Abstract
We propose an empirical estimator of the preferential attachment function $f$ in the setting of general sublinear preferential attachment trees. Using a supercritical continuous-time branching process framework, we prove the almost sure consistency of the proposed estimator. We perform simulations to study the empirical properties of our estimators.
Highlights
After the conception of the scale-free phenomenon in Barabasi’s series of seminal work (Barabasi and Albert (1999); Barabasi et al (1999, 2000)), scientists from numerous disciplines have made discoveries that support the ubiquity of realworld scale-free networks
We propose an empirical estimator of the preferential attachment function f in the setting of general sublinear preferential attachment trees
Using a supercritical continuous-time branching process framework, we prove the almost sure consistency of the proposed estimator
Summary
After the conception of the scale-free phenomenon in Barabasi’s series of seminal work (Barabasi and Albert (1999); Barabasi et al (1999, 2000)), scientists from numerous disciplines have made discoveries that support the ubiquity of realworld scale-free networks. The limiting degree distribution may be much lighter tailed than in the affine case, corresponding to rarer occurrence of nodes of high degrees (see Rudas et al (2007)) Such scenarios have been reported frequently in empirical work on real-world networks. The numbers of nodes of high degrees in an observed network usually exhibit large variations and irregular behavior as a function of the degree This is to be expected, as they are rare, but makes it hard to determine whether there is a power law at all. In the affine case that f (k) = k + δ, the total preference is deterministic and takes the form ni=1(di(n) + δ) = nδ + 2n This property allows to study the limiting degree distribution with simple recursions on the degree evolution, and is handy for the study of statistical estimators, as shown in. Several simulation studies are carried out in order to uncover a more detailed picture of the properties of the empirical estimator, the most intere√sting one being that the estimator seems to be asymptotically normal with a n rate
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