(joint work with Sebastiano Vigna et al.)

Axioms for centrality: rank monotonicity for PageRank

Paolo BoldiUniversità degli Studi di Milano

TOC❖ Why centrality

❖ Orienteering in the centrality jungle

❖ Some important centrality indices

❖ Why axioms

❖ Why axioms❖ Orienteering in the axiom jungle

❖ Some important axioms

❖ Focus on rank monotonicity for PageRank

❖ Conclusions

IR System

Document Repertoire- pages retrieved by Google- items on sale on Amazon- members of facebook- tweets posted by your friends- photographs on Instagram

IR System

Set of Queries (query language)- SE query - product recommendation- new-friend suggestion- tweets to be shown

IR System

Result- a selected

subset S⊆D- with a score

(typically: a non-negative real number) assigned to every element of S

IR System

Result- a selected

IMPORTANT

Often D is endowed

with a graph structure

IR System: 1st simplification

Result- a selected

No selection, only scores

Result- a selected

Result- a score

assigned to every element of D

Result- a score

The system can be formally represented

as a function:c: Q × D → ℝ

IR System: 2nd simplification

Result- a score

Scores do not depend

on the query

Result- a score

on the query

Result- a score

on the query

as a function:c: D → ℝ

IR System: 3rd simplification

Result- a score

Scores depend

only on the linkage

structure on D

Result- a score

Scores depend

only on the linkage

structure on D

Centrality

❖ The system, given a graph G assigns a score to every node of G: cG: VG → ℝ

Centrality

❖ The nodes of G are precisely our documents (VG =D)

Centrality

❖ 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”)

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)

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)

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

The path tribe: betweenness and Katz (Anthonisse 1971; Katz 1953)

❖ Betweenness centrality: cbetw(x) =

y,z 6=x

�yz(x)

❖ Betweenness centrality:

Fraction of shortest paths from y to z passing through x

cbetw(x) =X

y,z 6=x

�yz(x)

❖ Betweenness centrality: cbetw(x) =

y,z 6=x

�yz(x)

❖ Katz centrality:

cbetw(x) =X

y,z 6=x

�yz(x)

cbetw(x) =X

y,z 6=x

�yz(x)

cKatz(x) =1X

↵t⇧x(t) = 11X

↵tGt

❖ Katz centrality: # of paths of length t ending in x

cbetw(x) =X

y,z 6=x

�yz(x)

cKatz(x) =1X

↵t⇧x(t) = 11X

↵tGt

cbetw(x) =X

y,z 6=x

�yz(x)

cKatz(x) =1X

↵t⇧x(t) = 11X

↵tGt

Spectral centralities (1)

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

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:

❖ 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) =

cSeeley(y)

❖ 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

cSeeley(y)

Spectral centralities (2)

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

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

cpr(x) = α∑y→x

cpr(x)

d+(x)+ (1 − α)vx

❖ … or the dominant eigenvector of the Google matrix

cpr(x) = α∑y→x

cpr(x)

d+(x)+ (1 − α)vx

αGr + (1 − α)1Tv

Geometric centralities and neighbourhood functions

❖ Define the distance-count function

DG(x, t) = #{z ∣ dG(z, x) = t}

❖ DG(x,-) is the distance-count vector of x

DG(x, t) = #{z ∣ dG(z, x) = t}

❖ BTW: in 1-to-1 correspondence with the better known “neighbourhood function”

DG(x, t) = #{z ∣ dG(z, x) = t}

❖ 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)

DG(x, t) = #{z ∣ dG(z, x) = t}

The geometric tribe: closeness and harmonic (Bavelas 1946; B., Vigna 2013)

❖ Closeness centrality:

cclos(x) =1P

y d(y, x)

Distance from y to x

cclos(x) =1P

y d(y, x)

cclos(x) =1P

y d(y, x)

❖ The summation is over all y such that d(y,x)<∞

cclos(x) =1P

y d(y, x)

❖ Harmonic centrality:

cclos(x) =1P

y d(y, x)

charm(x) =X

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

d(y, x)

Making sense of centrality

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

❖ comparatively

❖ external source of ground truth

❖ comparatively

❖ external source of ground truth

❖ axioms (abstract desirable/undesirable properties)

Axioms for Centrality

❖ Various attempts, with different flavours: (Sabidussi 1966), (Nieminen 1973), (Kitti 2012), (Brandes et al. 2012), (B. & Vigna 2014)

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)

Orienteering in the jungle of axioms

❖ Invariance properties

❖ Score-dominance properties

❖ Rank-dominance properties

❖ Many other axioms that still need a classification

Invariance properties

❖ Two graphs G and G’…

❖ …and two nodes x∈G and x’∈G’…

❖ …satisfying some constraints

❖ …satisfying some constraints cG(x) = cG′�(x′�)

Example: Invariance by isomorphism

❖ If G and G’ are isomorphic (via isomorphism f: G → G’)

❖ and f(x)=x’

cG(x) = cG′�(x′�)

❖ 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!

