(joint work with sebastiano vigna et al.) · spectral centralities (1) eigenvector centrality:...
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(joint work with Sebastiano Vigna et al.)
Axioms for centrality: rank monotonicity for PageRank
Paolo BoldiUniversità degli Studi di Milano
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TOC
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TOC❖ Why centrality
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TOC❖ Why centrality
❖ Orienteering in the centrality jungle
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TOC❖ Why centrality
❖ Orienteering in the centrality jungle
❖ Some important centrality indices
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TOC❖ Why centrality
❖ Orienteering in the centrality jungle
❖ Some important centrality indices
❖ Why axioms
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TOC❖ Why centrality
❖ Orienteering in the centrality jungle
❖ Some important centrality indices
❖ Why axioms❖ Orienteering in the axiom jungle
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TOC❖ Why centrality
❖ Orienteering in the centrality jungle
❖ Some important centrality indices
❖ Why axioms❖ Orienteering in the axiom jungle
❖ Some important axioms
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TOC❖ Why centrality
❖ Orienteering in the centrality jungle
❖ Some important centrality indices
❖ Why axioms❖ Orienteering in the axiom jungle
❖ Some important axioms
❖ Focus on rank monotonicity for PageRank
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TOC❖ Why centrality
❖ Orienteering in the centrality jungle
❖ Some important centrality indices
❖ Why axioms❖ Orienteering in the axiom jungle
❖ Some important axioms
❖ Focus on rank monotonicity for PageRank
❖ Conclusions
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IR System
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IR System
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IR System
D
Document Repertoire- pages retrieved by Google- items on sale on Amazon- members of facebook- tweets posted by your friends- photographs on Instagram
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IR System
D
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IR System
D
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IR System
D
Set of Queries (query language)- SE query - product recommendation- new-friend suggestion- tweets to be shown
Q
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IR System
D
Q
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IR System
D
Q
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IR System
D
Q
Result- a selected
subset S⊆D- with a score
(typically: a non-negative real number) assigned to every element of S
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IR System
D
Q
Result- a selected
subset S⊆D- with a score
(typically: a non-negative real number) assigned to every element of S
IMPORTANT
Often D is endowed
with a graph structure
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IR System: 1st simplification
D
Q
Result- a selected
subset S⊆D- with a score
(typically: a non-negative real number) assigned to every element of S
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IR System: 1st simplification
D
Q
No selection, only scores
Result- a selected
subset S⊆D- with a score
(typically: a non-negative real number) assigned to every element of S
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IR System: 1st simplification
D
Q
Result- a score
assigned to every element of D
No selection, only scores
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IR System: 1st simplification
D
Q
Result- a score
assigned to every element of D
No selection, only scores
The system can be formally represented
as a function:c: Q × D → ℝ
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IR System: 2nd simplification
D
Q
Result- a score
assigned to every element of D
The system can be formally represented
as a function:c: Q × D → ℝ
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IR System: 2nd simplification
D
Q
Result- a score
assigned to every element of D
Scores do not depend
on the query
The system can be formally represented
as a function:c: Q × D → ℝ
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IR System: 2nd simplification
D
Q
Result- a score
assigned to every element of D
Scores do not depend
on the query
The system can be formally represented
as a function:c: Q × D → ℝ
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IR System: 2nd simplification
D
Q
Result- a score
assigned to every element of D
Scores do not depend
on the query
The system can be formally represented
as a function:c: D → ℝ
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IR System: 3rd simplification
D
Result- a score
assigned to every element of D
The system can be formally represented
as a function:c: D → ℝ
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IR System: 3rd simplification
D
Result- a score
assigned to every element of D
Scores depend
only on the linkage
structure on D
The system can be formally represented
as a function:c: D → ℝ
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IR System: 3rd simplification
D
Result- a score
assigned to every element of D
Scores depend
only on the linkage
structure on D
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Centrality
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Centrality
❖ The system, given a graph G assigns a score to every node of G: cG: VG → ℝ
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Centrality
❖ The system, given a graph G assigns a score to every node of G: cG: VG → ℝ
❖ The nodes of G are precisely our documents (VG =D)
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Centrality
❖ The system, given a graph G assigns a score to every node of G: cG: VG → ℝ
❖ The nodes of G are precisely our documents (VG =D)
❖ This is what people refers to as a centrality index (or measure, or score, or just “centrality”)
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Centrality in social sciences
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Centrality in social sciences
❖ First works by Bavelas at MIT (1946)
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Centrality in social sciences
❖ First works by Bavelas at MIT (1946)
❖ This sparked countless works (Bavelas 1951; Katz 1953; Shaw 1954; Beauchamp 1965; Mackenzie 1966; Burgess 1969; Anthonisse 1971; Czapiel 1974…)
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Centrality in social sciences
❖ First works by Bavelas at MIT (1946)
❖ This sparked countless works (Bavelas 1951; Katz 1953; Shaw 1954; Beauchamp 1965; Mackenzie 1966; Burgess 1969; Anthonisse 1971; Czapiel 1974…)
❖ Brought to CS through IR
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Centrality in social sciences
❖ First works by Bavelas at MIT (1946)
❖ This sparked countless works (Bavelas 1951; Katz 1953; Shaw 1954; Beauchamp 1965; Mackenzie 1966; Burgess 1969; Anthonisse 1971; Czapiel 1974…)
❖ Brought to CS through IR
❖ Key role in modern IR (=search engines)
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Orienteering in the jungle of centrality indices
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Orienteering in the jungle of centrality indices
❖ Path-based indices, based on the number of paths or shortest paths (geodesics) passing through a vertex [betweenness, Katz, …]
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Orienteering in the jungle of centrality indices
❖ Path-based indices, based on the number of paths or shortest paths (geodesics) passing through a vertex [betweenness, Katz, …]
❖ Spectral indices, based on some linear-algebra construction [PageRank, Seeley, …]
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Orienteering in the jungle of centrality indices
❖ Path-based indices, based on the number of paths or shortest paths (geodesics) passing through a vertex [betweenness, Katz, …]
