of Bush, Hamelman & Staaf (1974) :
φi = λ
∑j cij/ai∑
k
∑j ckj/ak
.
value is proportional to the average number of citations madeto its articles
The counting modified method :
φi = λ
∑j cij/ai × φj∑
k
∑j ckj/ak × φj
.
value is proportional to the average number of citations madeto its articles
Measuring scientific production Ranking
Influence
A unified writing of centrality
A = (ai ) with ai the number of articles published in/of i .
C = (cij) with cij the number of citations i received from j (orthe number of references made in j to i).
The counting method of Bush, Hamelman & Staaf (1974) :
φi = λ
∑j cij/ai∑
k
∑j ckj/ak
.
value is proportional to the average number of citations madeto its articles
The counting modified method :
φi = λ
∑j cij/ai × φj∑
k
∑j ckj/ak × φj
.
value is proportional to the average number of citations madeto its articles
Measuring scientific production Ranking
Influence
A unified writing of centrality
A = (ai ) with ai the number of articles published in/of i .
C = (cij) with cij the number of citations i received from j (orthe number of references made in j to i).
The counting method of Bush, Hamelman & Staaf (1974) :
φi = λ
∑j cij/ai∑
k
∑j ckj/ak
.
value is proportional to the average number of citations madeto its articles
The counting modified method :
φi = λ
∑j cij/ai × φj∑
k
∑j ckj/ak × φj
.
value is proportional to the average number of citations madeto its articles
Measuring scientific production Ranking
Influence
A unified writing of centrality
The Leibowitz and Palmer (1986) method :
φi = λ
∑j 6=i cij/ai∑
k
∑j 6=k ckj/ak
,
the value is proportional to the average number of citationsmade to its articles.
In matrix form, this becomes :
φ = λA−1Cφ
‖A−1Cφ‖
with ‖X‖ =∑
k ‖xk‖ the 1 norm of X .
Measuring scientific production Ranking
Influence
A unified writing of centrality
The Leibowitz and Palmer (1986) method :
φi = λ
∑j 6=i cij/ai∑
k
∑j 6=k ckj/ak
,
the value is proportional to the average number of citationsmade to its articles.
In matrix form, this becomes :
φ = λA−1Cφ
‖A−1Cφ‖
with ‖X‖ =∑
k ‖xk‖ the 1 norm of X .
Measuring scientific production Ranking
Influence
A unified writing of centrality
The Narin and Pinski (1976) method :
φi = λ∑j 6=i
cij/ai
cj/ajφj ,
the value is a weighted average of the centrality of thejournals citing it.
In matrix form, this becomes :
φ = λA−1CD−1C Aφ
with DC the diagonal matrix having the sum of the of thereferences in the diagonal (ci ). The matrix CD−1
C is thenormalized matrix of C and is a stochastic matrix (sums toone in each column).
Measuring scientific production Ranking
Influence
A unified writing of centrality
The Narin and Pinski (1976) method :
φi = λ∑j 6=i
cij/ai
cj/ajφj ,
the value is a weighted average of the centrality of thejournals citing it.
In matrix form, this becomes :
φ = λA−1CD−1C Aφ
with DC the diagonal matrix having the sum of the of thereferences in the diagonal (ci ). The matrix CD−1
C is thenormalized matrix of C and is a stochastic matrix (sums toone in each column).
Measuring scientific production Ranking
Influence
Prestige centrality (reminder)
Prestige depends of the average prestige of its neighbors asystem of equations :
pi (g) =∑j 6=i
gij
ηj (g)pj (g) .
Measuring scientific production Ranking
Measures for both quality and quantity
Quantity and quality
The number of articles published ni , the average 〈s〉i and thetotal number 〈s〉i ni of citations received.
The first synthetic measure of both quality and quantity wasproposed by (Lindsay, 1978) :
Qi = 〈s〉i × (〈s〉i ni )1/2 = 〈s〉3/2
i n1/2i .
Measuring scientific production Ranking
Measures for both quality and quantity
Quantity and quality
The number of articles published ni , the average 〈s〉i and thetotal number 〈s〉i ni of citations received.
The first synthetic measure of both quality and quantity wasproposed by (Lindsay, 1978) :
Qi = 〈s〉i × (〈s〉i ni )1/2 = 〈s〉3/2
i n1/2i .
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index
Each article published is denoted by an index a ∈ N\ {0} andis characterized by an associated impact measure (herecitations) sa ∈ R+ (here sa ∈ N).
The structured set of agent i ’s publications isSi := (s1, s2, ..., s
′a, ..., sa, ..., sni ), a vector assumed to be
(decreasingly) ordered according to the number of citationsreceived : a > a′ → s ′a ≤ sa.
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index
Each article published is denoted by an index a ∈ N\ {0} andis characterized by an associated impact measure (herecitations) sa ∈ R+ (here sa ∈ N).
The structured set of agent i ’s publications isSi := (s1, s2, ..., s
′a, ..., sa, ..., sni ), a vector assumed to be
(decreasingly) ordered according to the number of citationsreceived : a > a′ → s ′a ≤ sa.
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index
Hirsch (2005) proposes the h-index as a synthetic a uniquemeasure for both the # of citations and the # ofpublications. More, it a basic measure for the wholepublication/citation sequence.
The h-index of agent i is a measure computed from her/hisstructured set Si as follows :
h := maxa
(a ≤ sa).
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index
Hirsch (2005) proposes the h-index as a synthetic a uniquemeasure for both the # of citations and the # ofpublications. More, it a basic measure for the wholepublication/citation sequence.
The h-index of agent i is a measure computed from her/hisstructured set Si as follows :
h := maxa
(a ≤ sa).
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index
Example : ni = 7, Si = (8, 6, 5, 4, 2, 2, 1).
S
rank 1 2 3 4 5 6 7
8
S
rank 1 2 3 4 5 6 7
8
h=4
sh=
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index
Example : ni = 7, Si = (8, 6, 5, 4, 2, 2, 1).
S
rank 1 2 3 4 5 6 7
8
S
rank 1 2 3 4 5 6 7
8
h=4
sh=
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index
Example : ni = 7, Si = (8, 6, 5, 4, 2, 2, 1).
S
rank 1 2 3 4 5 6 7
8
S
rank 1 2 3 4 5 6 7
8
h=4
sh=
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Main shortcoming of the h-index : citations have no impact onthe index outside the h-core
The derivatives :
the g-index (Egghe, 2006) : g := maxa a2 ≤∑
j=1...a sj
the r-index (Jin et al., 2007)the tapered h-index (Anderson et al., 2008)the w-index (Woeringer, 2008)...
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Main shortcoming of the h-index : citations have no impact onthe index outside the h-core
The derivatives :
the g-index (Egghe, 2006) : g := maxa a2 ≤∑
j=1...a sj
the r-index (Jin et al., 2007)the tapered h-index (Anderson et al., 2008)the w-index (Woeringer, 2008)...
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Main shortcoming of the h-index : citations have no impact onthe index outside the h-core
The derivatives :
the g-index (Egghe, 2006) : g := maxa a2 ≤∑
j=1...a sj
the r-index (Jin et al., 2007)the tapered h-index (Anderson et al., 2008)the w-index (Woeringer, 2008)...
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Main shortcoming of the h-index : citations have no impact onthe index outside the h-core
The derivatives :
the g-index (Egghe, 2006) : g := maxa a2 ≤∑
j=1...a sj
the r-index (Jin et al., 2007)
the tapered h-index (Anderson et al., 2008)the w-index (Woeringer, 2008)...
