Models based on preferential attachment
Analysis of PA-models
Conclusion
Local clustering coefficient in preferential
attach...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Plan
1 Models based on the preferential attachmen...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
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Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
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Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
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Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
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Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
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Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
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Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
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Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
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Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
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Properties of interest
Generalized preferential a...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Degree distribution
Clustering coefficient
Local cl...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Generalized preferential attachment models:
Power...
Models based on preferential attachment
Analysis of PA-models
Conclusion
Thank You!
Questions?
Liudmila Ostroumova, Egor S...
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Alexander Krot – Limits of Local Algorithms for Randomly Generated Constraint Satisfaction Problems

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In this talk we discuss some properties of generalized preferential attachment models. A general approach to preferential attachment was introduced in [1], where a wide class of models (PA-class) was defined in terms of constraints that are sufficient for the study of the degree distribution and the clustering coefficient.

It was shown in [1] that the degree distribution in all models of the PA-class follows the power law. Also, the global clustering coefficient was analyzed and a lower bound for the average local clustering coefficient was obtained. It was also shown that in preferential attachment models global and average local clustering coefficients behave differently.

In our study we expand the results of [1] by analyzing the local clustering coefficient for the PA-class of models. We analyze the behavior of C(d) which is the average local clustering for vertices of degree d. The value C(d) is defined in the following way. First, the local clustering of a given vertex is defined as the ratio of the number of edges between the neighbors of this vertex to the number of pairs of such neighbors. Then the obtained values are averaged over all vertices of degree d.

[1] L. Ostroumova, A. Ryabchenko, E. Samosvat, Generalized Preferential Attachment: Tunable Power-Law Degree Distribution and Clustering Coefficient, Algorithms and Models for the Web Graph, Lecture Notes in Computer Science Volume 8305, 2013, pp 185-202.

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Alexander Krot – Limits of Local Algorithms for Randomly Generated Constraint Satisfaction Problems

