1. VIP-club phenomenon: emergence
of elites and masterminds in social
networks
Naoki Masuda, Norio Konno,
Social Networks, 28, 297-309 (2006)
Takashi Umeda,
Deguchi Lab.,
Department of Computational Intelligence
and Systems Science,
1

2. Outline
1. My Objectives
2. Introduction to graph theory
3. Introduction(chap.1-2) Paper Introduction
4. Model(chap.3)
5. VIP-Club phenomenon(chap.4)
6. Conclusions(chap.6)
2

3. 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
3

4. 2. The introduction to the “Graph theory”
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)
4

5. 2-1. Definition of Term(1/2)
• Vertex
– Adjacent vertices of V1
v1 is {v2,v3,v4}
V2
– Adjacent vertex is also
called „neighbor‟
• Edge
• k:Vertex Degree V4
• p(k): Probability density V3
function of k
– p(3) = 0.5, p(2)= 0.5
5

6. 2-1. Definition of Term(2/2)
• C:Clustering coefficient
– The probability of the Graph A
case that
a friend's friend is my
friend
• L:
– The average distance Graph B
between any two
vertices
6

7. 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'
7

8. 3.Introduction
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)
8

9. 3-1.Definition of Hub
• Definition: vertex directly linking to a major
part of networks
– Example: Opinion leader
Hub A major part of
networks
k=1
k=3
k=1
k=1
Opinion Leader mass 9

10. 3-2. Definition of Elite(1/2)
• Elite: a vertex with a large utility value
• Utility Value: utility function such as Eq.(1)

• kl : the k 
l 1
l
l
 Ck
(1)
number of
vertices at
distance l Benefit Cost
• C : cost Direct and indirect Trade Being exposed to
• δ: discount connecting to ohters off others
factor
10

11. 3-2. Definition of Elite(2/2)
A vertex is not directly
but indirectly The majority
linking to the majority
hub
Elite
hub
k 2 5:Larger 1
Utility 10:Larger 1 8 11

12. 3-3.Example of Elites
• There are a lots of examples of elites in the real world
Cost
Trade To expose
Benefit
off themselves to a
manipulatin
major part of
g hubs
networks
System Crackers (Elite)
Objectives
•To invade a major part of networks
•Not to be detected by the authority 12

13. 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
13

14. 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
14

15. 3-6.Thresholdings
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
– Example : Diffusion of computer virus at a host
• Computer virus often invade the host with low
security level
• w : security level
15

16. 3-7.Homophily
Definition
The property that similar agents tend to flock
together
– 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

17. 4.Model
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)
17

18. 4-1. The Outline of Model
n vertices
7 5 1
are prepared
1 2 Wi
Wi
are randomly and independently
chosen from a distribution f(w)
+
Rule 1: thresholdings
Edges Rule 2:homophily
are formed by rule1-3 Rule 3 : thresholdings
18

19. 4-2.Rule1(1/2)
Rule 1: Two vertices with weights w and
w’ are connected if w + w’ > θ
Example: θ = 10
11 1
10 2
19

20. 4-2. Rule1(2/2)
– By rule 1, the model becomes Threshold Graph
– Scale-free networks with the small-world
properties result from various f(w)
20

21. 4-3. Rule2
Rule 2: Homophily rule
making the connection probability
decreasing with | w'– w| < c
Example: c = 2
11 1
10 2
21

22. 4-4.Rule3 (1/2)
Rule 3: Directed edge w →w' may form only
when w> w'
Example:
11 1
10 2
22

23. 4-4.Rule3(2/2)
– By rule 1,2 and 3, a vertex with w sends directed edges
to ones with

  
, w
2


w'   w, w   w    c
, ( 6)
 2 2
  c
w  c, w w 
,
 2
– By rule 1,2 and 3, a vertex with w'' satisfies following
formula
– w'': The Weight of neighbor's neighbor

 
 '  2
, w


w' '   w' , w'   w' 
,
  c
(9)
 2 2
  c
w' c, w' w' 
, 23
 2

24. 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 w
24

25. 4-6. Concrete Case(1/2)
• Case: f(w) = λe-λw V1 V2 V3
• k and k2 can be derived
V4
• k2(w):
– k2:This is the number of the V5
vertices within two hops from
vertex with weight w
– k2(w) is derived from
Eq.(6)and (9)
25

26. 5. VIP-club Phenomenon
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)
26

27. 5-1. Results
Model Result
Model A
Rule Rule
Rich-club phenomenon
1 3
Rich-club: cluster made of hubs
Threshold Graph
Rule Rule Rule
VIP-club phenomenon (New)
1 2 3
Model B VIP-club: cluster made of elites
27

28. 5-2. Rich-club phenomenon(1/2)
Hub is directly linked to the majority
1 2 3 Majority
1 2 3
5 6 Hubs
Elites
28

29. 5-2. Rich-club Phenomenon(2/2)
k(w) k2(w) Elites: none
•Hubs : w > whub
•k , k2: larger
Hub
Wmajority < whub
whub
• Rich-club phenomenon is shown in this figure 29
• Elites don‟t exist but hub exist

30. 5-3. VIP-club phenomenon(1/2)
Hub is directly linked to the majority
1 2 3 Majority
1 2 3
5 5 Hubs
7 6 Elites
• Elite is indirectly linked to the majority
• k: smaller, k2: larger 30

31. 5-3. VIP-club Phenomenon(2/2)
•Elites : vertices indirectly
• Hub : vertices
linked to the majority
directly linked to
• k2 : larger
the majority
• k,k2 : larger
k2(w)
•Elites :vertices Majority Hub Elite
not directly k(w)
linked to the
majority
• k : smaller
• wmajority < whub < welite whub welite 31

32. 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
32