❖ Spectral indices, based on some linear-algebra construction [PageRank, Seeley, …]
❖ Geometric indices, based on distances from a vertex to other vertices [closeness, harmonic, …]
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Orienteering in the jungle of centrality indices
❖ Path-based indices, based on the number of paths or shortest paths (geodesics) passing through a vertex [betweenness, Katz, …]
❖ Spectral indices, based on some linear-algebra construction [PageRank, Seeley, …]
❖ Geometric indices, based on distances from a vertex to other vertices [closeness, harmonic, …]
❖ (Actually, the first two families are largely the same, even if that wasn’t fully understood for a long time)
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Path-based centralities
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Path-based centralities
❖ Centrality depends on the paths entering (or passing through) the node
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Path-based centralities
❖ Centrality depends on the paths entering (or passing through) the node
❖ Katz’s index is a paradigmatic example
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Path-based centralities
❖ Centrality depends on the paths entering (or passing through) the node
❖ Katz’s index is a paradigmatic example
❖ Among them: betweenness (Anthonisse 1971), über-popular among social scientists
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The path tribe: betweenness and Katz (Anthonisse 1971; Katz 1953)
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❖ Betweenness centrality: cbetw(x) =
X
y,z 6=x
�yz(x)
�yz
The path tribe: betweenness and Katz (Anthonisse 1971; Katz 1953)
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❖ Betweenness centrality:
Fraction of shortest paths from y to z passing through x
cbetw(x) =X
y,z 6=x
�yz(x)
�yz
The path tribe: betweenness and Katz (Anthonisse 1971; Katz 1953)
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❖ Betweenness centrality: cbetw(x) =
X
y,z 6=x
�yz(x)
�yz
The path tribe: betweenness and Katz (Anthonisse 1971; Katz 1953)
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❖ Betweenness centrality:
❖ Katz centrality:
cbetw(x) =X
y,z 6=x
�yz(x)
�yz
The path tribe: betweenness and Katz (Anthonisse 1971; Katz 1953)
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❖ Betweenness centrality:
❖ Katz centrality:
cbetw(x) =X
y,z 6=x
�yz(x)
�yz
cKatz(x) =1X
t=0
↵t⇧x(t) = 11X
t=0
↵tGt
The path tribe: betweenness and Katz (Anthonisse 1971; Katz 1953)
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❖ Betweenness centrality:
❖ Katz centrality: # of paths of length t ending in x
cbetw(x) =X
y,z 6=x
�yz(x)
�yz
cKatz(x) =1X
t=0
↵t⇧x(t) = 11X
t=0
↵tGt
The path tribe: betweenness and Katz (Anthonisse 1971; Katz 1953)
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❖ Betweenness centrality:
❖ Katz centrality:
cbetw(x) =X
y,z 6=x
�yz(x)
�yz
cKatz(x) =1X
t=0
↵t⇧x(t) = 11X
t=0
↵tGt
The path tribe: betweenness and Katz (Anthonisse 1971; Katz 1953)
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Spectral centralities (1)
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Spectral centralities (1)❖ Eigenvector centrality: consider the left or right dominant eigenvector
of the adjacency matrix
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Spectral centralities (1)❖ Eigenvector centrality: consider the left or right dominant eigenvector
of the adjacency matrix
❖ First works by Edmund Landau in 1895 on matrices coming from chess tournaments
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Spectral centralities (1)❖ Eigenvector centrality: consider the left or right dominant eigenvector
of the adjacency matrix
❖ First works by Edmund Landau in 1895 on matrices coming from chess tournaments
❖ Motivation: take a matrix M that in entry Mxy has 1 if x won playing with y, 0 if he/she lost, 1/2 for a draw
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Spectral centralities (1)❖ Eigenvector centrality: consider the left or right dominant eigenvector
of the adjacency matrix
❖ First works by Edmund Landau in 1895 on matrices coming from chess tournaments
❖ Motivation: take a matrix M that in entry Mxy has 1 if x won playing with y, 0 if he/she lost, 1/2 for a draw
❖ M1T (the row sums) are a nice score; M21T even better, but it oscillates: so take score s such that MsT = λsT
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Spectral centralities (1)❖ Eigenvector centrality: consider the left or right dominant eigenvector
of the adjacency matrix
❖ First works by Edmund Landau in 1895 on matrices coming from chess tournaments
❖ Motivation: take a matrix M that in entry Mxy has 1 if x won playing with y, 0 if he/she lost, 1/2 for a draw
❖ M1T (the row sums) are a nice score; M21T even better, but it oscillates: so take score s such that MsT = λsT
❖ Berge (1958) extends to general social graphs and develops the theory
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Spectral centralities (1)❖ Eigenvector centrality: consider the left or right dominant eigenvector
of the adjacency matrix
❖ First works by Edmund Landau in 1895 on matrices coming from chess tournaments
❖ Motivation: take a matrix M that in entry Mxy has 1 if x won playing with y, 0 if he/she lost, 1/2 for a draw
❖ M1T (the row sums) are a nice score; M21T even better, but it oscillates: so take score s such that MsT = λsT
❖ Berge (1958) extends to general social graphs and develops the theory
❖ A similar idea was proposed by Seeley to evaluate children popularity
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The spectral tribe: Seeley index (Seeley 1949)
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The spectral tribe: Seeley index (Seeley 1949)
❖ Basic idea: in a group of children, a child is as popular as the sum of the popularities of the children who like him, but popularities are divided evenly among friends:
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The spectral tribe: Seeley index (Seeley 1949)
❖ Basic idea: in a group of children, a child is as popular as the sum of the popularities of the children who like him, but popularities are divided evenly among friends: cSeeley(x) =
X
y!x
cSeeley(y)
d+(y)
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The spectral tribe: Seeley index (Seeley 1949)
❖ Basic idea: in a group of children, a child is as popular as the sum of the popularities of the children who like him, but popularities are divided evenly among friends:
❖ In general it is a left dominant eigenvector of Gr
cSeeley(x) =X
y!x
cSeeley(y)
d+(y)
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Spectral centralities (2)
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Spectral centralities (2)❖ In 1998, Page, Brin, Motwani and Winograd propose a
spectral ranking for the web: PageRank
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Spectral centralities (2)❖ In 1998, Page, Brin, Motwani and Winograd propose a
spectral ranking for the web: PageRank
❖ After some changes in the definition, it stabilizes to a Markov chain αGr + (1 – α)1Tv
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Spectral centralities (2)❖ In 1998, Page, Brin, Motwani and Winograd propose a
spectral ranking for the web: PageRank
❖ After some changes in the definition, it stabilizes to a Markov chain αGr + (1 – α)1Tv
❖ Gr is Seeley’s matrix, α is the damping factor and v the preference vector
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Spectral centralities (2)❖ In 1998, Page, Brin, Motwani and Winograd propose a
spectral ranking for the web: PageRank
❖ After some changes in the definition, it stabilizes to a Markov chain αGr + (1 – α)1Tv
❖ Gr is Seeley’s matrix, α is the damping factor and v the preference vector
❖ This is just Katz’s index with ℓ1-normalization, i.e., (1 – α)v∑t≥0 αtGrt = (1 – α)v(1 – αGr)–1
