Assortativity and act degree distribution of some collaboration networks

Empirical investigation results on weighted and un-weighted assortativity, act degree distribution, degree distribution and node strength distribution of nine real world collaboration networks have been presented. The investigations propose that act degree distribution, degree distribution and node strength distribution usually show so-called “shifted power law” (SPL) function forms, which can continuously vary from an ideal power law to an ideal exponential decay. Two parameters, α and η, can be used for description of the distribution functions. Another conclusion is that assortativity coefficient and the parameter, α or η, is monotonously dependent on each other. By the collaboration network evolution model introduced in a reference [P. Zhang et al., Physica A 360 (2006) 599], we analytically derived the SPL distributions, which typically appeared in general situations where nodes are selected partially randomly, with a probability p, and partially by linear preferential principle, with the probability 1-p. The analytic discussion gives an explicit expression on the relationship between the random selection proportion p and the parameters α and η. The numerical simulation results by the model show a monotonic dependence of assortativity on the random selection proportion p. The empirically obtained assortativity versus α or η curve for the four collaboration networks with small maximal act size, Tmax, shows a good agreement with the model prediction. According to the curves, one can qualitatively judge the random selection proportion of the real world network in its evolution process.