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Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
Summerseminar 2007
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Summerseminar 2007

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  • 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

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