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The spectral tribe: PageRank (Brin, Page, Motwani, Winograd 1999)
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The spectral tribe: PageRank (Brin, Page, Motwani, Winograd 1999)
❖ The recursive version of the definition (for uniform preference) is
cpr(x) = α∑y→x
cpr(x)
d+(x)+ (1 − α)vx
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The spectral tribe: PageRank (Brin, Page, Motwani, Winograd 1999)
❖ The recursive version of the definition (for uniform preference) is
cpr(x) = α∑y→x
cpr(x)
d+(x)+ (1 − α)vx
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The spectral tribe: PageRank (Brin, Page, Motwani, Winograd 1999)
❖ The recursive version of the definition (for uniform preference) is
❖ … or the dominant eigenvector of the Google matrix
cpr(x) = α∑y→x
cpr(x)
d+(x)+ (1 − α)vx
αGr + (1 − α)1Tv
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Geometric centralities and neighbourhood functions
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❖ Define the distance-count function
Geometric centralities and neighbourhood functions
DG(x, t) = #{z ∣ dG(z, x) = t}
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❖ Define the distance-count function
❖ DG(x,-) is the distance-count vector of x
Geometric centralities and neighbourhood functions
DG(x, t) = #{z ∣ dG(z, x) = t}
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❖ Define the distance-count function
❖ DG(x,-) is the distance-count vector of x
❖ BTW: in 1-to-1 correspondence with the better known “neighbourhood function”
Geometric centralities and neighbourhood functions
DG(x, t) = #{z ∣ dG(z, x) = t}
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❖ Define the distance-count function
❖ DG(x,-) is the distance-count vector of x
❖ BTW: in 1-to-1 correspondence with the better known “neighbourhood function”
❖ A geometric centrality is a function of the distance-count vector (i.e., two nodes with the same distance-count vector have the same centrality)
Geometric centralities and neighbourhood functions
DG(x, t) = #{z ∣ dG(z, x) = t}
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The geometric tribe: closeness and harmonic (Bavelas 1946; B., Vigna 2013)
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❖ Closeness centrality:
cclos(x) =1P
y d(y, x)
The geometric tribe: closeness and harmonic (Bavelas 1946; B., Vigna 2013)
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❖ Closeness centrality:
Distance from y to x
cclos(x) =1P
y d(y, x)
The geometric tribe: closeness and harmonic (Bavelas 1946; B., Vigna 2013)
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❖ Closeness centrality:
cclos(x) =1P
y d(y, x)
The geometric tribe: closeness and harmonic (Bavelas 1946; B., Vigna 2013)
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❖ Closeness centrality:
❖ The summation is over all y such that d(y,x)<∞
cclos(x) =1P
y d(y, x)
The geometric tribe: closeness and harmonic (Bavelas 1946; B., Vigna 2013)
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❖ Closeness centrality:
❖ The summation is over all y such that d(y,x)<∞
❖ Harmonic centrality:
cclos(x) =1P
y d(y, x)
charm(x) =X
y 6=x
1
d(y, x)
The geometric tribe: closeness and harmonic (Bavelas 1946; B., Vigna 2013)
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❖ Closeness centrality:
❖ The summation is over all y such that d(y,x)<∞
❖ Harmonic centrality:
❖ Inspired by (Marchiori, Latora 2000), but may be dated back to (Harris 1954)
cclos(x) =1P
y d(y, x)
charm(x) =X
y 6=x
1
d(y, x)
The geometric tribe: closeness and harmonic (Bavelas 1946; B., Vigna 2013)
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Making sense of centrality
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Making sense of centrality❖ Centrality indices can be studied
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Making sense of centrality❖ Centrality indices can be studied
❖ individually (each single centrality index is a world in its own right)
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Making sense of centrality❖ Centrality indices can be studied
❖ individually (each single centrality index is a world in its own right)
❖ comparatively
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Making sense of centrality❖ Centrality indices can be studied
❖ individually (each single centrality index is a world in its own right)
❖ comparatively
❖ Both kinds of studies can be based on
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Making sense of centrality❖ Centrality indices can be studied
❖ individually (each single centrality index is a world in its own right)
❖ comparatively
❖ Both kinds of studies can be based on
❖ external source of ground truth
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Making sense of centrality❖ Centrality indices can be studied
❖ individually (each single centrality index is a world in its own right)
❖ comparatively
❖ Both kinds of studies can be based on
❖ external source of ground truth
❖ axioms (abstract desirable/undesirable properties)
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Axioms for Centrality
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Axioms for Centrality
❖ Various attempts, with different flavours: (Sabidussi 1966), (Nieminen 1973), (Kitti 2012), (Brandes et al. 2012), (B. & Vigna 2014)
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Axioms for Centrality
❖ Various attempts, with different flavours: (Sabidussi 1966), (Nieminen 1973), (Kitti 2012), (Brandes et al. 2012), (B. & Vigna 2014)
❖ Sometimes aimed at specific indices (e.g. PageRank): (Chien et al. 2004), (Altman and Tennenholtz 2005)
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Orienteering in the jungle of axioms
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Orienteering in the jungle of axioms
❖ Invariance properties
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Orienteering in the jungle of axioms
❖ Invariance properties
❖ Score-dominance properties
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Orienteering in the jungle of axioms
❖ Invariance properties
❖ Score-dominance properties
❖ Rank-dominance properties
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Orienteering in the jungle of axioms
❖ Invariance properties
❖ Score-dominance properties
❖ Rank-dominance properties
❖ Many other axioms that still need a classification
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Invariance properties
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Invariance properties
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Invariance properties
❖ Two graphs G and G’…
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Invariance properties
❖ Two graphs G and G’…
❖ …and two nodes x∈G and x’∈G’…
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Invariance properties
❖ Two graphs G and G’…
❖ …and two nodes x∈G and x’∈G’…
❖ …satisfying some constraints
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Invariance properties
❖ Two graphs G and G’…
❖ …and two nodes x∈G and x’∈G’…
❖ …satisfying some constraints cG(x) = cG′�(x′�)
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Example: Invariance by isomorphism
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Example: Invariance by isomorphism
❖ If G and G’ are isomorphic (via isomorphism f: G → G’)
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Example: Invariance by isomorphism
❖ If G and G’ are isomorphic (via isomorphism f: G → G’)
❖ and f(x)=x’
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Example: Invariance by isomorphism
❖ If G and G’ are isomorphic (via isomorphism f: G → G’)
❖ and f(x)=x’
cG(x) = cG′�(x′�)
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Example: Invariance by isomorphism
❖ If G and G’ are isomorphic (via isomorphism f: G → G’)
❖ and f(x)=x’
cG(x) = cG′�(x′�)
This is so fundamental that it is often given for granted aspart of the notion of centrality!