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Main shortcoming of the h-index : citations have no impact onthe index outside the h-core
The derivatives :
the g-index (Egghe, 2006) : g := maxa a2 ≤∑
j=1...a sj
the r-index (Jin et al., 2007)the tapered h-index (Anderson et al., 2008)
the w-index (Woeringer, 2008)...
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Main shortcoming of the h-index : citations have no impact onthe index outside the h-core
The derivatives :
the g-index (Egghe, 2006) : g := maxa a2 ≤∑
j=1...a sj
the r-index (Jin et al., 2007)the tapered h-index (Anderson et al., 2008)the w-index (Woeringer, 2008)
...
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Main shortcoming of the h-index : citations have no impact onthe index outside the h-core
The derivatives :
the g-index (Egghe, 2006) : g := maxa a2 ≤∑
j=1...a sj
the r-index (Jin et al., 2007)the tapered h-index (Anderson et al., 2008)the w-index (Woeringer, 2008)...
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Hirsch self defense :
He argues his aim is precisely to NOT consider both i) lowimpact papers (is intended to capture a maintained flow ofinfluential contributions) AND ii) the high volume of citationsthat some papers sometime receive (could unduly grant someco-authors of highly cited papers).Hirsch argues the h-index has a strong predictive power on thearrival of future citations (Hirsch, 2007).
Main implicit advantage : easy to compute and thereforerobust to errors.
From a theoretical point of view, it is not a defense of theimplicit value judgements it incorporates.
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Hirsch self defense :
He argues his aim is precisely to NOT consider both i) lowimpact papers (is intended to capture a maintained flow ofinfluential contributions) AND ii) the high volume of citationsthat some papers sometime receive (could unduly grant someco-authors of highly cited papers).
Hirsch argues the h-index has a strong predictive power on thearrival of future citations (Hirsch, 2007).
Main implicit advantage : easy to compute and thereforerobust to errors.
From a theoretical point of view, it is not a defense of theimplicit value judgements it incorporates.
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Hirsch self defense :
He argues his aim is precisely to NOT consider both i) lowimpact papers (is intended to capture a maintained flow ofinfluential contributions) AND ii) the high volume of citationsthat some papers sometime receive (could unduly grant someco-authors of highly cited papers).Hirsch argues the h-index has a strong predictive power on thearrival of future citations (Hirsch, 2007).
Main implicit advantage : easy to compute and thereforerobust to errors.
From a theoretical point of view, it is not a defense of theimplicit value judgements it incorporates.
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Hirsch self defense :
He argues his aim is precisely to NOT consider both i) lowimpact papers (is intended to capture a maintained flow ofinfluential contributions) AND ii) the high volume of citationsthat some papers sometime receive (could unduly grant someco-authors of highly cited papers).Hirsch argues the h-index has a strong predictive power on thearrival of future citations (Hirsch, 2007).
Main implicit advantage : easy to compute and thereforerobust to errors.
From a theoretical point of view, it is not a defense of theimplicit value judgements it incorporates.
Measuring scientific production Ranking
Measures for both quality and quantity
The h-index and beyond
Hirsch self defense :
He argues his aim is precisely to NOT consider both i) lowimpact papers (is intended to capture a maintained flow ofinfluential contributions) AND ii) the high volume of citationsthat some papers sometime receive (could unduly grant someco-authors of highly cited papers).Hirsch argues the h-index has a strong predictive power on thearrival of future citations (Hirsch, 2007).
Main implicit advantage : easy to compute and thereforerobust to errors.
From a theoretical point of view, it is not a defense of theimplicit value judgements it incorporates.
Measuring scientific production Ranking
Sections :
1 Measuring scientific productionVolumeImpactInfluenceMeasures for both quality and quantity
2 RankingThe axiomatic approachThe extended stochastic dominance approach
Core background theoryIllustrative Results
Measuring scientific production Ranking
Ranking : what for ?
Any ranking incorporate value judgements.
An axiomatic approach which renders explicit the valuejudgements incorporated in any ranking based on some index(as Arrow 1955).
A normative approach directly applied to the structured set ofpublications (as Rothshild and Stiglitz 1969, Atkinson, 1970).
Measuring scientific production Ranking
Ranking : what for ?
Any ranking incorporate value judgements.
An axiomatic approach which renders explicit the valuejudgements incorporated in any ranking based on some index(as Arrow 1955).
A normative approach directly applied to the structured set ofpublications (as Rothshild and Stiglitz 1969, Atkinson, 1970).
Measuring scientific production Ranking
Ranking : what for ?
Any ranking incorporate value judgements.
An axiomatic approach which renders explicit the valuejudgements incorporated in any ranking based on some index(as Arrow 1955).
A normative approach directly applied to the structured set ofpublications (as Rothshild and Stiglitz 1969, Atkinson, 1970).
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
→ An axiomatic approach which renders explicit the valuejudgements incorporated in measures :
Palacios Huerta & Volij 2004 (econometrica) →axiomatization of Pinski & Narin influence measure and someothers
Woeringer 2008 (math soc science) & Marchant 2009(scientometrics) → axiomatization of h-index and some othermeasures
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
→ An axiomatic approach which renders explicit the valuejudgements incorporated in measures :
Palacios Huerta & Volij 2004 (econometrica) →axiomatization of Pinski & Narin influence measure and someothers
Woeringer 2008 (math soc science) & Marchant 2009(scientometrics) → axiomatization of h-index and some othermeasures
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Palacios Huerta & Volij 2004 :
Let J be the set of ranked agents
Axioms :
1 Invariance with respect to reference intensity : the ranking isnot modified by any modification of the reference intensity ofagents (each one has a vote one) : for any diagonal matrix Λwith strictly positive diagonal entries λj , if C ′ = C Λ, theranking is unchanged.
2 Weak homogeneity : in a two agents ranking problem issueonly, for any two agents with identical number of references(ai = aj) : φi
φj=
cij
cji.
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Palacios Huerta & Volij 2004 :
Let J be the set of ranked agents
Axioms :
1 Invariance with respect to reference intensity : the ranking isnot modified by any modification of the reference intensity ofagents (each one has a vote one) : for any diagonal matrix Λwith strictly positive diagonal entries λj , if C ′ = C Λ, theranking is unchanged.
2 Weak homogeneity : in a two agents ranking problem issueonly, for any two agents with identical number of references(ai = aj) : φi
φj=
cij
cji.
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Palacios Huerta & Volij 2004 :
Let J be the set of ranked agents
Axioms :
1 Invariance with respect to reference intensity : the ranking isnot modified by any modification of the reference intensity ofagents (each one has a vote one) : for any diagonal matrix Λwith strictly positive diagonal entries λj , if C ′ = C Λ, theranking is unchanged.
2 Weak homogeneity : in a two agents ranking problem issueonly, for any two agents with identical number of references(ai = aj) : φi
φj=
cij
cji.
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Palacios Huerta & Volij 2004 :
Let J be the set of ranked agents
Axioms :
1 Invariance with respect to reference intensity : the ranking isnot modified by any modification of the reference intensity ofagents (each one has a vote one) : for any diagonal matrix Λwith strictly positive diagonal entries λj , if C ′ = C Λ, theranking is unchanged.
2 Weak homogeneity : in a two agents ranking problem issueonly, for any two agents with identical number of references(ai = aj) : φi
φj=
cij
cji.
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Palacios Huerta & Volij 2004 :
Let J be the set of ranked agents
Axioms :
1 Invariance with respect to reference intensity : the ranking isnot modified by any modification of the reference intensity ofagents (each one has a vote one) : for any diagonal matrix Λwith strictly positive diagonal entries λj , if C ′ = C Λ, theranking is unchanged.