Example: Invariance by neighbours

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”

Invariance by neighbours…

❖ It is easy to verify that all geometric centralities are invariant by neighbours

Invariance by neighbours…

❖ It is easy to verify that all geometric centralities are invariant by neighbours

❖ Same for spectral centralities

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′�)

❖ NG-(x)=NG-(x’)

cG(x) = cG(x′�)

“Two nodes with the same in-neighbours have the same centrality”

Invariance by in-neighbours…

❖ A superficial observer may believe that geometric centralities satisfy invariance by in-neighbours

x x’

NG-(x)=NG-(x’)

x x’

A shortest path from z to x

x x’

A shortest path from z to xcan be turned into a

shortest path from z to x’

x x’

∀z ∉ {x, x′�} dG(z, x) = dG(z, x′�)

x x’

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 …

x x’

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 …

❖ 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

Score-dominance properties

cG(x) ≥ cG′�(x′�)

❖ Sometimes > is required (strict dominance)

cG(x) ≥ cG′�(x′�)

Example: Dominance by in-neighbours(Schoch and Brandes, 2016)

❖ NG-(x)⊆NG-(x’)

❖ NG-(x)⊆NG-(x’) cG(x) ≤ cG(x′�)

❖ NG-(x)⊆NG-(x’)

❖ Observe: if a measure satisfies this property, it is also invariant by in-neighbours

cG(x) ≤ cG(x′�)

❖ 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′�)

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)

❖ And G’=G ∪ {x→y}

❖ Then cG′�(y) > cG(y) “Adding one arc towards y

(strictly) increases its score”

❖ 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”

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

Score monotonicity (“Axioms for Centrality”, B. & Vigna 2014)

Seeley no yes

PageRank yes yes

betweenness no no

Katz yes yes

closeness no yes

harmonic yes yes

PageRank satisfies score monotonicity

❖ Proved by (Chien, Dwork, Kumar, Simon and Sivakumar 2003) for the case when all nodes have nonzero PageRank

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

Closeness does not satisfy score monotonicity

cG(y) =1

∑t dG(t, y)=

cG(y) =1

∑t dG(t, y)=

cG(y) =1

∑t dG(t, y)=

cG′�(y) =1

∑t dG′ �(t, y)=

11 + 1

Closeness satisfies score monotonicity in the strongly connected case

cclos(x) =1P

y d(y, x)

❖ On strongly connected graphscclos(x) =

1Py d(y, x)

❖ On strongly connected graphs

❖ the summation includes all nodes

cclos(x) =1P

y d(y, x)

❖ the distances do not increase after adding the new arc

cclos(x) =1P

y d(y, x)

❖ at least one distance strictly decreases

cclos(x) =1P

y d(y, x)

❖ at least one distance strictly decreases

❖ So closeness centrality is score monotone on strongly connected graphs!

cclos(x) =1P

y d(y, x)

Rank-dominance properties

❖ The rank of x’ in G’ is “not less” than the rank of x in G

❖ 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

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)

∀z . cG(x) > cG(z) ⟹ cG′�(x) > cG′�(z)

∀y . cG(x) = cG(y) ⟹ cG′�(x) ≥ cG′�(y)

Rank-dominance properties revised(strict version)

∀z . cG(x) ≥ cG(z) ⟹ cG′�(x) > cG′�(z)

Example: Rank-dominance by arc addition (a.k.a. rank monotonicity)

❖ 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)

❖ 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)

Rank monotonicity (“Rank monotonicity in centrality measures.”, B. & Luongo & Vigna 2017)

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

Seeley no yes

betweenness no no

Katz† yes* yes*

closeness no yes

harmonic yes* yes*

Seeley no yes

betweenness no no

Katz† yes* yes*

closeness no yes

harmonic yes* yes*

Seeley no yes

betweenness no no

Katz† yes* yes*

closeness no yes

harmonic yes* yes*

PageRank and rank monotonicity

Loose (non-strict) rank monotonicity

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)

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)

Loose vs. strict

❖ The proof in (Chien, Dwork, Kumar, Simon & Sivakumar 2004) exploits the fact that the Google matrix is a regular Markov chain

Loose vs. strict

❖ (B., Luongo, Vigna 2017) is based on some properties of M-matrices…

Loose vs. strict

❖ (B., Luongo, Vigna 2017) is based on some properties of M-matrices…

❖ …the results have wider applicability (e.g., Katz)

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

❖ 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

Lemma (ext. Willoughby, 1977)C = (I − αM)−1 r = vC

Lemma (ext. Willoughby, 1977)z

C = (I − αM)−1 r = vCy

cwz÷cwy

C = (I − αM)−1 r = vCy

cwz÷cwy

C = (I − αM)−1 r = vC

cwz÷cwy

C = (I − αM)−1 r = vC

cwz÷cwy

÷ cyzcyy

Lemma (ext. Willoughby, 1977)