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Example: Invariance by neighbours
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Example: Invariance by neighbours❖ Let G be a graph and x, x’ be two nodes such that
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Example: Invariance by neighbours❖ Let G be a graph and x, x’ be two nodes such that
❖ NG-(x)=NG-(x’) and NG+(x)=NG+(x’)
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Example: Invariance by neighbours❖ Let G be a graph and x, x’ be two nodes such that
❖ NG-(x)=NG-(x’) and NG+(x)=NG+(x’)
cG(x) = cG(x′�)
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Example: Invariance by neighbours❖ Let G be a graph and x, x’ be two nodes such that
❖ NG-(x)=NG-(x’) and NG+(x)=NG+(x’)
cG(x) = cG(x′�)
“Two nodes with the same (in- and out-)neighbours have
the same centrality”
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Invariance by neighbours…
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Invariance by neighbours…
❖ It is easy to verify that all geometric centralities are invariant by neighbours
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Invariance by neighbours…
❖ It is easy to verify that all geometric centralities are invariant by neighbours
❖ Same for spectral centralities
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Example: Invariance by in-neighbours
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Example: Invariance by in-neighbours
❖ Let G be a graph and x, x’ be two nodes such that
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Example: Invariance by in-neighbours
❖ Let G be a graph and x, x’ be two nodes such that
❖ NG-(x)=NG-(x’)
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Example: Invariance by in-neighbours
❖ Let G be a graph and x, x’ be two nodes such that
❖ NG-(x)=NG-(x’)
cG(x) = cG(x′�)
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Example: Invariance by in-neighbours
❖ Let G be a graph and x, x’ be two nodes such that
❖ NG-(x)=NG-(x’)
cG(x) = cG(x′�)
“Two nodes with the same in-neighbours have the same centrality”
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Invariance by in-neighbours…
❖ A superficial observer may believe that geometric centralities satisfy invariance by in-neighbours
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Invariance by in-neighbours…
❖ A superficial observer may believe that geometric centralities satisfy invariance by in-neighbours
x x’
NG-(x)=NG-(x’)
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Invariance by in-neighbours…
❖ A superficial observer may believe that geometric centralities satisfy invariance by in-neighbours
x x’
z
A shortest path from z to x
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Invariance by in-neighbours…
❖ A superficial observer may believe that geometric centralities satisfy invariance by in-neighbours
x x’
z
A shortest path from z to xcan be turned into a
shortest path from z to x’
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Invariance by in-neighbours…
❖ A superficial observer may believe that geometric centralities satisfy invariance by in-neighbours
x x’
z
∀z ∉ {x, x′�} dG(z, x) = dG(z, x′�)
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Invariance by in-neighbours…
❖ A superficial observer may believe that geometric centralities satisfy invariance by in-neighbours
x x’
z
DG(x, − ) and DG(x′�, − ) are almost the same...
5 0 3 7 2 0 0 4 1 …DG(x, − )
5 0 3 6 2 0 0 5 1 …DG(x′�, − )
1 2 3 4 5 6 7 8 9 …
1 2 3 4 5 6 7 8 9 …
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Invariance by in-neighbours…
❖ A superficial observer may believe that geometric centralities satisfy invariance by in-neighbours
x x’
z
DG(x, − ) and DG(x′�, − ) are almost the same...
5 0 3 7 2 0 0 4 1 …DG(x, − )
5 0 3 6 2 0 0 5 1 …DG(x′�, − )
The difference (+1 in one position, -1 in another position) depends on the values of d(x,x’) and d(x’,x)
1 2 3 4 5 6 7 8 9 …
1 2 3 4 5 6 7 8 9 …
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Invariance by in-neighbours…
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Invariance by in-neighbours…
❖ So, in general, geometric centralities are not invariant by in-neighbours
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Invariance by in-neighbours…
❖ So, in general, geometric centralities are not invariant by in-neighbours
❖ They are on symmetric (i.e. undirected) graphs, though
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Invariance by in-neighbours…
❖ So, in general, geometric centralities are not invariant by in-neighbours
❖ They are on symmetric (i.e. undirected) graphs, though
❖ But spectral centralities (e.g. PageRank) are invariant by in-neighbours
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Score-dominance properties
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Score-dominance properties
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Score-dominance properties
❖ Two graphs G and G’…
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Score-dominance properties
❖ Two graphs G and G’…
❖ …and two nodes x∈G and x’∈G’…
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Score-dominance properties
❖ Two graphs G and G’…
❖ …and two nodes x∈G and x’∈G’…
❖ …satisfying some constraints
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Score-dominance properties
❖ Two graphs G and G’…
❖ …and two nodes x∈G and x’∈G’…
❖ …satisfying some constraints
cG(x) ≥ cG′�(x′�)
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Score-dominance properties
❖ Two graphs G and G’…
❖ …and two nodes x∈G and x’∈G’…
❖ …satisfying some constraints
❖ Sometimes > is required (strict dominance)
cG(x) ≥ cG′�(x′�)
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Example: Dominance by in-neighbours(Schoch and Brandes, 2016)
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Example: Dominance by in-neighbours(Schoch and Brandes, 2016)
❖ Let G be a graph and x, x’ be two nodes such that
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Example: Dominance by in-neighbours(Schoch and Brandes, 2016)
❖ Let G be a graph and x, x’ be two nodes such that
❖ NG-(x)⊆NG-(x’)
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Example: Dominance by in-neighbours(Schoch and Brandes, 2016)
❖ Let G be a graph and x, x’ be two nodes such that
❖ NG-(x)⊆NG-(x’) cG(x) ≤ cG(x′�)
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Example: Dominance by in-neighbours(Schoch and Brandes, 2016)
❖ Let G be a graph and x, x’ be two nodes such that
❖ NG-(x)⊆NG-(x’)
❖ Observe: if a measure satisfies this property, it is also invariant by in-neighbours