2 Weak homogeneity : in a two agents ranking problem issueonly, for any two agents with identical number of references(ai = aj) : φi
φj=
cij
cji.
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
3 Weak consistency : if for any set of agents with identicalnumber of references (ai = aj ,∀i , j ∈ J) :
∀k ∈ J, φiφj
=φk
i
φkj
, ∀i , j ∈ J\ {k}, with φki the rank of i with the
modified citation matrix C k such thatckij = cij + ckj
cik∑h∈J\{k} chk
. (the relative valuations shall be
proportional of the relative influences without any otheragent, the influences of which are redistributed to the agentsit cites (proportionally).
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
4 Invariance to the splitting of agents : if for all rankingproblems, and any splitting of agent i∈ J into nj identicalagents (same profile of citation and references :
anj
j = aj/nj , cninj
ij = cij/(ninj)) : φiφj
=φ
nii
φnjj
, ∀ni , nj .
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Theorem
The Narin and Pinski (1976) measure of centrality is the uniquemeasure that may rank any set of agents while satisfying the 4axioms.
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Woeringer 2008 axiomatization of h − index and some othermeasures
Si := (s1, s2, ..., s′a, ..., sa, ..., sni )
S ∈ Ψ the set of all possible positive entry vectors ofdimension 1× n(n > 0).
An index is a function φ : Ψ→ N with φ(0, 0, ..., 0) = 0
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Woeringer 2008 axiomatization of h − index and some othermeasures
Si := (s1, s2, ..., s′a, ..., sa, ..., sni )
S ∈ Ψ the set of all possible positive entry vectors ofdimension 1× n(n > 0).
An index is a function φ : Ψ→ N with φ(0, 0, ..., 0) = 0
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Woeringer 2008 axiomatization of h − index and some othermeasures
Si := (s1, s2, ..., s′a, ..., sa, ..., sni )
S ∈ Ψ the set of all possible positive entry vectors ofdimension 1× n(n > 0).
An index is a function φ : Ψ→ N with φ(0, 0, ..., 0) = 0
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Woeringer 2008 axiomatization of h − index and some othermeasures
Si := (s1, s2, ..., s′a, ..., sa, ..., sni )
S ∈ Ψ the set of all possible positive entry vectors ofdimension 1× n(n > 0).
An index is a function φ : Ψ→ N with φ(0, 0, ..., 0) = 0
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Axioms
a Monotonicity : for all vectors S of size 1× n and S ′ of size m,such that m ≤ n and sk ≥ s ′k ,∀k ≤ m then φ(S) > φ(S ′).
b If the vector S of size 1× (n + 1) can be built from vector S ′
of size 1× n by simply adding a paper with φ(S) citations,then φ(S ′) ≤ φ(S).
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Axioms
a Monotonicity : for all vectors S of size 1× n and S ′ of size m,such that m ≤ n and sk ≥ s ′k ,∀k ≤ m then φ(S) > φ(S ′).
b If the vector S of size 1× (n + 1) can be built from vector S ′
of size 1× n by simply adding a paper with φ(S) citations,then φ(S ′) ≤ φ(S).
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Axioms
a Monotonicity : for all vectors S of size 1× n and S ′ of size m,such that m ≤ n and sk ≥ s ′k ,∀k ≤ m then φ(S) > φ(S ′).
b If the vector S of size 1× (n + 1) can be built from vector S ′
of size 1× n by simply adding a paper with φ(S) citations,then φ(S ′) ≤ φ(S).
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Axioms
c If the vector S of size 1× n can be built from vector S ′ of size1× n by simply adding a citation to a paper (∃h ≤ n sts ′h = sh + 1 and ∀k ≤ n st k 6= h then s ′h = sh) thenφ(S) ≤ φ(S ′) + 1.
d If the vector S of size 1× (n + 1) can be built from vector S ′
of size 1× n by simply adding an article with φ(S) citations,and then increasing the number of citations of paper by atleast one (∀h ≤ n st sh ≥ s ′h) then φ(S) > φ(S ′).
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Axioms
c If the vector S of size 1× n can be built from vector S ′ of size1× n by simply adding a citation to a paper (∃h ≤ n sts ′h = sh + 1 and ∀k ≤ n st k 6= h then s ′h = sh) thenφ(S) ≤ φ(S ′) + 1.
d If the vector S of size 1× (n + 1) can be built from vector S ′
of size 1× n by simply adding an article with φ(S) citations,and then increasing the number of citations of paper by atleast one (∀h ≤ n st sh ≥ s ′h) then φ(S) > φ(S ′).
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Axioms
c If the vector S of size 1× n can be built from vector S ′ of size1× n by simply adding a citation to a paper (∃h ≤ n sts ′h = sh + 1 and ∀k ≤ n st k 6= h then s ′h = sh) thenφ(S) ≤ φ(S ′) + 1.
d If the vector S of size 1× (n + 1) can be built from vector S ′
of size 1× n by simply adding an article with φ(S) citations,and then increasing the number of citations of paper by atleast one (∀h ≤ n st sh ≥ s ′h) then φ(S) > φ(S ′).
Measuring scientific production Ranking
The axiomatic approach
The axiomatic approach
Theorem
An index φ : Ψ→ N satisfies the four axioms a, b, c and d if andonly if it is the h-index.
Measuring scientific production Ranking
The extended stochastic dominance approach
→ A normative approach directly applied to the structured set ofpublications.
Measuring scientific production Ranking
The extended stochastic dominance approach
→ A normative approach directly applied to the structured set ofpublications.
Measuring scientific production Ranking
The extended stochastic dominance approach
The main goal
There is a well known concept often used in economics :stochastic dominance.
Applied to the theory of choice under uncertainty (Stiglitz &Rothschild, 1970) and income distribution (Atkinson 1970).
But :
This is designed to compare distributions only (care only aboutquality and quality distribution).The assumptions associated to second order stochasticdominance (concavity) will not be systematically consistent.
→ A need an adaptation of the technique which would value bothimpact and volume of publication.
Measuring scientific production Ranking
The extended stochastic dominance approach
The main goal
There is a well known concept often used in economics :stochastic dominance.
Applied to the theory of choice under uncertainty (Stiglitz &Rothschild, 1970) and income distribution (Atkinson 1970).
But :
This is designed to compare distributions only (care only aboutquality and quality distribution).The assumptions associated to second order stochasticdominance (concavity) will not be systematically consistent.
→ A need an adaptation of the technique which would value bothimpact and volume of publication.
Measuring scientific production Ranking
The extended stochastic dominance approach
The main goal
There is a well known concept often used in economics :stochastic dominance.
Applied to the theory of choice under uncertainty (Stiglitz &Rothschild, 1970) and income distribution (Atkinson 1970).
But :
This is designed to compare distributions only (care only aboutquality and quality distribution).The assumptions associated to second order stochasticdominance (concavity) will not be systematically consistent.
→ A need an adaptation of the technique which would value bothimpact and volume of publication.
Measuring scientific production Ranking
The extended stochastic dominance approach
The main goal
There is a well known concept often used in economics :stochastic dominance.
Applied to the theory of choice under uncertainty (Stiglitz &Rothschild, 1970) and income distribution (Atkinson 1970).
But :
This is designed to compare distributions only (care only aboutquality and quality distribution).
The assumptions associated to second order stochasticdominance (concavity) will not be systematically consistent.
→ A need an adaptation of the technique which would value bothimpact and volume of publication.
Measuring scientific production Ranking
The extended stochastic dominance approach
The main goal
There is a well known concept often used in economics :stochastic dominance.
Applied to the theory of choice under uncertainty (Stiglitz &Rothschild, 1970) and income distribution (Atkinson 1970).