Assume cyz> 0 and let q = cyy/cyz

zC = (I − αM)−1 r = vC

cwz÷cwy

÷ cyzcyy

zC = (I − αM)−1 r = vC

cwz÷cwy

÷ cyzcyy

zC = (I − αM)−1 r = vC

• Then cwy/cwz ≤ q

cwz÷cwy

÷ cyzcyy

zC = (I − αM)−1 r = vC

cwz÷cwy

÷ cyzcyy

• Then cwy ≤ q cwz

As a consequence if q < 1…

ry = ∑w

vwcwy ≤

C = (I − αM)−1 r = vC

cwz÷cwy

÷ cyzcyy

C = (I − αM)−1 r = vC

cwz÷cwy

÷ cyzcyy

ry = ∑w

vwcwy ≤ ∑w

vwcwzcyy

C = (I − αM)−1 r = vC

cwz÷cwy

÷ cyzcyy

ry = ∑w

vwcwy ≤ ∑w

vwcwzcyy

cyz≤ ∑

vwcwz = rz

Hence:

C = (I − αM)−1 r = vC

cwz÷cwy

÷ cyzcyy

ry = ∑w

vwcwy ≤ ∑w

vwcwzcyy

cyz≤ ∑

vwcwz = rz

Hence:

C = (I − αM)−1 r = vC

• If rz ≤ ry then q ≥ 1• If rz < ry then q > 1

cwz÷cwy

÷ cyzcyy

ry = ∑w

vwcwy ≤ ∑w

vwcwzcyy

cyz≤ ∑

vwcwz = rz

Hence:

THIS CONDITION IS NECESSARY!

C = (I − αM)−1 r = vC

• If rz ≤ ry then q ≥ 1• If rz < ry then q > 1

cwz÷cwy

÷ cyzcyy

ry = ∑w

vwcwy ≤ ∑w

vwcwzcyy

cyz≤ ∑

vwcwz = rz

PageRank as a special case ofdamped spectral ranking

r = v(I − αM)−1

❖ In the case of PageRank, M=Gr

r = v(I − αM)−1

❖ When adding the arc x→y we obtain a new matrix M’ and

r = v(I − αM)−1

M 0 �M =

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

CCCCCCA

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r = v(I − αM)−1

M 0 �M =

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

CCCCCCA

only row x is nonzero

r = v(I − αM)−1

M 0 �M =

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

CCCCCCA

r = v(I − αM)−1

M 0 �M =

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

CCCCCCA

“old” outneighbours of x

r = v(I − αM)−1

M 0 �M =

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

CCCCCCA

r = v(I − αM)−1

M 0 �M =

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

CCCCCCA

only the y-th column is positive

r = v(I − αM)−1

M 0 �M =

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

CCCCCCA

r = v(I − αM)−1

M 0 �M =

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

CCCCCCA

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

❖ We need to prove that

r′�− r = κδ (I − αM)−1

0 < rz ≤ ry implies r′�z < r′�y

❖ We will in fact prove that

r′�− r = κδ (I − αM)−1

0 < rz ≤ ry implies [r′�− r]z < [r′�− r]y

❖ We will in fact prove that

r′�− r = κδ (I − αM)−1

0 < rz ≤ ry implies [δ(I − αM)−1]z < [δ(I − αM)−1]y

❖ We aim at proving that

❖ Let cyz > 0 (the other case is easy), and q = cyy/cyz

[δ(1 − αM)−1]y = δycyy − ∑w≠y

|δw |cwy ≥ δyqcyz − ∑w≠y

q |δw |cwz

❖ By the Lemma,

[δ(1 − αM)−1]y = δycyy − ∑w≠y

q |δw |cwz

rz ≤ ry ⟹ q ≥ 1

❖ By the Lemma,

[δ(1 − αM)−1]y = δycyy − ∑w≠y

q |δw |cwz

rz ≤ ry ⟹ q ≥ 1

≥ δycyz − ∑w≠y

|δw |cwz = [δ(1 − αM)−1]z

❖ We in fact proved only

0 < rz ≤ ry implies [δ(I − αM)−1]z ≤ [δ(I − αM)−1]y

❖ 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

Take-home messages

❖ Centrality is important and ubiquitous

Take-home messages

❖ A jungle of indices: taxonomies (and generalizations) are of help

Take-home messages

❖ Axiomatization is a good way to make sense of so many indices

Take-home messages

❖ A jungle of axioms: taxonomies (and generalizations) are of help

Take-home messages

❖ Apparently trivial properties fail to hold, or require a lot of work to be proved

Take-home messages

❖ Apparently trivial properties fail to hold, or require a lot of work to be proved

❖ Beware, it’s a wild world out there

Thanks for your attention!

(joint work with Sebastiano Vigna et al.) · Spectral centralities (1) Eigenvector centrality:...

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