cG(x) ≤ cG(x′�)
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Example: Dominance by in-neighbours(Schoch and Brandes, 2016)
❖ Let G be a graph and x, x’ be two nodes such that
❖ NG-(x)⊆NG-(x’)
❖ Observe: if a measure satisfies this property, it is also invariant by in-neighbours
❖ ⇒ geometric centralities do not satisfy “dominance by in-neighbours”
cG(x) ≤ cG(x′�)
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Example: Score-dominance by arc addition (a.k.a. score monotonicity)
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Example: Score-dominance by arc addition (a.k.a. score monotonicity)
❖ If G is a graph not containing the arc x→y
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Example: Score-dominance by arc addition (a.k.a. score monotonicity)
❖ If G is a graph not containing the arc x→y
❖ And G’=G ∪ {x→y}
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Example: Score-dominance by arc addition (a.k.a. score monotonicity)
❖ If G is a graph not containing the arc x→y
❖ And G’=G ∪ {x→y}
❖ Then cG′�(y) > cG(y)
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Example: Score-dominance by arc addition (a.k.a. score monotonicity)
❖ If G is a graph not containing the arc x→y
❖ And G’=G ∪ {x→y}
❖ Then cG′�(y) > cG(y) “Adding one arc towards y
(strictly) increases its score”
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Example: Score-dominance by arc addition (a.k.a. score monotonicity)
❖ If G is a graph not containing the arc x→y
❖ And G’=G ∪ {x→y}
❖ Then
❖ The weak version (with ≥) also makes sense
cG′�(y) > cG(y) “Adding one arc towards y (strictly) increases its score”
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Score monotonicity (“Axioms for Centrality”, B. & Vigna 2014)
General Strongly connected
Seeley no yes
PageRank yes yes
betweenness no no
Katz yes yes
closeness no yes
harmonic yes yes
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Score monotonicity (“Axioms for Centrality”, B. & Vigna 2014)
General Strongly connected
Seeley no yes
PageRank yes yes
betweenness no no
Katz yes yes
closeness no yes
harmonic yes yes
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PageRank satisfies score monotonicity
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PageRank satisfies score monotonicity
❖ Proved by (Chien, Dwork, Kumar, Simon and Sivakumar 2003) for the case when all nodes have nonzero PageRank
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PageRank satisfies score monotonicity
❖ Proved by (Chien, Dwork, Kumar, Simon and Sivakumar 2003) for the case when all nodes have nonzero PageRank
❖ Generalized in (B. and Vigna, 2014) to the case rx > 0
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Closeness does not satisfy score monotonicity
z y
x
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Closeness does not satisfy score monotonicity
z y
x
cG(y) =1
∑t dG(t, y)=
11
= 1
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Closeness does not satisfy score monotonicity
z y
x
cG(y) =1
∑t dG(t, y)=
11
= 1
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Closeness does not satisfy score monotonicity
z y
x
cG(y) =1
∑t dG(t, y)=
11
= 1
cG′�(y) =1
∑t dG′ �(t, y)=
11 + 1
=12
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Closeness satisfies score monotonicity in the strongly connected case
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Closeness satisfies score monotonicity in the strongly connected case
cclos(x) =1P
y d(y, x)
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Closeness satisfies score monotonicity in the strongly connected case
❖ On strongly connected graphscclos(x) =
1Py d(y, x)
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Closeness satisfies score monotonicity in the strongly connected case
❖ On strongly connected graphs
❖ the summation includes all nodes
cclos(x) =1P
y d(y, x)
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Closeness satisfies score monotonicity in the strongly connected case
❖ On strongly connected graphs
❖ the summation includes all nodes
❖ the distances do not increase after adding the new arc
cclos(x) =1P
y d(y, x)
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Closeness satisfies score monotonicity in the strongly connected case
❖ On strongly connected graphs
❖ the summation includes all nodes
❖ the distances do not increase after adding the new arc
❖ at least one distance strictly decreases
cclos(x) =1P
y d(y, x)
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Closeness satisfies score monotonicity in the strongly connected case
❖ On strongly connected graphs
❖ the summation includes all nodes
❖ the distances do not increase after adding the new arc
❖ at least one distance strictly decreases
❖ So closeness centrality is score monotone on strongly connected graphs!
cclos(x) =1P
y d(y, x)
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Rank-dominance properties
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Rank-dominance properties
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Rank-dominance properties
❖ Two graphs G and G’…
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Rank-dominance properties
❖ Two graphs G and G’…
❖ …and two nodes x∈G and x’∈G’…
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Rank-dominance properties
❖ Two graphs G and G’…
❖ …and two nodes x∈G and x’∈G’…
❖ …satisfying some constraints
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Rank-dominance properties
❖ Two graphs G and G’…
❖ …and two nodes x∈G and x’∈G’…
❖ …satisfying some constraints
❖ The rank of x’ in G’ is “not less” than the rank of x in G
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Rank-dominance properties
❖ Two graphs G and G’…
❖ …and two nodes x∈G and x’∈G’…
❖ …satisfying some constraints
❖ The rank of x’ in G’ is “not less” than the rank of x in GTypically stated on two graphs with the same
set of nodes, and for a single node
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Rank-dominance properties revised(weak version)
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Rank-dominance properties revised(weak version)
❖ Two graphs G and G’ with the same node set V and node x∈V
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Rank-dominance properties revised(weak version)
❖ Two graphs G and G’ with the same node set V and node x∈V
1. ∀z . cG(x) > cG(z) ⟹ cG′�(x) > cG′�(z)
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Rank-dominance properties revised(weak version)
❖ Two graphs G and G’ with the same node set V and node x∈V
1.
2.2.