But :
This is designed to compare distributions only (care only aboutquality and quality distribution).The assumptions associated to second order stochasticdominance (concavity) will not be systematically consistent.
→ A need an adaptation of the technique which would value bothimpact and volume of publication.
Measuring scientific production Ranking
The extended stochastic dominance approach
The main goal
There is a well known concept often used in economics :stochastic dominance.
Applied to the theory of choice under uncertainty (Stiglitz &Rothschild, 1970) and income distribution (Atkinson 1970).
But :
This is designed to compare distributions only (care only aboutquality and quality distribution).The assumptions associated to second order stochasticdominance (concavity) will not be systematically consistent.
→ A need an adaptation of the technique which would value bothimpact and volume of publication.
Measuring scientific production Ranking
The extended stochastic dominance approach
The main goal
There is a well known concept often used in economics :stochastic dominance.
Applied to the theory of choice under uncertainty (Stiglitz &Rothschild, 1970) and income distribution (Atkinson 1970).
But :
This is designed to compare distributions only (care only aboutquality and quality distribution).The assumptions associated to second order stochasticdominance (concavity) will not be systematically consistent.
→ A need an adaptation of the technique which would value bothimpact and volume of publication.
Measuring scientific production Ranking
The extended stochastic dominance approach
The main goal
→ we hereby introduce a theory and develop a methodologyto compare the scientific production of institutions.
1 Dominance relations are established.
2 Dominance relations are used to build dominance networks,rankings and reference classes.
Measuring scientific production Ranking
The extended stochastic dominance approach
The main goal
→ we hereby introduce a theory and develop a methodologyto compare the scientific production of institutions.
1 Dominance relations are established.
2 Dominance relations are used to build dominance networks,rankings and reference classes.
Measuring scientific production Ranking
The extended stochastic dominance approach
The main goal
→ we hereby introduce a theory and develop a methodologyto compare the scientific production of institutions.
1 Dominance relations are established.
2 Dominance relations are used to build dominance networks,rankings and reference classes.
Measuring scientific production Ranking
The extended stochastic dominance approach
General problem
→ in principle that theory could apply to the comparison ofthe outcome of a set agents for which :
1 The outcome is composite,
2 Each element can be described by a “quality” index,
3 The modeler is concerned by both quantity and quality,
4 The modeler does not know exactly the function whichtransforms the “quality” in valued outcome, and thusrefers to classes of functions.
Measuring scientific production Ranking
The extended stochastic dominance approach
General problem
→ in principle that theory could apply to the comparison ofthe outcome of a set agents for which :
1 The outcome is composite,
2 Each element can be described by a “quality” index,
3 The modeler is concerned by both quantity and quality,
4 The modeler does not know exactly the function whichtransforms the “quality” in valued outcome, and thusrefers to classes of functions.
Measuring scientific production Ranking
The extended stochastic dominance approach
General problem
→ in principle that theory could apply to the comparison ofthe outcome of a set agents for which :
1 The outcome is composite,
2 Each element can be described by a “quality” index,
3 The modeler is concerned by both quantity and quality,
4 The modeler does not know exactly the function whichtransforms the “quality” in valued outcome, and thusrefers to classes of functions.
Measuring scientific production Ranking
The extended stochastic dominance approach
General problem
→ in principle that theory could apply to the comparison ofthe outcome of a set agents for which :
1 The outcome is composite,
2 Each element can be described by a “quality” index,
3 The modeler is concerned by both quantity and quality,
4 The modeler does not know exactly the function whichtransforms the “quality” in valued outcome, and thusrefers to classes of functions.
Measuring scientific production Ranking
The extended stochastic dominance approach
General problem
→ in principle that theory could apply to the comparison ofthe outcome of a set agents for which :
1 The outcome is composite,
2 Each element can be described by a “quality” index,
3 The modeler is concerned by both quantity and quality,
4 The modeler does not know exactly the function whichtransforms the “quality” in valued outcome, and thusrefers to classes of functions.
Measuring scientific production Ranking
The extended stochastic dominance approach
General problem : Potential applictions
Departments of economics
Museums
Social clubs
Night clubs
...
Measuring scientific production Ranking
The extended stochastic dominance approach
General problem : Potential applictions
Departments of economics
Museums
Social clubs
Night clubs
...
Measuring scientific production Ranking
The extended stochastic dominance approach
General problem : Potential applictions
Departments of economics
Museums
Social clubs
Night clubs
...
Measuring scientific production Ranking
The extended stochastic dominance approach
General problem : Potential applictions
Departments of economics
Museums
Social clubs
Night clubs
...
Measuring scientific production Ranking
The extended stochastic dominance approach
General problem : Potential applictions
Departments of economics
Museums
Social clubs
Night clubs
...
Measuring scientific production Ranking
The extended stochastic dominance approach
Implicit to explicit valuation of the scientific output
Let f ki (s) :=
∑j=1,...,ni
1 {sj = s} be the publicationperformance of i with “impact” s in discipline k .
0 1 2 3 4 s
f(0)
f(1) f(2)
f(3)
Measuring scientific production Ranking
The extended stochastic dominance approach
The scientific output : articles and citations
Then the vector of structured publication outcome ofinstitution i in domain k is computed by summing over allarticles a and filtering out the article scores in the sum on theright side of previous equation. It gives :
f ki (s) =
∑j=1,...,ni
pki ,a × 1 {sj = s} .
Assume the exists some s such that ∀i , f ki (s) = 0 if s ≥ s.
The “value” (for the agent, the consumer, the holder, thesponsor...) of the whole publication production of institution iin domain k is assumed to be symmetric and additiveseparable given by :
V k (Si ) = ωk
∫ s
0v(s)f k
i (s) ds,
with ωk the discipline dependent normalization parameter.Computation of the “value” of the whole publicationproduction of institution i in domain k is conditional to thevaluation function v(s).
Measuring scientific production Ranking
The extended stochastic dominance approach
The scientific output : articles and citations
Then the vector of structured publication outcome ofinstitution i in domain k is computed by summing over allarticles a and filtering out the article scores in the sum on theright side of previous equation. It gives :
f ki (s) =
∑j=1,...,ni
pki ,a × 1 {sj = s} .
Assume the exists some s such that ∀i , f ki (s) = 0 if s ≥ s.
The “value” (for the agent, the consumer, the holder, thesponsor...) of the whole publication production of institution iin domain k is assumed to be symmetric and additiveseparable given by :
V k (Si ) = ωk
∫ s
0v(s)f k
i (s) ds,
with ωk the discipline dependent normalization parameter.Computation of the “value” of the whole publicationproduction of institution i in domain k is conditional to thevaluation function v(s).
Measuring scientific production Ranking
The extended stochastic dominance approach
The scientific output : articles and citations
Then the vector of structured publication outcome ofinstitution i in domain k is computed by summing over allarticles a and filtering out the article scores in the sum on theright side of previous equation. It gives :
f ki (s) =
∑j=1,...,ni
pki ,a × 1 {sj = s} .
Assume the exists some s such that ∀i , f ki (s) = 0 if s ≥ s.
The “value” (for the agent, the consumer, the holder, thesponsor...) of the whole publication production of institution iin domain k is assumed to be symmetric and additiveseparable given by :
V k (Si ) = ωk
∫ s
0v(s)f k
i (s) ds,
with ωk the discipline dependent normalization parameter.
Computation of the “value” of the whole publicationproduction of institution i in domain k is conditional to thevaluation function v(s).
