Elite, Periphery and Symmetry in Social Networks_Periphery... · Elite, Periphery and Symmetry in...

Elite, Periphery and Symmetry in Social Networks:

An Axiomatic Approach

Joint work with Zvi Lotker ,David Peleg, Yvonne-Anne Pignolet and Itzik Turkel

Chen Avin

Monday, March 24, 2014

Elite, Periphery and Symmetry in Social Networks: An Axiomatic Approach - Stochastic Graph Models - ICERM, March 18, 2014

“Every people is governed by an elite, by a chosen element

of the population”

V. Pareto, Mind and Society, 1935



Who is the Elite?



Who is the Elite?

• “The richest, most powerful, best educated or best trained group in a society.” (Cambridge Dictionary)



Who is the Elite?


• “... is a small group of people who control a disproportionate amount of wealth or political power” (Wikipedia)



Who is the Elite?


• “... is a small group of people who control a disproportionate amount of wealth or political power” (Wikipedia)

• What about the network?



Our Goal

• Elites and masses in (social) networks

• Most basic high-level structure of society

• Study core-periphery relations and dynamics from a macro perspective

• Are these universal properties?

• The asymptotic size of the elite

• The “relation” with periphery



Universal Properties

• Small World - [Milgram 67]

• Clustering - [Watts and Strogatz 98]

• Degree distribution [Albert, Jeong, Barabasi 99]

• Navigability - [Kleinberg 00]

• Densification - [Leskovec, Kleinberg, Faloutsos 05]

• Shrinking Diameter - [LKF05].


Elite, Periphery and Symmetry in Social Networks: An Axiomatic Approach - Stochastic Graph Models - ICERM, March 18, 2014Monday, March 24, 2014


• “Six Degree of separation”

NE

MA

Small World




• The friends of my friends are my friends

NE

MA

Clustering

Small World





• Power law degree distribution

NE

MA

Clustering

Degree dist.

Small World

hubsP (k) ≈ k−α






• Local greedy routing

NE

MA

Clustering

Navigability

Degree dist.

Small World








• The graph is getting denser (over time)

NE

MA

Clustering

Navigability

Degree dist.

Densification

Small World








• The graph is getting denser (over time)

• The distance is getting smaller (over time)

NE

MA

Clustering

Navigability

Degree dist.

Densification

Shrinking Diameter

Small World




Are core-periphery properties universal?


Elite, Periphery and Symmetry in Social Networks: An Axiomatic Approach - Stochastic Graph Models - ICERM, March 18, 2014Monday, March 24, 2014


139 nodes924 edges



CAPTAIN AMERICA

THING

BEAST THOR

IRON MAN

VISION

HUMAN TORCH

SPIDERMAN

MR. FANTASTIC

SCARLET WITCH

WOLVERINE INVISIBLE WOMAN

STORM HULK

WASP

CYCLOPS

PROFESSOR X

ANTMAN

HAWK

WONDER MAN

COLOSSUS II

SUBMARINER

SHEHULK

ICEMAN

MARVEL GIRL

HERCULES GREEK GOD

ANGEL

139 nodes924 edgesC:252 edges P:249 edges



20 40 60 80 100 120 140Core Size

200

400

600

800

Number of Edges

I,I,I,




20 40 60 80 100 120 140Core Size

200

400

600

800

Number of Edges

I,I,I,


Symmetry Point

27 nodes 112 nodes



20 40 60 80 100 120 140Core Size

200

400

600

800

Number of Edges

I,I,I,


Symmetry Point

1 50 100 139

1

50

100

139

1 50 100 139

1

50

100

139

27 nodes 112 nodes



139 nodes924 edgesC:252 edges27 nodesP:249 edges112 nodes




• Core-Periphery universal properties???

• Core-Periphery “symmetry”






• Core size of 19% (27/139)? linear in network size







• Core size of ≈ ? sub-linear!√m (

√924)







• Core size of ≈ ? sub-linear!

• US population: 324M people. Elite of 3M (1%) or 18K ???

