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TableĀ 4 The parameters of fitting functions

From: Feature analysis of multidisciplinary scientific collaboration patterns based on PNAS

Degree distribution a b c d s B E G P p-value
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
  1. The ranges of generalized Poisson \(f_{1}(x)\), cross-over, and power-law \(f_{2}(x)\) are [1,E], [B,E], and [B,max(x)] respectively. The fitting function is \(f(x)=q(x) s f_{1}(x) +(1-q(x))f_{2}(x)\), where \(q(x)=\mathrm{e}^{ - (x -B)/(E-x ) }\). 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 (p-value>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.
Hyperdegree distribution a b c d s B E G P p-value
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
  1. The ranges of generalized Poisson \(f_{1}(x)\), cross-over, and power-law \(f_{2}(x)\) are [1,E], [B,E], and [B,max(x)] respectively. The fitting function is \(f(x)=q(x) s f_{1}(x) +(1-q(x))f_{2}(x)\), where \(q(x)=\mathrm{e}^{ - (x -B)/(E-x ) }\). 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 (p-value>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.