Measuring scientific production Ranking
The extended stochastic dominance approach
The scientific output : articles and citations
Then the vector of structured publication outcome ofinstitution i in domain k is computed by summing over allarticles a and filtering out the article scores in the sum on theright side of previous equation. It gives :
f ki (s) =
∑j=1,...,ni
pki ,a × 1 {sj = s} .
Assume the exists some s such that ∀i , f ki (s) = 0 if s ≥ s.
The “value” (for the agent, the consumer, the holder, thesponsor...) of the whole publication production of institution iin domain k is assumed to be symmetric and additiveseparable given by :
V k (Si ) = ωk
∫ s
0v(s)f k
i (s) ds,
with ωk the discipline dependent normalization parameter.Computation of the “value” of the whole publicationproduction of institution i in domain k is conditional to thevaluation function v(s).
Measuring scientific production Ranking
The extended stochastic dominance approach
Implicit valuation : examples
0 1 2 3 4 T s
A
B C D
E F
Measuring scientific production Ranking
The extended stochastic dominance approach
Various assumptions
Assumption 1. v ≥ 0 (A1)
Assumption 2. v ′ ≥ 0 (A2)
Assumption 3. v ′′ ≥ 0 (A3)
Assumption 4. v ′′ ≤ 0 (A4)
According to your goals and/or beliefs, you can rely uponwell defined dominance relations that can be used to producerankings and references classes.
Measuring scientific production Ranking
The extended stochastic dominance approach
Various assumptions
Assumption 1. v ≥ 0 (A1)
Assumption 2. v ′ ≥ 0 (A2)
Assumption 3. v ′′ ≥ 0 (A3)
Assumption 4. v ′′ ≤ 0 (A4)
According to your goals and/or beliefs, you can rely uponwell defined dominance relations that can be used to producerankings and references classes.
Measuring scientific production Ranking
The extended stochastic dominance approach
Various assumptions
Assumption 1. v ≥ 0 (A1)
Assumption 2. v ′ ≥ 0 (A2)
Assumption 3. v ′′ ≥ 0 (A3)
Assumption 4. v ′′ ≤ 0 (A4)
According to your goals and/or beliefs, you can rely uponwell defined dominance relations that can be used to producerankings and references classes.
Measuring scientific production Ranking
The extended stochastic dominance approach
Various assumptions
Assumption 1. v ≥ 0 (A1)
Assumption 2. v ′ ≥ 0 (A2)
Assumption 3. v ′′ ≥ 0 (A3)
Assumption 4. v ′′ ≤ 0 (A4)
According to your goals and/or beliefs, you can rely uponwell defined dominance relations that can be used to producerankings and references classes.
Measuring scientific production Ranking
The extended stochastic dominance approach
Various assumptions
Assumption 1. v ≥ 0 (A1)
Assumption 2. v ′ ≥ 0 (A2)
Assumption 3. v ′′ ≥ 0 (A3)
Assumption 4. v ′′ ≤ 0 (A4)
According to your goals and/or beliefs, you can rely uponwell defined dominance relations that can be used to producerankings and references classes.
Measuring scientific production Ranking
The extended stochastic dominance approach
Various assumptions
Assumption 1. v ≥ 0 (A1)
Assumption 2. v ′ ≥ 0 (A2)
Assumption 3. v ′′ ≥ 0 (A3)
Assumption 4. v ′′ ≤ 0 (A4)
According to your goals and/or beliefs, you can rely uponwell defined dominance relations that can be used to producerankings and references classes.
Measuring scientific production Ranking
The extended stochastic dominance approach
Standard measures using publications and citations
The number of articles published : v(s) = θ > 0
The total number of citations : v(s) = s
The average number of citations per article :vi (s) = s/
∫fi (s)ds
The h-index respects (A1) & (A2) but violates (A3) &(A4).
Measuring scientific production Ranking
The extended stochastic dominance approach
Standard measures using publications and citations
The number of articles published : v(s) = θ > 0
The total number of citations : v(s) = s
The average number of citations per article :vi (s) = s/
∫fi (s)ds
The h-index respects (A1) & (A2) but violates (A3) &(A4).
Measuring scientific production Ranking
The extended stochastic dominance approach
Standard measures using publications and citations
The number of articles published : v(s) = θ > 0
The total number of citations : v(s) = s
The average number of citations per article :vi (s) = s/
∫fi (s)ds
The h-index respects (A1) & (A2) but violates (A3) &(A4).
Measuring scientific production Ranking
The extended stochastic dominance approach
Standard measures using publications and citations
The number of articles published : v(s) = θ > 0
The total number of citations : v(s) = s
The average number of citations per article :vi (s) = s/
∫fi (s)ds
The h-index respects (A1) & (A2) but violates (A3) &(A4).
Measuring scientific production Ranking
The extended stochastic dominance approach
Strong dominance
Definition
The scientific production of institution i in field k stronglydominates the one of institution j , noted i Ik j , if, for anypositive function v (·) (A1),
∫ s0 v(s)f k
i (s) ds ≥∫ s
0 v(s)f kj (s) ds.
Lemma
i Ik j if and only if ∀x ∈ [0,∞[ , f ki (x)− f k
j (x) ≥ 0.
Measuring scientific production Ranking
The extended stochastic dominance approach
Strong dominance
Definition
The scientific production of institution i in field k stronglydominates the one of institution j , noted i Ik j , if, for anypositive function v (·) (A1),
∫ s0 v(s)f k
i (s) ds ≥∫ s
0 v(s)f kj (s) ds.
Lemma
i Ik j if and only if ∀x ∈ [0,∞[ , f ki (x)− f k
j (x) ≥ 0.
Measuring scientific production Ranking
The extended stochastic dominance approach
Implicit valuation of strong dominance : positive functions
0 1 2 3 4 T s
A
B C D
E F
Measuring scientific production Ranking
The extended stochastic dominance approach
Strong dominance
Measuring scientific production Ranking
The extended stochastic dominance approach
But often :
Measuring scientific production Ranking
The extended stochastic dominance approach
Dominance
Definition
The scientific production of institution i in field k dominates theone of institution j , noted i Bk j , if, for any positive and nondecreasing function v (·) (A1 & A2),∫ s
0 v(s)f ki (s) ds ≥
∫ s0 v(s)f k
j (s) ds.
Lemma
i Bk j , if and only if ∀x ∈ [0, s[ ,∫ sx
[f ki (s)− f k
j (s)]
ds ≥ 0.
Measuring scientific production Ranking
The extended stochastic dominance approach
Dominance
Definition
The scientific production of institution i in field k dominates theone of institution j , noted i Bk j , if, for any positive and nondecreasing function v (·) (A1 & A2),∫ s
0 v(s)f ki (s) ds ≥
∫ s0 v(s)f k
j (s) ds.
Lemma
i Bk j , if and only if ∀x ∈ [0, s[ ,∫ sx
[f ki (s)− f k
j (s)]
ds ≥ 0.
Measuring scientific production Ranking
The extended stochastic dominance approach
Implicit valuation of dominance : positive and increasingfunctions
0 1 2 3 4 T s
B C
E
Measuring scientific production Ranking
The extended stochastic dominance approach
Dominance
Measuring scientific production Ranking
The extended stochastic dominance approach
Dominance
Measuring scientific production Ranking
The extended stochastic dominance approach
Dominance
Measuring scientific production Ranking
The extended stochastic dominance approach
Dominance
Measuring scientific production Ranking
The extended stochastic dominance approach
Dominance
Measuring scientific production Ranking
The extended stochastic dominance approach
Weak dominance
Definition
The scientific production of institution i in field k weaklydominates the one of institution j , noted i Dk j , if, for anypositive, non decreasing function and weakly convex functionv (·) (A1, A2 & A3),
∫ s0 v(s)f k
i (s) ds ≥∫ s
0 v(s)f kj (s) ds.
