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QUANTITATIVE LAWS June 13 -June 24

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- 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. 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. 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. 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. 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. Continuous-time branching processes
- 7. Continuous-time branching processes Idea P (paper of age t and k past citations is cited) ≈ Yf (k)g(t) increasing function of degree f decreasing function of age g ﬁtness Y
- 8. Continuous-time branching processes Idea P (paper of age t and k past citations is cited) ≈ Yf (k)g(t) increasing function of degree f decreasing function of age g ﬁtness Y Individuals produce children according to independent copies of a stochastic process (Vt )t≥0.
- 9. Continuous-time branching processes Idea P (paper of age t and k past citations is cited) ≈ Yf (k)g(t) increasing function of degree f decreasing function of age g 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. Continuous-time branching processes Idea P (paper of age t and k past citations is cited) ≈ Yf (k)g(t) increasing function of degree f decreasing function of age g 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. Continuous-time branching processes Idea P (paper of age t and k past citations is cited) ≈ Yf (k)g(t) increasing function of degree f decreasing function of age g 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. Simulations
- 13. Simulations Exponential growth of the graph 1980 1985 1990 1995 2000 2005 2010 2000 4000 6000 8000 10000 12000 CBP1 CBP2
- 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. 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. 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. 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. 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. Open problems how to get a power-law distribution consequences of collapsing together individuals with diﬀerent birth times
- 20. Thank you!

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