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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