Lemma
i Dk j , if and only if ∀x ∈ [0, s[ ,∫ sx s[f ki (s)− f k
j (s)]
ds ≥ 0.
Measuring scientific production Ranking
The extended stochastic dominance approach
Weak dominance
Definition
The scientific production of institution i in field k weaklydominates the one of institution j , noted i Dk j , if, for anypositive, non decreasing function and weakly convex functionv (·) (A1, A2 & A3),
∫ s0 v(s)f k
i (s) ds ≥∫ s
0 v(s)f kj (s) ds.
Lemma
i Dk j , if and only if ∀x ∈ [0, s[ ,∫ sx s[f ki (s)− f k
j (s)]
ds ≥ 0.
Measuring scientific production Ranking
The extended stochastic dominance approach
Implicit valuation of weak dominance : positive, increasingand weakly convex functions
0 1 2 3 4 T s
B
F
Measuring scientific production Ranking
The extended stochastic dominance approach
Weak Dominance
fik (s)
fjk (s)
Measuring scientific production Ranking
The extended stochastic dominance approach
Weak Dominance
fik (s)
fjk (s)
Measuring scientific production Ranking
The extended stochastic dominance approach
Weak Dominance
fik (s)
fjk (s)
Measuring scientific production Ranking
The extended stochastic dominance approach
Weak Dominance
Measuring scientific production Ranking
The extended stochastic dominance approach
Weak concave dominance
Definition
The scientific production of institution i in field k weaklyconcavely dominates the one of institution j , noted iBk j , if, forany positive, non decreasing function and weakly concavefunction v (·) (A1, A2 & A4),
∫ s0 v(s)f k
i (s) ds ≥∫ s
0 v(s)f kj (s) ds.
Lemma
iBk j if and only if ∀x ∈ [0, s[ ,∫ x
0 s[f ki (s)− f k
j (s)]
ds ≥ 0.
Measuring scientific production Ranking
The extended stochastic dominance approach
Weak concave dominance
Definition
The scientific production of institution i in field k weaklyconcavely dominates the one of institution j , noted iBk j , if, forany positive, non decreasing function and weakly concavefunction v (·) (A1, A2 & A4),
∫ s0 v(s)f k
i (s) ds ≥∫ s
0 v(s)f kj (s) ds.
Lemma
iBk j if and only if ∀x ∈ [0, s[ ,∫ x
0 s[f ki (s)− f k
j (s)]
ds ≥ 0.
Measuring scientific production Ranking
The extended stochastic dominance approach
Upward dominance
Definition
Let sφk be the smallest visibility a paper may exhibit among the φ%most visible papers in the field k (in the world).
Strong dominance at order φ : i Iφk j .
Dominance at order φ : i Bφk j .
Weak dominance at order φ : i Dφk j .
Upward dominance relations are obviously generalization ofthe dominance relations which are equivalent to upwarddominance relations at order 1.
Measuring scientific production Ranking
The extended stochastic dominance approach
Upward dominance
Definition
Let sφk be the smallest visibility a paper may exhibit among the φ%most visible papers in the field k (in the world).
Strong dominance at order φ : i Iφk j .
Dominance at order φ : i Bφk j .
Weak dominance at order φ : i Dφk j .
Upward dominance relations are obviously generalization ofthe dominance relations which are equivalent to upwarddominance relations at order 1.
Measuring scientific production Ranking
The extended stochastic dominance approach
Upward dominance
Definition
Let sφk be the smallest visibility a paper may exhibit among the φ%most visible papers in the field k (in the world).
Strong dominance at order φ : i Iφk j .
Dominance at order φ : i Bφk j .
Weak dominance at order φ : i Dφk j .
Upward dominance relations are obviously generalization ofthe dominance relations which are equivalent to upwarddominance relations at order 1.
Measuring scientific production Ranking
The extended stochastic dominance approach
Upward dominance
Definition
Let sφk be the smallest visibility a paper may exhibit among the φ%most visible papers in the field k (in the world).
Strong dominance at order φ : i Iφk j .
Dominance at order φ : i Bφk j .
Weak dominance at order φ : i Dφk j .
Upward dominance relations are obviously generalization ofthe dominance relations which are equivalent to upwarddominance relations at order 1.
Measuring scientific production Ranking
The extended stochastic dominance approach
Upward dominance
Definition
Let sφk be the smallest visibility a paper may exhibit among the φ%most visible papers in the field k (in the world).
Strong dominance at order φ : i Iφk j .
Dominance at order φ : i Bφk j .
Weak dominance at order φ : i Dφk j .
Upward dominance relations are obviously generalization ofthe dominance relations which are equivalent to upwarddominance relations at order 1.
Measuring scientific production Ranking
The extended stochastic dominance approach
Upward Dominance
fik (s)
fjk (s)
fik (s)
fjk (s)
sφ
Measuring scientific production Ranking
The extended stochastic dominance approach
Upward Dominance
fik (s)
fjk (s)
fik (s)
fjk (s)
sφ
Measuring scientific production Ranking
The extended stochastic dominance approach
The Interdisciplinary scientific output
Θi =(Θk
i
)k=1,...|K |, the vector of publication informed by
citations performance in all disciplines/domains k ∈ K
The value of the publication performance of institution i asthe weighted and valued sum of the articles of thatinstitution :
V (Θi ) =∑k
ωk
∫ s
0v(s)f k
i (s) ds.
We propose to set ωk as the inverse of the average number ofcitations made by articles in field k .
The value function is not symmetric anymore.
Measuring scientific production Ranking
The extended stochastic dominance approach
The Interdisciplinary scientific output
Θi =(Θk
i
)k=1,...|K |, the vector of publication informed by
citations performance in all disciplines/domains k ∈ K
The value of the publication performance of institution i asthe weighted and valued sum of the articles of thatinstitution :
V (Θi ) =∑k
ωk
∫ s
0v(s)f k
i (s) ds.
We propose to set ωk as the inverse of the average number ofcitations made by articles in field k .
The value function is not symmetric anymore.
Measuring scientific production Ranking
The extended stochastic dominance approach
The Interdisciplinary scientific output
Θi =(Θk
i
)k=1,...|K |, the vector of publication informed by
citations performance in all disciplines/domains k ∈ K
The value of the publication performance of institution i asthe weighted and valued sum of the articles of thatinstitution :
V (Θi ) =∑k
ωk
∫ s
0v(s)f k
i (s) ds.
We propose to set ωk as the inverse of the average number ofcitations made by articles in field k .
The value function is not symmetric anymore.
Measuring scientific production Ranking
The extended stochastic dominance approach
The Interdisciplinary scientific output
Θi =(Θk
i
)k=1,...|K |, the vector of publication informed by
citations performance in all disciplines/domains k ∈ K
The value of the publication performance of institution i asthe weighted and valued sum of the articles of thatinstitution :
V (Θi ) =∑k
ωk
∫ s
0v(s)f k
i (s) ds.
We propose to set ωk as the inverse of the average number ofcitations made by articles in field k .
The value function is not symmetric anymore.
Measuring scientific production Ranking
The extended stochastic dominance approach
The scientific output : articles and citations
Definition
The scientific production of institution i inter-disciplinary (a)strongly dominates
(i Iφ j
), (b) dominates
(i Bφ j
)or (c) weakly
dominates(i Dφ j
)at order φ ∈ ]0, 1] the one of institution j , if∫ s
sφk
v(s)f ki (s) ds ≥
∑k
∫ ssφk
v(s)f kj (s) ds for any (a) positive
function (b) positive and non-decreasing (c) positive andnon-decreasing and weakly convex function v (·) .
