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Random graph model for citation networks - Alessandro Garavaglia

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Random graph model for citation networks - Alessandro Garavaglia

  1. 1. Random graph model for citation networks Alessandro Garavaglia work with Remco van der Hofstad and Gerhard Woeginger Eindhoven University of Technology 22-06-2016, Como Lake School
  2. 2. Citation networks data (Web of Science) Citation networks are directed graphs. An edge between papers A and B is present when A cites B.
  3. 3. Citation networks data (Web of Science) Citation networks are directed graphs. An edge between papers A and B is present when A cites B. (PS = probability and statistics, EE = electrical engineering, BT = biotechnology and applied microbiology) Exponential growth of the graph (Log Y axis) 1980 1985 1990 1995 2000 2005 2010 104 PS EE BT
  4. 4. Citation networks data (Web of Science) Citation networks are directed graphs. An edge between papers A and B is present when A cites B. (PS = probability and statistics, EE = electrical engineering, BT = biotechnology and applied microbiology) Exponential growth of the graph (Log Y axis) 1980 1985 1990 1995 2000 2005 2010 104 PS EE BT Power-law tail for degree distribution 100 101 102 103 10-6 10-4 10-2 100 PS EE BT
  5. 5. Citation networks data (Web of Science) Citation networks are directed graphs. An edge between papers A and B is present when A cites B. (PS = probability and statistics, EE = electrical engineering, BT = biotechnology and applied microbiology) Exponential growth of the graph (Log Y axis) 1980 1985 1990 1995 2000 2005 2010 104 PS EE BT Power-law tail for degree distribution 100 101 102 103 10-6 10-4 10-2 100 PS EE BT Inhomogeneity among papers (samples of 20 papers per dataset) 1980 1990 2000 2010 0 10 20 30 40 50 60 PS 1980 1990 2000 2010 0 5 10 15 20 25 30 35 EE 1990 2000 2010 0 50 100 150 BT
  6. 6. Continuous-time branching processes
  7. 7. Continuous-time branching processes Idea P (paper of age t and k past citations is cited) ≈ Yf (k)g(t) ˆ increasing function of degree f ˆ decreasing function of age g ˆ fitness Y
  8. 8. Continuous-time branching processes Idea P (paper of age t and k past citations is cited) ≈ Yf (k)g(t) ˆ increasing function of degree f ˆ decreasing function of age g ˆ fitness Y Individuals produce children according to independent copies of a stochastic process (Vt )t≥0.
  9. 9. Continuous-time branching processes Idea P (paper of age t and k past citations is cited) ≈ Yf (k)g(t) ˆ increasing function of degree f ˆ decreasing function of age g ˆ fitness Y Individuals produce children according to independent copies of a stochastic process (Vt )t≥0. 1 2 6 9 3 4 7 12 5 8 10 11 1 2 3 4
  10. 10. Continuous-time branching processes Idea P (paper of age t and k past citations is cited) ≈ Yf (k)g(t) ˆ increasing function of degree f ˆ decreasing function of age g ˆ fitness Y Individuals produce children according to independent copies of a stochastic process (Vt )t≥0. 1 2 6 9 3 4 7 12 5 8 10 11
  11. 11. Continuous-time branching processes Idea P (paper of age t and k past citations is cited) ≈ Yf (k)g(t) ˆ increasing function of degree f ˆ decreasing function of age g ˆ fitness Y Individuals produce children according to independent copies of a stochastic process (Vt )t≥0. 1 2 6 9 3 4 7 12 5 8 10 11 1 2 3 4
  12. 12. Simulations
  13. 13. Simulations Exponential growth of the graph 1980 1985 1990 1995 2000 2005 2010 2000 4000 6000 8000 10000 12000 CBP1 CBP2
  14. 14. Simulations Exponential growth of the graph 1980 1985 1990 1995 2000 2005 2010 2000 4000 6000 8000 10000 12000 CBP1 CBP2 Inhomogeneity among papers 1980 1990 2000 2010 0 10 20 30 40 50 60 70 80 90 CBP1 1980 1990 2000 2010 0 50 100 150 200 CBP2
  15. 15. Simulations Exponential growth of the graph 1980 1985 1990 1995 2000 2005 2010 2000 4000 6000 8000 10000 12000 CBP1 CBP2 Inhomogeneity among papers 1980 1990 2000 2010 0 10 20 30 40 50 60 70 80 90 CBP1 1980 1990 2000 2010 0 50 100 150 200 CBP2 Not a power-law tail for degree distribution 100 101 102 103 10-4 10-2 100 CBP1 CBP2
  16. 16. Simulations Exponential growth of the graph 1980 1985 1990 1995 2000 2005 2010 2000 4000 6000 8000 10000 12000 CBP1 CBP2 Inhomogeneity among papers 1980 1990 2000 2010 0 10 20 30 40 50 60 70 80 90 CBP1 1980 1990 2000 2010 0 50 100 150 200 CBP2 Not a power-law tail for degree distribution 100 101 102 103 10-4 10-2 100 CBP1 CBP2 (α∗ = Malthusian parameter) In-degree pk = α∗ α∗ + fk k−1 ∏ i=0 fi α∗ + fi (Power-law when fk = ak + b)
  17. 17. Simulations Exponential growth of the graph 1980 1985 1990 1995 2000 2005 2010 2000 4000 6000 8000 10000 12000 CBP1 CBP2 Inhomogeneity among papers 1980 1990 2000 2010 0 10 20 30 40 50 60 70 80 90 CBP1 1980 1990 2000 2010 0 50 100 150 200 CBP2 Not a power-law tail for degree distribution 100 101 102 103 10-4 10-2 100 CBP1 CBP2 (α∗ = Malthusian parameter) In-degree, aging pk = α∗ α∗ + fk c(k) k−1 ∏ i=0 fi c(i) α∗ + fi c(i)
  18. 18. Simulations Exponential growth of the graph (Log Y axis) 1980 1985 1990 1995 2000 2005 2010 2000 4000 6000 8000 10000 12000 CBP1 CBP2 Inhomogeneity among papers (samples of 20 vertices) 1980 1990 2000 2010 0 10 20 30 40 50 60 70 80 90 CBP1 1980 1990 2000 2010 0 50 100 150 200 CBP2 Not a power-law tail for degree distribution 100 101 102 103 10-4 10-2 100 CBP1 CBP2 (α∗ = Malthusian parameter) In-degree, aging, fitness pk = E [ α∗ α∗ + Y fk c(k, Y ) k−1 ∏ i=0 Y fi c(i, Y ) α∗ + Y fi c(i, Y ) ]
  19. 19. Open problems ˆ how to get a power-law distribution ˆ consequences of collapsing together individuals with different birth times
  20. 20. Thank you!

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