∀z . cG(x) > cG(z) ⟹ cG′�(x) > cG′�(z)
∀y . cG(x) = cG(y) ⟹ cG′�(x) ≥ cG′�(y)
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Rank-dominance properties revised(strict version)
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Rank-dominance properties revised(strict version)
❖ Two graphs G and G’ with the same node set V and node x∈V
∀z . cG(x) ≥ cG(z) ⟹ cG′�(x) > cG′�(z)
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Example: Rank-dominance by arc addition (a.k.a. rank monotonicity)
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Example: Rank-dominance by arc addition (a.k.a. rank monotonicity)
❖ If G is a graph not containing the arc x→y
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Example: Rank-dominance by arc addition (a.k.a. rank monotonicity)
❖ If G is a graph not containing the arc x→y
❖ And G’=G ∪ {x→y}
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Example: Rank-dominance by arc addition (a.k.a. rank monotonicity)
❖ If G is a graph not containing the arc x→y
❖ And G’=G ∪ {x→y}
❖ Then, for all z ∀z . cG(y) > cG(z) ⟹ cG′�(y) > cG′�(z)
∀z . cG(y) = cG(z) ⟹ cG′�(y) ≥ cG′�(z)
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Example: Rank-dominance by arc addition (a.k.a. rank monotonicity)
❖ If G is a graph not containing the arc x→y
❖ And G’=G ∪ {x→y}
❖ Then, for all z
❖ For the strict version, the last ≥ should become a >
∀z . cG(y) > cG(z) ⟹ cG′�(y) > cG′�(z)∀z . cG(y) = cG(z) ⟹ cG′�(y) ≥ cG′�(z)
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Rank monotonicity (“Rank monotonicity in centrality measures.”, B. & Luongo & Vigna 2017)
General Strongly connected
Seeley no yes
PageRank† yes* yes*
betweenness no no
Katz† yes* yes*
closeness no yes
harmonic yes* yes*
† provided that no node has null preference
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Rank monotonicity (“Rank monotonicity in centrality measures.”, B. & Luongo & Vigna 2017)
General Strongly connected
Seeley no yes
PageRank† yes* yes*
betweenness no no
Katz† yes* yes*
closeness no yes
harmonic yes* yes*
(no)
(yes)
(no)
(yes)
(no)
(yes)
(yes)
(yes)
(no)
(yes)
(yes)
(yes)
† provided that no node has null preference
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Rank monotonicity (“Rank monotonicity in centrality measures.”, B. & Luongo & Vigna 2017)
General Strongly connected
Seeley no yes
PageRank† yes* yes*
betweenness no no
Katz† yes* yes*
closeness no yes
harmonic yes* yes*
† provided that no node has null preference
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Rank monotonicity (“Rank monotonicity in centrality measures.”, B. & Luongo & Vigna 2017)
General Strongly connected
Seeley no yes
PageRank† yes* yes*
betweenness no no
Katz† yes* yes*
closeness no yes
harmonic yes* yes*
†
† provided that no node has null preference
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PageRank and rank monotonicity
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Loose (non-strict) rank monotonicity
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Loose (non-strict) rank monotonicity❖ For PageRank, with G’=G ∪ {x→y} holds (Chien, Dwork, Kumar, Simon & Sivakumar 2004) for everywhere nonzero score
∀y . cG(y) > cG(z) ⟹ cG′�(y) > cG′�(z)∀y . cG(y) = cG(z) ⟹ cG′�(y) ≥ cG′�(z)
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Loose (non-strict) rank monotonicity❖ For PageRank, with G’=G ∪ {x→y} holds (Chien, Dwork, Kumar, Simon & Sivakumar 2004) for everywhere nonzero score
❖ The strict version was proved in (B., Luongo, Vigna 2017) for everywhere nonzero preference
∀y . cG(y) > cG(z) ⟹ cG′�(y) > cG′�(z)∀y . cG(y) = cG(z) ⟹ cG′�(y) ≥ cG′�(z)
∀y . cG(y) ≥ cG(z) ⟹ cG′�(y) > cG′�(z)
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Loose vs. strict
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Loose vs. strict
❖ The proof in (Chien, Dwork, Kumar, Simon & Sivakumar 2004) exploits the fact that the Google matrix is a regular Markov chain
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Loose vs. strict
❖ The proof in (Chien, Dwork, Kumar, Simon & Sivakumar 2004) exploits the fact that the Google matrix is a regular Markov chain
❖ (B., Luongo, Vigna 2017) is based on some properties of M-matrices…
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Loose vs. strict
❖ The proof in (Chien, Dwork, Kumar, Simon & Sivakumar 2004) exploits the fact that the Google matrix is a regular Markov chain
❖ (B., Luongo, Vigna 2017) is based on some properties of M-matrices…
❖ …the results have wider applicability (e.g., Katz)
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Damped spectral ranking
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Damped spectral ranking
❖ Let M be a nonnegative matrix, 0 < α < 1/ρ(M), v a strictly positive vector
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Damped spectral ranking
❖ Let M be a nonnegative matrix, 0 < α < 1/ρ(M), v a strictly positive vector
❖ Then, the centrality vector r defined by satisfies strict rank monotonicity, suitably generalised to matrices (see below)
r = v(I − αM)−1
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Damped spectral ranking
❖ Let M be a nonnegative matrix, 0 < α < 1/ρ(M), v a strictly positive vector
❖ Then, the centrality vector r defined by satisfies strict rank monotonicity, suitably generalised to matrices (see below)
❖ Applies to PageRank, Katz, …
r = v(I − αM)−1
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Lemma (ext. Willoughby, 1977)C = (I − αM)−1 r = vC
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Lemma (ext. Willoughby, 1977)z
C = (I − αM)−1 r = vCy
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Lemma (ext. Willoughby, 1977)z
C = (I − αM)−1 r = vCy
cwz÷cwy
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Lemma (ext. Willoughby, 1977)z
C = (I − αM)−1 r = vCy
cwz÷cwy
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Lemma (ext. Willoughby, 1977)z
C = (I − αM)−1 r = vC
y
y
cwz÷cwy
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Lemma (ext. Willoughby, 1977)z
C = (I − αM)−1 r = vC
y
y
cwz÷cwy
÷ cyzcyy
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Lemma (ext. Willoughby, 1977)
Assume cyz> 0 and let q = cyy/cyz
zC = (I − αM)−1 r = vC
y
y
cwz÷cwy
÷ cyzcyy
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Lemma (ext. Willoughby, 1977)
Assume cyz> 0 and let q = cyy/cyz
zC = (I − αM)−1 r = vC
y
y
cwz÷cwy
÷ cyzcyy
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Lemma (ext. Willoughby, 1977)
Assume cyz> 0 and let q = cyy/cyz
zC = (I − αM)−1 r = vC
• Then cwy/cwz ≤ q
y
y
cwz÷cwy
÷ cyzcyy
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Lemma (ext. Willoughby, 1977)
Assume cyz> 0 and let q = cyy/cyz
zC = (I − αM)−1 r = vC
y
y
cwz÷cwy
÷ cyzcyy
• Then cwy ≤ q cwz
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Lemma (ext. Willoughby, 1977)
Assume cyz> 0 and let q = cyy/cyz
z
As a consequence if q < 1…
ry = ∑w
vwcwy ≤
C = (I − αM)−1 r = vC
y
y
cwz÷cwy
÷ cyzcyy
• Then cwy ≤ q cwz
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Lemma (ext. Willoughby, 1977)
Assume cyz> 0 and let q = cyy/cyz
z
As a consequence if q < 1…
C = (I − αM)−1 r = vC
y
y
cwz÷cwy
÷ cyzcyy
• Then cwy ≤ q cwz
ry = ∑w
vwcwy ≤ ∑w
vwcwzcyy
cyz
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Lemma (ext. Willoughby, 1977)
Assume cyz> 0 and let q = cyy/cyz
z
As a consequence if q < 1…
C = (I − αM)−1 r = vC
y
y
cwz÷cwy
÷ cyzcyy
• Then cwy ≤ q cwz
ry = ∑w
vwcwy ≤ ∑w
vwcwzcyy
cyz≤ ∑
w
vwcwz = rz
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Lemma (ext. Willoughby, 1977)
Assume cyz> 0 and let q = cyy/cyz
z
As a consequence if q < 1…
Hence:
C = (I − αM)−1 r = vC
y
y
cwz÷cwy
÷ cyzcyy
• Then cwy ≤ q cwz
ry = ∑w
vwcwy ≤ ∑w
vwcwzcyy
cyz≤ ∑
w
vwcwz = rz
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Lemma (ext. Willoughby, 1977)
Assume cyz> 0 and let q = cyy/cyz
z
As a consequence if q < 1…
Hence:
C = (I − αM)−1 r = vC
• If rz ≤ ry then q ≥ 1• If rz < ry then q > 1
y
y
cwz÷cwy
÷ cyzcyy
• Then cwy ≤ q cwz
ry = ∑w
vwcwy ≤ ∑w
vwcwzcyy
cyz≤ ∑
w
vwcwz = rz
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Lemma (ext. Willoughby, 1977)
Assume cyz> 0 and let q = cyy/cyz
z
As a consequence if q < 1…
Hence:
THIS CONDITION IS NECESSARY!