Measuring scientific production Ranking
The extended stochastic dominance approach
The scientific output : articles and citations
Lemma
The three following statements hold :i) i Iφ j iff, ∀x = (xk)k=1...|K | st
∀k , xk ∈[sφk , s
[,∑
k ωk
[f ki (xk)− f k
j (xk)]
ds ≥ 0;
ii) i Bφ j iff, ∀x = (xk)k=1...|K | st
∀k, xk ∈[sφk , s
[,∑
k ωk
∫ sxk
[f ki (s)− f k
j (s)]
ds ≥ 0;
iii) i Dφ j iff ∀x = (xk)k=1...|K | st
∀k, xk ∈[sφk , s
[,∑
k ωk
∫ sxk
s[f ki (s)− f k
j (s)]
ds ≥ 0.
Measuring scientific production Ranking
The extended stochastic dominance approach
Properties of dominance relations
Lemma
All the dominance relations introduced (upward strong dominance,upward dominance and upward weak dominance) are transitive : ifi � j and j � h, then i � h, where � potentially accounts forIφ
k ,Bφk or Dφ
k , with ∀φ ∈ ]0, 1].
Measuring scientific production Ranking
The extended stochastic dominance approach
Relations between dominance relations
Definition
A dominance relation � is stronger than dominance relation �′,noted ���′, if, ∀i , j , i � j implies i �′ j .
Theorem
∀φ, φ′ ∈ [0, 1] such that φ ≥ φ′, then
- Iφk�Bφ
k�Dφk ,
- Iφk�Iφ′
k , Bφk�Bφ′
k and Dφk�Dφ′
k ,- Iφ�Bφ�Dφ, and- Iφ�Iφ′ , Bφ�Bφ′ and Dφ�Dφ′ .
Thus the weaker the dominance relation, the more complete,that is the more dominance relations it is possible to establisha given set of institutions.
Measuring scientific production Ranking
The extended stochastic dominance approach
Relations between dominance relations
Definition
A dominance relation � is stronger than dominance relation �′,noted ���′, if, ∀i , j , i � j implies i �′ j .
Theorem
∀φ, φ′ ∈ [0, 1] such that φ ≥ φ′, then
- Iφk�Bφ
k�Dφk ,
- Iφk�Iφ′
k , Bφk�Bφ′
k and Dφk�Dφ′
k ,- Iφ�Bφ�Dφ, and- Iφ�Iφ′ , Bφ�Bφ′ and Dφ�Dφ′ .
Thus the weaker the dominance relation, the more complete,that is the more dominance relations it is possible to establisha given set of institutions.
Measuring scientific production Ranking
The extended stochastic dominance approach
Relations between dominance relations
Definition
A dominance relation � is stronger than dominance relation �′,noted ���′, if, ∀i , j , i � j implies i �′ j .
Theorem
∀φ, φ′ ∈ [0, 1] such that φ ≥ φ′, then
- Iφk�Bφ
k�Dφk ,
- Iφk�Iφ′
k , Bφk�Bφ′
k and Dφk�Dφ′
k ,- Iφ�Bφ�Dφ, and- Iφ�Iφ′ , Bφ�Bφ′ and Dφ�Dφ′ .
Thus the weaker the dominance relation, the more complete,that is the more dominance relations it is possible to establisha given set of institutions.
Measuring scientific production Ranking
The extended stochastic dominance approach
Publication data
OST in-house ISI-WOS database recording of publications andcitations
Large disciplines : 1/ Fund. bio., 2/ Medicine, 3/ Ap.bio/ecol., 4/ Chem., 5/ Physics, 6/ Science univ., 7/ Eng.sciences, 8/ Maths (Humanities and social sciences areexcluded.
Multidisciplinary sciences papers published in PNAS, Science& Nature have been allocated to their reference discipline fordisciplinary comparisons.
3-years citations moving window for all indexes.
Linearization of the distribution.
Measuring scientific production Ranking
The extended stochastic dominance approach
Publication data
OST in-house ISI-WOS database recording of publications andcitations
Large disciplines : 1/ Fund. bio., 2/ Medicine, 3/ Ap.bio/ecol., 4/ Chem., 5/ Physics, 6/ Science univ., 7/ Eng.sciences, 8/ Maths (Humanities and social sciences areexcluded.
Multidisciplinary sciences papers published in PNAS, Science& Nature have been allocated to their reference discipline fordisciplinary comparisons.
3-years citations moving window for all indexes.
Linearization of the distribution.
Measuring scientific production Ranking
The extended stochastic dominance approach
Publication data
OST in-house ISI-WOS database recording of publications andcitations
Large disciplines : 1/ Fund. bio., 2/ Medicine, 3/ Ap.bio/ecol., 4/ Chem., 5/ Physics, 6/ Science univ., 7/ Eng.sciences, 8/ Maths (Humanities and social sciences areexcluded.
Multidisciplinary sciences papers published in PNAS, Science& Nature have been allocated to their reference discipline fordisciplinary comparisons.
3-years citations moving window for all indexes.
Linearization of the distribution.
Measuring scientific production Ranking
The extended stochastic dominance approach
Publication data
OST in-house ISI-WOS database recording of publications andcitations
Large disciplines : 1/ Fund. bio., 2/ Medicine, 3/ Ap.bio/ecol., 4/ Chem., 5/ Physics, 6/ Science univ., 7/ Eng.sciences, 8/ Maths (Humanities and social sciences areexcluded.
Multidisciplinary sciences papers published in PNAS, Science& Nature have been allocated to their reference discipline fordisciplinary comparisons.
3-years citations moving window for all indexes.
Linearization of the distribution.
Measuring scientific production Ranking
The extended stochastic dominance approach
Publication data
OST in-house ISI-WOS database recording of publications andcitations
Large disciplines : 1/ Fund. bio., 2/ Medicine, 3/ Ap.bio/ecol., 4/ Chem., 5/ Physics, 6/ Science univ., 7/ Eng.sciences, 8/ Maths (Humanities and social sciences areexcluded.
Multidisciplinary sciences papers published in PNAS, Science& Nature have been allocated to their reference discipline fordisciplinary comparisons.
3-years citations moving window for all indexes.
Linearization of the distribution.
Measuring scientific production Ranking
The extended stochastic dominance approach
Data set
French universities :
129 institutionsThe institutions checked the validity of signing patterns.
US research universities :
112 best ranked in the ARWU Shanghaı ranking (30% of PhDgranting Univ.)
Measuring scientific production Ranking
The extended stochastic dominance approach
Data set
French universities :
129 institutions
The institutions checked the validity of signing patterns.
US research universities :
112 best ranked in the ARWU Shanghaı ranking (30% of PhDgranting Univ.)
Measuring scientific production Ranking
The extended stochastic dominance approach
Data set
French universities :
129 institutionsThe institutions checked the validity of signing patterns.
US research universities :
112 best ranked in the ARWU Shanghaı ranking (30% of PhDgranting Univ.)
Measuring scientific production Ranking
The extended stochastic dominance approach
Data set
French universities :
129 institutionsThe institutions checked the validity of signing patterns.
US research universities :
112 best ranked in the ARWU Shanghaı ranking (30% of PhDgranting Univ.)
Measuring scientific production Ranking
The extended stochastic dominance approach
Data set
French universities :
129 institutionsThe institutions checked the validity of signing patterns.
US research universities :
112 best ranked in the ARWU Shanghaı ranking (30% of PhDgranting Univ.)
