Summerseminar 2007

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VIP-club phenomenon: emergence

of elites and masterminds in social

networksNaoki Masuda, Norio Konno,

Social Networks, 28, 297-309 (2006)

Takashi Umeda,Deguchi Lab.,

Department of Computational Intelligence

and Systems Science,

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Outline

1. My Objectives

2. Introduction to graph theory

3. Introduction(chap.1-2)

4. Model(chap.3)

5. VIP-Club phenomenon(chap.4)

6. Conclusions(chap.6)

Paper Introduction

1.My Objectives

• My Interest: social phenomenon on the

online network

– Example: Information diffusion on electric

bulletin boards

• Building a model for that, it will be useful

to study about various models of social

networks

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2. The introduction to the “Graph theory”

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1. My Objectives



4. Model(chap.3)



2-1. Definition of Term(1/2)

• Vertex– Adjacent vertices of

v1 is {v2,v3,v4}

– Adjacent vertex is alsocalled „neighbor‟

• Edge

• k:Vertex Degree

• p(k): Probability density function of k– p(3) = 0.5, p(2)= 0.5

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V1

V2

V4

V3

2-1. Definition of Term(2/2)

• C:Clustering coefficient

– The probability of the

case that

a friend's friend is my

friend

• L:

– The average distance

between any two

vertices6

Graph A

Graph B

2-2.Introduction to Complex

Network Theory• Scale-free

– p(k) follows the power-law distribution

– p(k) ∝ k-γ , γ > 0

– Example: WWW(γ ∈ [1.9 ,2.7])

• Small-world – L : smaller

– C : larger

– Example: Six Degrees of Separations

• A network in the real world often satisfies the property of both 'scale-free' and 'small-world'

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3.Introduction

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1. My Objectives



4. Model(chap.3)



3-1.Definition of Hub

• Definition: vertex directly linking to a major

part of networks

– Example: Opinion leader

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k=3

k=1

A major part of

networksHub

Opinion Leader

k=1

k=1

mass

3-2. Definition of Elite(1/2)

• Elite: a vertex with a large utility value

• Utility Value: utility function such as Eq.(1)

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Benefit

Direct and indirect

connecting to ohters

Cost

Being exposed to

others

Trade

off

• kl : the

number of

vertices at

distance l

• C : cost

• δ: discount

factor

)1(1

　　Ckkl

l

l

3-2. Definition of Elite(2/2)

11

hub

Elite

hub

A vertex is not directly

but indirectly

linking to the majority

k 2 5:Larger 1

110:LargerUtility 8

The majority

3-3.Example of Elites

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Benefit

manipulatin

g hubs

Cost

To expose

themselves to a

major part of

networks

System Crackers (Elite)

Trade

off

Objectives

•To invade a major part of networks

•Not to be detected by the authority

• There are a lots of examples of elites in the real world

3-4.Purpose of This Paper

• Revealing how hubs and elites emerge

– Existence of elites has been neglected in past

years

– Existence of hubs has been researched

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3-5.Intrinsic Weight of Each Vertex

• The intrinsic weight of individual

vertices(w) is introduced

– Probability of linking to arbitrary two vertices

is based on w

– w is individual attribute

• fame, social status ,asset..

• Weight of the i-th vertex is denoted by wi

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3-6.Thresholdings

– Example : Diffusion of computer virus at a host

• Computer virus often invade the host with low

security level

• w : security level15

Definition

•Property that a edge is assumed to form based

on a threshold conditions about w

•Property that a edge has a direction from

vertices with larger w to ones with smaller w

3-7.Homophily

– Similar agents: agents having a near value of w

– Example: In the human relation, a cluster will

be made of people that has near household

income

• w: Household income 16

Definition

The property that similar agents tend to flock

together

4.Model

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1. My Objectives



4. Model(chap.3)



4-1. The Outline of Model

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1 2

5

Wi

7 1

n vertices are prepared

Wiare randomly and independently

chosen from a distribution f(w)

Rule 1: thresholdings

Rule 2:homophily

Rule 3 : thresholdings

Edges are formed by rule1-3

+

4-2.Rule1(1/2)

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Rule 1: Two vertices with weights w and

w’ are connected if w + w’ > θ

10

11

2

1

Example: θ = 10

4-2. Rule1(2/2)

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– By rule 1, the model becomes Threshold Graph

– Scale-free networks with the small-world

properties result from various f(w)

4-3. Rule2

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Rule 2: Homophily rule

making the connection probability

decreasing with | w'– w| < c

10

11

2

1

Example: c = 2

4-4.Rule3 (1/2)

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Rule 3: Directed edge w →w' may form only

when w> w'

10

11

2

1

Example:

4-4.Rule3(2/2)

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– By rule 1,2 and 3, a vertex with w sends directed edges

to ones with

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– By rule 1,2 and 3, a vertex with w'' satisfies following

formula

– w'': The Weight of neighbor's neighbor

2,,

)6(22

,,

2,

'

cwwcw

cwww

w

w

　　

　　

2',','

)9(2

'2

,','

2',

''

cwwcw

cwww

w

w

　　

　　

4-5. Properties of the model

• k is obtained analytically as a function of w

– k is derived by integrating f(w’) over the range

given in Eq.(6)

• Hubs: vertices with w = wc

– K(w) takes maximum at w = wc

• Elites: vertices with w >> wc

– These are not exposed via direct edges to the

major group of vertices with small w24

4-6. Concrete Case(1/2)

• Case: f(w) = λe-λw

• k and k2 can be derived

• k2(w):

– k2:This is the number of the

vertices within two hops from

vertex with weight w

– k2(w) is derived from

Eq.(6)and (9)

2525

V1 V2

V4

V3

V5

5. VIP-club Phenomenon

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1. My Objectives



4. Model(chap.3)



5-1. Results

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Rule

1

Model Result

Rule

3

Rule

1

Rule

2

Rule

3Rich-club phenomenon

VIP-club phenomenon (New)

Threshold Graph

VIP-club: cluster made of elites

Rich-club: cluster made of hubs

Model A

Model B

5-2. Rich-club phenomenon(1/2)

2828

1 2

5

3 1 2

6

3Majority

Hubs

Elites

Hub is directly linked to the majority

5-2. Rich-club Phenomenon(2/2)

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k(w) k2(w)

•Hubs : w > whub

•k , k2: larger

Hub

Elites: none

Wmajority < whub

whub

• Rich-club phenomenon is shown in this figure

• Elites don‟t exist but hub exist

5-3. VIP-club phenomenon(1/2)

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1 2

5

3

7

1 2

5

3

6

Majority

Hubs

Elites

• Elite is indirectly linked to the majority

• k: smaller, k2: larger

Hub is directly linked to the majority

5-3. VIP-club Phenomenon(2/2)

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k(w)

k2(w)

Hub Elite•Elites :vertices

not directly

linked to the

majority

• k : smaller

•Elites : vertices indirectly

linked to the majority

• k2 : larger

• Hub : vertices

directly linked to

the majority

• k,k2 : larger

whub welite• wmajority < whub < welite

Majority

6.Conclusions

• The Combination of homophily and thresholding induces networks with elites– Loss of homophily leads to the rich-club

phenomenon

• Intrinsic properties of individual vertices is very important– Elite and the majority of vertices with small

weights remain undistinguished if based on vertex properties such as k or C

– To understand the nature of a network, intrinsic properties of each vertex are essential

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Summerseminar 2007

Technology

Transcript of Summerseminar 2007