√m (

√924)

(√324M)



Axiomatic Approach



Axiomatic Approach

• Traditional approach

• Empirical results → random model



Axiomatic Approach

• Traditional approach

• Empirical results → random model

• We propose axiomatic approach:

• Small set of axioms → additional implications

• If agreed - any future models should satisfy axioms

• Asymptotic behavior from empirical data



Elite Axioms



Elite Axioms

• Axioms about influence & power



Elite Axioms


• Dominance - The elite has large influence on the society



Elite Axioms



• Robustness - The elite protects its member from outside influence



Elite Axioms




• Elite is beneficial to its members



Elite Axioms





• Minimality



Elite Axioms





• Minimality

• Conflicting expansion & reduction forces



Influence



Influence

• Social Network as a graph G(V,E)



Influence

• Social Network as a graph

• Influence between sets

G(V,E)

I(X,Y ), X, Y ⊆ V



Influence



• partition:

G(V,E)

I(X,Y ), X, Y ⊆ V

I(E , E), I(E ,P), I(P, E), I(P,P)(E ,P)



Influence



• partition:

• edges as influence

G(V,E)

I(X,Y ), X, Y ⊆ V

E(X,Y )

I(E , E), I(E ,P), I(P, E), I(P,P)(E ,P)



Influence



• partition:


• Adjacency matrix

G(V,E)

I(X,Y ), X, Y ⊆ V

E(X,Y )

I(E , E), I(E ,P), I(P, E), I(P,P)(E ,P)

E(E , E)

E(E ,P)

E(P, E) E(P,P)

1 50 100 139

1

50

100

139

1 50 100 139

1

50

100

139



Influence



• partition:


• Adjacency matrix

•

G(V,E)

I(X,Y ), X, Y ⊆ V

E(X,Y )

I(X,Y ) = |E(X,Y )|

I(E , E), I(E ,P), I(P, E), I(P,P)(E ,P)

E(E , E)

E(E ,P)

E(P, E) E(P,P)

1 50 100 139

1

50

100

139

1 50 100 139

1

50

100

139



Symmetry Point

•

• Partition is at symmetry point if:

• For undirected graphs:

I(X) = I(X,X) + I(X,V \X)

(E ,P)

I(E) = I(P)

I(E) = I(P) =⇒ I(E , E) = I(P,P)



Elite Axioms(A1) Elite-Dominance:

(A2) Elite-Robustness:

(A3) Elite-compactness:

The elite is a minimal set of agents that satisfies the dominance and robustness axioms (A1) and (A2).

I(E ,P) ≥ cd · I(P,P)

I(E , E) ≥ cr · I(P, E)



Axioms and SymmetryTheorem: For every partition such that the elite satisfies the dominance, robustness and compactness axioms (A1), (A2) and (A3),

1.

2.

3. is near its symmetry point, i.e.,

If the elite is not over-dominant,

(E ,P)

I(E) = Θ(m)

I(P) = Θ(m)

(E ,P) I(E) ≈ I(P)

I(P,P) = Θ(m)



Axioms and SymmetryTheorem: For every partition such that the elite satisfies the dominance, robustness and compactness axioms (A1), (A2) and (A3),

1.

2.

3. is near its symmetry point, i.e.,

If the elite is not over-dominant,

(E ,P)

I(E) = Θ(m)

I(P) = Θ(m)

(E ,P) I(E) ≈ I(P)

I(P,P) = Θ(m)

Balance



20 40 60 80 100 120 140Core Size

200

400

600

800

Number of Edges

I,I,I,

Core-Periphery balance

Theorem: In a random configuration graph and any positive degree sequence d the expected cut is maximized at the symmetry point.



20 40 60 80 100 120 140Core Size

200

400

600

800

Number of Edges

I,I,I,



Symmetry Point



20 40 60 80 100 120 140Core Size

200

400

600

800

Number of Edges

I,I,I,



Symmetry Point

Max cut



Axioms and Elite Size



Axioms and Elite Size Theorem: If the elite satisfies Axioms (A1) and (A2) the size of the elite is . |E| = Ω(

√m)



Axioms and Elite Size Theorem: If the elite satisfies Axioms (A1) and (A2) the size of the elite is .

