QUANTITATIVE LAWS June 13 -June 24

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)
ˆ increasing function of degree f
ˆ decreasing function of age g
ˆ fitness Y

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. 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. 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. 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. 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
ˆ how to get a power-law distribution
ˆ consequences of collapsing together individuals with different birth
times

20. Thank you!