C = (I − αM)−1 r = vC
• If rz ≤ ry then q ≥ 1• If rz < ry then q > 1
y
y
cwz÷cwy
÷ cyzcyy
• Then cwy ≤ q cwz
ry = ∑w
vwcwy ≤ ∑w
vwcwzcyy
cyz≤ ∑
w
vwcwz = rz
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PageRank as a special case ofdamped spectral ranking
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PageRank as a special case ofdamped spectral ranking
r = v(I − αM)−1
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PageRank as a special case ofdamped spectral ranking
❖ In the case of PageRank, M=Gr
r = v(I − αM)−1
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PageRank as a special case ofdamped spectral ranking
❖ In the case of PageRank, M=Gr
❖ When adding the arc x→y we obtain a new matrix M’ and
r = v(I − αM)−1
![Page 229: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/229.jpg)
PageRank as a special case ofdamped spectral ranking
❖ In the case of PageRank, M=Gr
❖ When adding the arc x→y we obtain a new matrix M’ and
r = v(I − αM)−1
M 0 �M =
0
BBBBBB@
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0. . .0 � 1
d(d+1) . . . 1d . . . 0 0
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0
1
CCCCCCA
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PageRank as a special case ofdamped spectral ranking
❖ In the case of PageRank, M=Gr
❖ When adding the arc x→y we obtain a new matrix M’ and
r = v(I − αM)−1
M 0 �M =
0
BBBBBB@
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0. . .0 � 1
d(d+1) . . . 1d . . . 0 0
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0
1
CCCCCCA
<latexit sha1_base64="PbRjgTce+w/F69vxEfT9UA/38uo=">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</latexit><latexit sha1_base64="OzaI/3hxNHfUSZsYzctOcZ+l83w=">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</latexit><latexit sha1_base64="OzaI/3hxNHfUSZsYzctOcZ+l83w=">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</latexit><latexit sha1_base64="kot7j8Jcpi2jdWj1qixZ5/gAMZY=">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</latexit>
only row x is nonzero
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PageRank as a special case ofdamped spectral ranking
❖ In the case of PageRank, M=Gr
❖ When adding the arc x→y we obtain a new matrix M’ and
r = v(I − αM)−1
M 0 �M =
0
BBBBBB@
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0. . .0 � 1
d(d+1) . . . 1d . . . 0 0
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0
1
CCCCCCA
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![Page 232: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/232.jpg)
PageRank as a special case ofdamped spectral ranking
❖ In the case of PageRank, M=Gr
❖ When adding the arc x→y we obtain a new matrix M’ and
r = v(I − αM)−1
M 0 �M =
0
BBBBBB@
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0. . .0 � 1
d(d+1) . . . 1d . . . 0 0
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0
1
CCCCCCA
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“old” outneighbours of x
![Page 233: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/233.jpg)
PageRank as a special case ofdamped spectral ranking
❖ In the case of PageRank, M=Gr
❖ When adding the arc x→y we obtain a new matrix M’ and
r = v(I − αM)−1
M 0 �M =
0
BBBBBB@
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0. . .0 � 1
d(d+1) . . . 1d . . . 0 0
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0
1
CCCCCCA
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![Page 234: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/234.jpg)
PageRank as a special case ofdamped spectral ranking
❖ In the case of PageRank, M=Gr
❖ When adding the arc x→y we obtain a new matrix M’ and
r = v(I − αM)−1
M 0 �M =
0
BBBBBB@
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0. . .0 � 1
d(d+1) . . . 1d . . . 0 0
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0
1
CCCCCCA
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only the y-th column is positive
![Page 235: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/235.jpg)
PageRank as a special case ofdamped spectral ranking
❖ In the case of PageRank, M=Gr
❖ When adding the arc x→y we obtain a new matrix M’ and
r = v(I − αM)−1
M 0 �M =
0
BBBBBB@
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0. . .0 � 1
d(d+1) . . . 1d . . . 0 0
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0
1
CCCCCCA
<latexit sha1_base64="PbRjgTce+w/F69vxEfT9UA/38uo=">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</latexit><latexit sha1_base64="OzaI/3hxNHfUSZsYzctOcZ+l83w=">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</latexit><latexit sha1_base64="OzaI/3hxNHfUSZsYzctOcZ+l83w=">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</latexit><latexit sha1_base64="kot7j8Jcpi2jdWj1qixZ5/gAMZY=">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</latexit>
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PageRank as a special case ofdamped spectral ranking
❖ In the case of PageRank, M=Gr
❖ When adding the arc x→y we obtain a new matrix M’ and
r = v(I − αM)−1
M 0 �M =
0
BBBBBB@
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0. . .0 � 1
d(d+1) . . . 1d . . . 0 0
0 0 . . . 0 . . . 0 00 0 . . . 0 . . . 0 0
1
CCCCCCA
<latexit sha1_base64="PbRjgTce+w/F69vxEfT9UA/38uo=">AAAC8nicjVLLTttAFB27PNLwCmXZzQgEDUJENhvYIKF20w0SSASQ4jQaj6+TEeOxNXONFFn+DDYsaFHFjq/prl/RX+gkzgICi1zpSkfnnrnPCTMpDHreX8f9MDe/sFj7WF9aXllda6x/ujRprjm0eSpTfR0yA1IoaKNACdeZBpaEEq7Cm2+j+NUtaCNSdYHDDLoJ6ysRC87QUr11p3b6Zf/0OJAQYzMIoS9UwbRmw7LglZV1j+7QkQdRiuYd7NEgmE00Jibi/SDWjPtF1Iz2/N3yhbwKRNMJZi0yWyegosmggRb9Ae72GlteyxsbfQv8Cdg6+fHv6Rch5KzX+GOz8jwBhVwyYzq+l2HXJkXBJZT1IDeQMX7D+tCxULEETLcYn6yk25aJaJxq6wrpmH35omCJMcMktMqE4cBMx0bke7FOjvFRtxAqyxEUrwrFuaSY0tH9aSQ0cJRDCxjXwvZK+YDZhaP9JXW7BH965Lfg8qDley3/3G7jK6msRj6TTdIkPjkkJ+Q7OSNtwp3UuXMenJ8uuvfuo/u7krrO5M0GeWXu839xsddF</latexit><latexit sha1_base64="OzaI/3hxNHfUSZsYzctOcZ+l83w=">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</latexit><latexit sha1_base64="OzaI/3hxNHfUSZsYzctOcZ+l83w=">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</latexit><latexit sha1_base64="kot7j8Jcpi2jdWj1qixZ5/gAMZY=">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</latexit>