Let

|E| = Ω(√m)

I(E , E) = |E|1+γ



Axioms and Elite Size Theorem: If the elite satisfies Axioms (A1) and (A2) the size of the elite is .

Let

Theorem: In a network where the elite satisfies axioms (A1) and (A2) and the elite has a density , then the elite size satisfies:

|E| = Ω(√m)

γ

|E| = Θm

11+γ

I(E , E) = |E|1+γ



Is the elite sub-linear?Assume:

Theorem: In a network where the elite satisfies axioms (A1), (A2) and then the elite size is sub-linear, i.e,

γ > µ

m = n1+µ I(E , E) = |E|1+γ

|E| = Θn

1+µ1+γ

= o(n)



Empirical Results

• About 150 networks all sizes (up to millions nodes and 10th millions edges)

• Networks with time information ∼10

• Measure: symmetry, axioms, elite size

• Sizes of interest: symmetry point and

• Two selection methods: k-rich-club, k-core

√m



Symmetry point

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Slashdot 851083, 116573<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Buzznet 8101163, 2763066<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Livemocha 8104103, 2193083<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Delicious8536408, 1366136<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Digg 8771229, 5907413<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

LastFm 81191812, 4519340<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Pokec 81632803, 22301964<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

LiveJournal 82238731, 12816184<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nxIHÿ,ÿLêm

Flixster82523386, 7918801<



Symmetry point

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Slashdot 851083, 116573<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Buzznet 8101163, 2763066<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Livemocha 8104103, 2193083<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Delicious8536408, 1366136<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Digg 8771229, 5907413<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

LastFm 81191812, 4519340<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

Pokec 81632803, 22301964<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

IHÿ,ÿLêm

LiveJournal 82238731, 12816184<

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nxIHÿ,ÿLêm

Flixster82523386, 7918801<Signat

ure



Symmetry point

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

nx

IHÿ,ÿLêm

Median

IHE,ELm

IHP,PLm

IHE,PLm

∼150 networks, median elite size ∼n0.75



Symmetry point

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

nx

IHÿ,ÿLêm

Median

IHE,ELm

IHP,PLm

IHE,PLm

∼150 networks, median elite size ∼n0.75

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

»E»ên

IHÿ,ÿLêm

Median

IHE,ELm

IHP,PLm

IHE,PLm



dominance and robustnessI(E ,P) ≥ cd · I(P,P) I(E , E) ≥ cr · I(P, E)



dominance and robustness

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

Slashdot

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

Buzznet

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

Livemocha

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

Delicious

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

Digg

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

LastFm

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

Pokec

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

LiveJournal

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

nx

Flixster

rd

rr

I(E ,P) ≥ cd · I(P,P) I(E , E) ≥ cr · I(P, E)



dominance and robustness

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

nx

rd

rr

∼150 networks, median dominance and robustness

I(E ,P) ≥ cd · I(P,P) I(E , E) ≥ cr · I(P, E)



Size over time

20 40 60 80 1000.00

0.01

0.02

0.03

0.04

0.05

0.06

timeH%L@31 monthsD

Epinions8131580, 711210<

20 40 60 80 1000.00

0.05

0.10

0.15

0.20

timeH%L@52 monthsD

Facebook 845813, 183412<

20 40 60 80 1000.00

0.01

0.02

0.03

0.04

0.05

0.06

timeH%L@7 monthsD

Flickr 82302925, 22838276<

20 40 60 80 1000.00

0.02

0.04

0.06

0.08

0.10

timeH%L@33 monthsD

Slashdot 851083, 116573<



Summary

• Core-Periphery structure - the basic hierarchy of society

• Axiomatic approach

• Universal properties

• Balance, and the role of periphery

• Scaling laws of the the elite size



Future Work

• Evolutionary explanation for the structure

• Power law degree distribution may not be sufficient

• “Algorithmic” implications of the structure

• Identifying the elite

• What can you learn about the network from its Elite?


Thank You! www.bgu.ac.il/~avin


http://www.bgu.ac.il/~avin

http://www.bgu.ac.il/~avin

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