Measuring scientific production Ranking
The extended stochastic dominance approach
Networks of dominance relations
Let us consider �, which could be any one of the dominancerelations examined above (Iφ
k ,Bφk or Dφ
k , with ∀φ ∈ ]0, 1]).
Let’s build the dominance directed network ~g associated todominance relation � and the institutions set I by writing adirect link from institution i to institution j if i � j .
In this network, transitive triplets are uninformative since thetransitivity property holds. Therefore, let’s build the network~g ′ derived from ~g by deleting all such triplets : ∀i , j , k ∈ I , ifij , ik, jk ∈ ~g and kj /∈ ~g then ik /∈ ~g ′.
Measuring scientific production Ranking
The extended stochastic dominance approach
Networks of dominance relations
Let us consider �, which could be any one of the dominancerelations examined above (Iφ
k ,Bφk or Dφ
k , with ∀φ ∈ ]0, 1]).
Let’s build the dominance directed network ~g associated todominance relation � and the institutions set I by writing adirect link from institution i to institution j if i � j .
In this network, transitive triplets are uninformative since thetransitivity property holds. Therefore, let’s build the network~g ′ derived from ~g by deleting all such triplets : ∀i , j , k ∈ I , ifij , ik, jk ∈ ~g and kj /∈ ~g then ik /∈ ~g ′.
Measuring scientific production Ranking
The extended stochastic dominance approach
Networks of dominance relations
Let us consider �, which could be any one of the dominancerelations examined above (Iφ
k ,Bφk or Dφ
k , with ∀φ ∈ ]0, 1]).
Let’s build the dominance directed network ~g associated todominance relation � and the institutions set I by writing adirect link from institution i to institution j if i � j .
In this network, transitive triplets are uninformative since thetransitivity property holds. Therefore, let’s build the network~g ′ derived from ~g by deleting all such triplets : ∀i , j , k ∈ I , ifij , ik , jk ∈ ~g and kj /∈ ~g then ik /∈ ~g ′.
Measuring scientific production Ranking
The extended stochastic dominance approach
Dominance network : an example
Measuring scientific production Ranking
The extended stochastic dominance approach
Reference classes
Definition
∀k ∈ I , j ∈ c�i (⊆ I ) the reference class of institution i associatedto dominance relation � if i � j and j � i or if i � j and j � i
Reading example : j ∈ cBki means it is not always possible to
rank strictly and in a unique manner the scientific productionof i and j in domain k relying on any implicit valuationfunction of articles according to their impact which would beboth positive and increasing with impact.
Measuring scientific production Ranking
The extended stochastic dominance approach
Reference classes
Definition
∀k ∈ I , j ∈ c�i (⊆ I ) the reference class of institution i associatedto dominance relation � if i � j and j � i or if i � j and j � i
Reading example : j ∈ cBki means it is not always possible to
rank strictly and in a unique manner the scientific productionof i and j in domain k relying on any implicit valuationfunction of articles according to their impact which would beboth positive and increasing with impact.
Measuring scientific production Ranking
The extended stochastic dominance approach
Dominance Networks : Top French universities in FundBio, φ = 1 - citations
Paris 6
Paris 11
Strasbourg 1
Paris 5 Montpellier 2
Aix Marseille 2
Paris 7 Lyon 1
Grenoble 1
Measuring scientific production Ranking
The extended stochastic dominance approach
Dominance Networks : Top French universities in FundBio, φ = .1 - citations
Paris 6
Paris 11
Strasbourg 1
Paris 5 Montpelier 2 Aix Marseille 2
Paris 7
Lyon 1 Grenoble 1
Measuring scientific production Ranking
The extended stochastic dominance approach
Dominance Networks : Top French universities in FundBio, φ = .1 - impact factor
Paris 6
Paris 11
Strasbourg 1
Paris 5
Montpellier 2
Aix Marseille 2
Paris 7
Lyon 1
Grenoble 1
Measuring scientific production Ranking
The extended stochastic dominance approach
Complete dominance relations and ranking
Definition
A dominance relation � is said to be I -complete if ∀i , j ∈ I , i � jor j � i .
Definition
A (complete) dominance ranking R�I can be constructed overinstitutions set I on the basis of dominance relation � if and onlyif � is an I -complete dominance relation.
Measuring scientific production Ranking
The extended stochastic dominance approach
Complete dominance relations and ranking
Definition
A dominance relation � is said to be I -complete if ∀i , j ∈ I , i � jor j � i .
Definition
A (complete) dominance ranking R�I can be constructed overinstitutions set I on the basis of dominance relation � if and onlyif � is an I -complete dominance relation.
Measuring scientific production Ranking
The extended stochastic dominance approach
Complete dominance ranking
Definition
For each type of dominance (strong dominance, dominance, weakdominance), the smallest and strictly positive value of φ for whichthe corresponding dominance relations are I -complete are calledmax-I -complete dominance relations.
1 2 3 4 5 6 7 8
I .009 .004 .005 .022 .016 .019 .002 .009
B .009 .004 .008 .024 .016 .026 .002 .009
D .009 .004 .008 .024 .052 .026 .007 .009
The largest φ such that such dominance relation is I -complete overthe set of 129 French higher Education and research institutions
Measuring scientific production Ranking
The extended stochastic dominance approach
Complete dominance ranking
Definition
For each type of dominance (strong dominance, dominance, weakdominance), the smallest and strictly positive value of φ for whichthe corresponding dominance relations are I -complete are calledmax-I -complete dominance relations.
1 2 3 4 5 6 7 8
I .009 .004 .005 .022 .016 .019 .002 .009
B .009 .004 .008 .024 .016 .026 .002 .009
D .009 .004 .008 .024 .052 .026 .007 .009
The largest φ such that such dominance relation is I -complete overthe set of 129 French higher Education and research institutions
Measuring scientific production Ranking
The extended stochastic dominance approach
Complete dominance ranking
Definition
For each type of dominance (strong dominance, dominance, weakdominance), the smallest and strictly positive value of φ for whichthe corresponding dominance relations are I -complete are calledmax-I -complete dominance relations.
1 2 3 4 5 6 7 8
I .009 .004 .005 .022 .016 .019 .002 .009
B .009 .004 .008 .024 .016 .026 .002 .009
D .009 .004 .008 .024 .052 .026 .007 .009
The largest φ such that such dominance relation is I -complete overthe set of 129 French higher Education and research institutions
Measuring scientific production Ranking
The extended stochastic dominance approach
Pseudo dominance ranking
Definition
For any dominance relation and any set of institutions I , a pseudodominance ranking can be established on the basis of the score ofeach institution i , si = # {j |ij ∈ ~g }, the number of dominancerelations which emanate from institution i .
Measuring scientific production Ranking
The extended stochastic dominance approach
Pseudo dominance ranking : : US universities
Fund Biology I1 B1 D1
citations ri si ri si ri si
Harvard 1 111 1 111 1 111
John Hopkins 3 95 2 106 4 108
UCSF 7 91 3 105 3 109
Pennsylvania 3 95 4 103 8 104
UCLA 2 96 4 103 9 103
UCSD 5 92 6 100 6 105
Yale 11 87 6 100 5 107
Stanford 15 81 6 100 2 110
UW Seattle 5 92 9 99 12 99
Columbia 11 87 10 98 6 105
Measuring scientific production Ranking
The extended stochastic dominance approach
Conclusion
Combinations of influence, quality and quality.
Ranking can also be good way of doing economics (and notjust playing a childish game).
Measuring scientific production Ranking
The extended stochastic dominance approach
Conclusion
Combinations of influence, quality and quality.
Ranking can also be good way of doing economics (and notjust playing a childish game).