δ
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Rank monotonicity of PageRank (1)
![Page 238: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/238.jpg)
Rank monotonicity of PageRank (1)
❖ By the Sherman-Morrison formula: for some suitable positive constant 𝜅. For simplicity I will assume 𝜅=1
r′�− r = κδ (I − αM)−1
![Page 239: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/239.jpg)
Rank monotonicity of PageRank (1)
❖ By the Sherman-Morrison formula: for some suitable positive constant 𝜅. For simplicity I will assume 𝜅=1
❖ We need to prove that
r′�− r = κδ (I − αM)−1
0 < rz ≤ ry implies r′�z < r′�y
![Page 240: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/240.jpg)
Rank monotonicity of PageRank (1)
❖ By the Sherman-Morrison formula: for some suitable positive constant 𝜅. For simplicity I will assume 𝜅=1
❖ We need to prove that
❖ We will in fact prove that
r′�− r = κδ (I − αM)−1
0 < rz ≤ ry implies r′�z < r′�y
0 < rz ≤ ry implies [r′�− r]z < [r′�− r]y
![Page 241: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/241.jpg)
Rank monotonicity of PageRank (1)
❖ By the Sherman-Morrison formula: for some suitable positive constant 𝜅. For simplicity I will assume 𝜅=1
❖ We need to prove that
❖ We will in fact prove that
r′�− r = κδ (I − αM)−1
0 < rz ≤ ry implies r′�z < r′�y
0 < rz ≤ ry implies [δ(I − αM)−1]z < [δ(I − αM)−1]y
![Page 242: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/242.jpg)
Rank monotonicity of PageRank (2)
![Page 243: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/243.jpg)
Rank monotonicity of PageRank (2)
❖ We aim at proving that
0 < rz ≤ ry implies [δ(I − αM)−1]z < [δ(I − αM)−1]y
![Page 244: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/244.jpg)
Rank monotonicity of PageRank (2)
❖ We aim at proving that
❖ Let cyz > 0 (the other case is easy), and q = cyy/cyz
0 < rz ≤ ry implies [δ(I − αM)−1]z < [δ(I − αM)−1]y
![Page 245: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/245.jpg)
Rank monotonicity of PageRank (2)
❖ We aim at proving that
❖ Let cyz > 0 (the other case is easy), and q = cyy/cyz
0 < rz ≤ ry implies [δ(I − αM)−1]z < [δ(I − αM)−1]y
[δ(1 − αM)−1]y = δycyy − ∑w≠y
|δw |cwy ≥ δyqcyz − ∑w≠y
q |δw |cwz
![Page 246: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/246.jpg)
Rank monotonicity of PageRank (2)
❖ We aim at proving that
❖ Let cyz > 0 (the other case is easy), and q = cyy/cyz
❖ By the Lemma,
0 < rz ≤ ry implies [δ(I − αM)−1]z < [δ(I − αM)−1]y
[δ(1 − αM)−1]y = δycyy − ∑w≠y
|δw |cwy ≥ δyqcyz − ∑w≠y
q |δw |cwz
rz ≤ ry ⟹ q ≥ 1
![Page 247: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/247.jpg)
Rank monotonicity of PageRank (2)
❖ We aim at proving that
❖ Let cyz > 0 (the other case is easy), and q = cyy/cyz
❖ By the Lemma,
0 < rz ≤ ry implies [δ(I − αM)−1]z < [δ(I − αM)−1]y
[δ(1 − αM)−1]y = δycyy − ∑w≠y
|δw |cwy ≥ δyqcyz − ∑w≠y
q |δw |cwz
rz ≤ ry ⟹ q ≥ 1
≥ δycyz − ∑w≠y
|δw |cwz = [δ(1 − αM)−1]z
![Page 248: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/248.jpg)
Rank monotonicity of PageRank (3)
![Page 249: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/249.jpg)
Rank monotonicity of PageRank (3)
❖ We in fact proved only
0 < rz ≤ ry implies [δ(I − αM)−1]z ≤ [δ(I − αM)−1]y
![Page 250: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/250.jpg)
Rank monotonicity of PageRank (3)
❖ We in fact proved only
❖ The strict inequality requires more work…
0 < rz ≤ ry implies [δ(I − αM)−1]z ≤ [δ(I − αM)−1]y
![Page 251: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/251.jpg)
Take-home messages
![Page 252: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/252.jpg)
Take-home messages
❖ Centrality is important and ubiquitous
![Page 253: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/253.jpg)
Take-home messages
❖ Centrality is important and ubiquitous
❖ A jungle of indices: taxonomies (and generalizations) are of help
![Page 254: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/254.jpg)
Take-home messages
❖ Centrality is important and ubiquitous
❖ A jungle of indices: taxonomies (and generalizations) are of help
❖ Axiomatization is a good way to make sense of so many indices
![Page 255: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/255.jpg)
Take-home messages
❖ Centrality is important and ubiquitous
❖ A jungle of indices: taxonomies (and generalizations) are of help
❖ Axiomatization is a good way to make sense of so many indices
❖ A jungle of axioms: taxonomies (and generalizations) are of help
![Page 256: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/256.jpg)
Take-home messages
❖ Centrality is important and ubiquitous
❖ A jungle of indices: taxonomies (and generalizations) are of help
❖ Axiomatization is a good way to make sense of so many indices
❖ A jungle of axioms: taxonomies (and generalizations) are of help
❖ Apparently trivial properties fail to hold, or require a lot of work to be proved
![Page 257: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/257.jpg)
Take-home messages
❖ Centrality is important and ubiquitous
❖ A jungle of indices: taxonomies (and generalizations) are of help
❖ Axiomatization is a good way to make sense of so many indices
❖ A jungle of axioms: taxonomies (and generalizations) are of help
❖ Apparently trivial properties fail to hold, or require a lot of work to be proved
❖ Beware, it’s a wild world out there
![Page 258: (joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality: consider the left or right dominant eigenvector of the adjacency matrix First works](https://reader030.vdocuments.us/reader030/viewer/2022011807/5c64ca1b09d3f2a36e8bca69/html5/thumbnails/258.jpg)
Thanks for your attention!