  1. 1. Models based on preferential attachment Analysis of PA-models Conclusion Local clustering coefficient in preferential attachment graphs Liudmila Ostroumova, Egor Samosvat, Alexander Krot Moscow Institute of Physics and Technology Moscow State University Yandex June, 2014 Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  2. 2. Models based on preferential attachment Analysis of PA-models Conclusion Plan 1 Models based on the preferential attachment 2 Theoretical analysis of the general case 3 Conclusion Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  3. 3. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Degree distribution Real-world networks often have the power law degree distribution: #{v : deg(v) = d} n ≈ c dγ , where 2 < γ < 3. Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  4. 4. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Clustering coefficient Global clustering coefficient of a graph G: C1(n) = 3#(triangles in G) #(pairs of adjacent edges in G) . Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  5. 5. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Clustering coefficient Global clustering coefficient of a graph G: C1(n) = 3#(triangles in G) #(pairs of adjacent edges in G) . Average local clustering coefficient Ti is the number of edges between the neighbors of a vertex i Pi 2 is the number of pairs of neighbors C(i) = Ti Pi 2 is the local clustering coefficient for a vertex i C2(n) = 1 n n i=1 C(i) – average local clustering coefficient Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  6. 6. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Preferential attachment Idea of preferential attachment [Barab´asi, Albert]: Start with a small graph At every step we add new vertex with m edges The probability that a new vertex will be connected to a vertex i is proportional to the degree of i Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  7. 7. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples PA-class of models Start from an arbitrary graph Gn0 m with n0 vertices and mn0 edges Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  8. 8. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples PA-class of models Start from an arbitrary graph Gn0 m with n0 vertices and mn0 edges We make Gn+1 m from Gn m by adding a new vertex n + 1 with m edges Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  9. 9. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples PA-class of models Start from an arbitrary graph Gn0 m with n0 vertices and mn0 edges We make Gn+1 m from Gn m by adding a new vertex n + 1 with m edges PA-condition: the probability that the degree of a vertex i increases by one equals A deg(i) n + B 1 n + O (deg(i))2 n2 Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  10. 10. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples PA-class of models Start from an arbitrary graph Gn0 m with n0 vertices and mn0 edges We make Gn+1 m from Gn m by adding a new vertex n + 1 with m edges PA-condition: the probability that the degree of a vertex i increases by one equals A deg(i) n + B 1 n + O (deg(i))2 n2 The probability of adding a multiple edge is O (deg(i))2 n2 Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  11. 11. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples PA-class of models Start from an arbitrary graph Gn0 m with n0 vertices and mn0 edges We make Gn+1 m from Gn m by adding a new vertex n + 1 with m edges PA-condition: the probability that the degree of a vertex i increases by one equals A deg(i) n + B 1 n + O (deg(i))2 n2 The probability of adding a multiple edge is O (deg(i))2 n2 2mA + B = m, 0 ≤ A ≤ 1 Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  12. 12. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples T-subclass Triangles property: The probability that the degree of two vertices i and j increases by one equals eij D mn + O dn i dn j n2 Here eij is the number of edges between vertices i and j in Gn m and D is a positive constant. Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  13. 13. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Bollob´as–Riordan, Buckley–Osthus, M´ori, etc. Fix some positive number a – "initial attractiveness". (Bollob´as–Riordan model: a = 1). Start with a graph with one vertex and m loops. At n-th step add one vertex with m edges. We add m edges one by one. The probability to add an edge n → i at each step is proportional to indeg(i) + am. Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  14. 14. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Bollob´as–Riordan, Buckley–Osthus, M´ori, etc. Fix some positive number a – "initial attractiveness". (Bollob´as–Riordan model: a = 1). Start with a graph with one vertex and m loops. At n-th step add one vertex with m edges. We add m edges one by one. The probability to add an edge n → i at each step is proportional to indeg(i) + am. Outdegree: m Triangles property: D = 0 PA-condition: A = 1 1+a Degree distribution: Power law with γ = 2 + a Global clustering: (log n)2 n (a = 1), log n n (a > 1) Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  15. 15. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Random Apollonian networks Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  16. 16. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Random Apollonian networks Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  17. 17. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Random Apollonian networks Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  18. 18. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Random Apollonian networks Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  19. 19. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Random Apollonian networks Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  20. 20. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Random Apollonian networks Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  21. 21. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Random Apollonian networks Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  22. 22. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Random Apollonian networks Outdegree: m = 3 Triangles property: D = 3 PA-condition: A = 1 2 Degree distribution: Power law with γ = 3 Average local clustering: constant Global clustering: tends to zero Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  23. 23. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Polynomial model Put m = 2p Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  24. 24. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Polynomial model Put m = 2p Fix α, β, δ ≥ 0 and α + β + δ = 1 Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  25. 25. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Polynomial model Put m = 2p Fix α, β, δ ≥ 0 and α + β + δ = 1 Add a new vertex i with m edges. We add m edges in p steps Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  26. 26. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Polynomial model Put m = 2p Fix α, β, δ ≥ 0 and α + β + δ = 1 Add a new vertex i with m edges. We add m edges in p steps α – probability of an indegree preferential step β – probability of an edge preferential step δ – probability of a random step Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  27. 27. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Polynomial model Put m = 2p Fix α, β, δ ≥ 0 and α + β + δ = 1 Add a new vertex i with m edges. We add m edges in p steps α – probability of an indegree preferential step β – probability of an edge preferential step δ – probability of a random step Edge preferential: cite two endpoints of a random edge Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  28. 28. Models based on preferential attachment Analysis of PA-models Conclusion Properties of interest Generalized preferential attachment Examples Polynomial model Put m = 2p Fix α, β, δ ≥ 0 and α + β + δ = 1 Add a new vertex i with m edges. We add m edges in p steps α – probability of an indegree preferential step β – probability of an edge preferential step δ – probability of a random step Edge preferential: cite two endpoints of a random edge Outdegree: 2p Triangles property: D = βp PA-condition: A = α + β 2 . Degree distribution: Power law with γ = 1 + 2 2α+β Average local clustering: constant Global clustering: constant for A > 1/2 (γ > 3), tends to zero for A ≤ 1/2 (2 < γ ≤ 3) Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  29. 29. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Degree distribution Let Nn(d) be the number of vertices with degree d in Gn m. Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  30. 30. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Degree distribution Let Nn(d) be the number of vertices with degree d in Gn m. Expectation For every d ≥ m we have ENn(d) = c(m, d) n + O d2+ 1 A , where c(m, d) = Γ d + B A Γ m + B+1 A A Γ d + B+A+1 A Γ m + B A ∼ Γ m + B+1 A d−1− 1 A A Γ m + B A and Γ(x) is the gamma function. Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  31. 31. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Degree distribution Concentration For every d = d(n) we have P |Nn(d) − ENn(d)| ≥ d √ n log n = O n− log n . Therefore, for any δ > 0 there exists a function ϕ(n) = o(1) such that lim n→∞ P ∃ d ≤ n A−δ 4A+2 : |Nn(d) − ENn(d)| ≥ ϕ(n) ENn(d) = 0 . Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  32. 32. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Global clustering Let P2(n) be the number of all path of length 2 in Gn m. P2(n) (1) If 2A < 1, then whp P2(n) ∼ 2m(A + B) + m(m−1) 2 n 1−2A . (2) If 2A = 1, then whp P2(n) ∝ n log(n) . (3) If 2A > 1, then whp P2(n) ∝ n2A . Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  33. 33. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Global clustering Let P2(n) be the number of all path of length 2 in Gn m. P2(n) (1) If 2A < 1, then whp P2(n) ∼ 2m(A + B) + m(m−1) 2 n 1−2A . (2) If 2A = 1, then whp P2(n) ∝ n log(n) . (3) If 2A > 1, then whp P2(n) ∝ n2A . Triangles Whp the number of triangles T(n) ∼ D n . Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  34. 34. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Global clustering Global clustering (1) If 2A < 1 then whp C1(n) ∼ 3(1−2A)D (2m(A+B)+ m(m−1) 2 ) . (2) If 2A = 1 then whp C1(n) ∝ (log n)−1 . (2) If 2A > 1 then whp C1(n) ∝ n1−2A . Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  35. 35. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Average local clustering Average local clustering Whp C2(n) ≥ 1 n i:deg(i)=m C(i) ≥ 2cD m(m + 1) . Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  36. 36. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Local clustering: definition Let Tn(d) be the number of triangles on the vertices of degree d in Gn m. Let Nn(d) be the number of vertices of degree d in Gn m. Local clustering coefficient: C(d) = Tn(d) Nn(d)d(d−1) 2 Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  37. 37. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Local clustering: number of triangles Tn(d) is the number of triangles on the vertices of degree d in Gn m Expectation ETn(d) = f(d) n + O d2+ 1 A , where f(d) ∼ d−1/A Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  38. 38. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Idea of the proof The following recurrent formula holds: ETn+1(d) = ETn(d) − ETn(d) · Ad + B n + O d2 n2 + +ETn(d − 1) · A(d − 1) + B n + O (d − 1)2 n2 + + j:j is a neighbor of a vertex of degree d − 1 D mn + O (d − 1) · dj n2 Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  39. 39. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Idea of the proof The following recurrent formula holds: ETn+1(d) = ETn(d) − ETn(d) · Ad + B n + O d2 n2 + +ETn(d − 1) · A(d − 1) + B n + O (d − 1)2 n2 + + j:j is a neighbor of a vertex of degree d − 1 D mn + O (d − 1) · dj n2 Next we prove that ETn(d) = f(d)(n + O(d2+ 1 A )) where f(d) ∼ d− 1 A . Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  40. 40. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Number of triangles Concentration For every d = d(n) we have P |Tn(d) − ETn(d)| ≥ d2 √ n log n = O n− log n . Therefore, for any δ > 0 there exists a function ϕ(n) = o(1) such that lim n→∞ P ∃ d ≤ n A−δ 4A+2 : |Tn(d) − ETn(d)| ≥ ϕ(n) ETn(d) = 0 . Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  41. 41. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Idea of the proof Azuma, Hoeffding Let (Xi)n i=0 be a martingale such that |Xi − Xi−1| ≤ ci for any 1 ≤ i ≤ n. Then P (|Xn − X0| ≥ x) ≤ 2e − x2 2 n i=1 c2 i for any x > 0. Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  42. 42. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Idea of the proof Azuma, Hoeffding Let (Xi)n i=0 be a martingale such that |Xi − Xi−1| ≤ ci for any 1 ≤ i ≤ n. Then P (|Xn − X0| ≥ x) ≤ 2e − x2 2 n i=1 c2 i for any x > 0. Xi(d) = E(Tn(d) | Gi m), i = 0, . . . , n. Note that X0(d) = ETn(d) and Xn(d) = Tn(d). Xn(d) is a martingale. For any i = 0, . . . , n − 1: |Xi+1(d) − Xi(d)| ≤ Md2, where M > 0 is some constant. Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  43. 43. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Idea of the proof Fix 0 ≤ i ≤ n − 1 and some graph Gi m. E Tn(d) | Gi+1 m − E Tn(d) | Gi m ≤ ≤ max ˜Gi+1 m ⊃Gi m E Tn(d) | ˜Gi+1 m − min ˜Gi+1 m ⊃Gi m E Tn(d) | ˜Gi+1 m . Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  44. 44. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Idea of the proof Fix 0 ≤ i ≤ n − 1 and some graph Gi m. E Tn(d) | Gi+1 m − E Tn(d) | Gi m ≤ ≤ max ˜Gi+1 m ⊃Gi m E Tn(d) | ˜Gi+1 m − min ˜Gi+1 m ⊃Gi m E Tn(d) | ˜Gi+1 m . ˆGi+1 m = arg max E(Tn(d) | ˜Gi+1 m ), ¯Gi+1 m = arg min E(Tn(d) | ˜Gi+1 m ). Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  45. 45. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Idea of the proof Fix 0 ≤ i ≤ n − 1 and some graph Gi m. E Tn(d) | Gi+1 m − E Tn(d) | Gi m ≤ ≤ max ˜Gi+1 m ⊃Gi m E Tn(d) | ˜Gi+1 m − min ˜Gi+1 m ⊃Gi m E Tn(d) | ˜Gi+1 m . ˆGi+1 m = arg max E(Tn(d) | ˜Gi+1 m ), ¯Gi+1 m = arg min E(Tn(d) | ˜Gi+1 m ). For i + 1 ≤ t ≤ n put δi t(d) = E(Tt(d) | ˆGi+1 m ) − E(Tt(d) | ¯Gi+1 m ) . Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  46. 46. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Idea of the proof Fix 0 ≤ i ≤ n − 1 and some graph Gi m. E Tn(d) | Gi+1 m − E Tn(d) | Gi m ≤ ≤ max ˜Gi+1 m ⊃Gi m E Tn(d) | ˜Gi+1 m − min ˜Gi+1 m ⊃Gi m E Tn(d) | ˜Gi+1 m . ˆGi+1 m = arg max E(Tn(d) | ˜Gi+1 m ), ¯Gi+1 m = arg min E(Tn(d) | ˜Gi+1 m ). For i + 1 ≤ t ≤ n put δi t(d) = E(Tt(d) | ˆGi+1 m ) − E(Tt(d) | ¯Gi+1 m ) . δi t+1(d) ≤ δi t(d) 1 − Ad + B t + O d2 t2 + +δi t(d − 1) A(d − 1) + B t + O d2 t2 + O d2 t . Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  47. 47. Models based on preferential attachment Analysis of PA-models Conclusion Degree distribution Clustering coefficient Local clustering Local clustering C(d) = Tn(d) Nn(d)d(d−1) 2 ∼ d−1/A · d1+1/A · d−2 ∼ 1 d Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  48. 48. Models based on preferential attachment Analysis of PA-models Conclusion Generalized preferential attachment models: Power law degree distribution with any exponent γ > 2 Constant global clustering coefficient only for γ > 3 Constant average local clustering coefficient Local clustering C(d) ∼ 1 d Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs
  49. 49. Models based on preferential attachment Analysis of PA-models Conclusion Thank You! Questions? Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs

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