Degree distribution

a

b

c

d

s

B

E

G

P

pvalue


Biological sciences

4.843

0.464

74.27

2.889

1.049

15

26

20

50

0.203

Physical sciences

3.958

0.477

49.31

2.798

1.037

7

14

20

53

0.178

Social sciences

3.292

0.513

20.78

2.657

1.046

11

28

20

35

0.111

Hyperdegree distribution

a

b

c

d

s

B

E

G

P

pvalue


Biological sciences

0.028

0.269

1.968

3.099

35.57

2

6

10

13

0.979

Physical sciences

0.021

0.320

2.977

2.916

47.15

2

11

10

10

0.625

Social sciences

0.022

0.375

19.48

3.665

46.24

3

20

10

11

0.206

 The ranges of generalized Poisson \(f_{1}(x)\), crossover, and powerlaw \(f_{2}(x)\) are [1,E], [B,E], and [B,max(x)] respectively. The fitting function is \(f(x)=q(x) s f_{1}(x) +(1q(x))f_{2}(x)\), where \(q(x)=\mathrm{e}^{  (x B)/(Ex ) }\). The fitting processes are: obverse proper G and P; calculate parameters of \(sf_{1}(x)\) (i.e. a, b, s) and \(f_{2}(x)\) (i.e. c, d) through regressing the empirical distribution in [1,G] and [P,max(x)] respectively; find B and E through exhaustion to make f(x) pass KS test (pvalue>0.05). The sum of each f(x) over [1,max(x)] is near unity, which means that f(x) can be regarded as a probability density function.