Accelerated growth in outgoing links in evolving networks: deterministic versus stochastic picture.

Abstract

In several real-world networks such as the Internet, World Wide Web, etc., the number of links grow in time in a nonlinear fashion. We consider growing networks in which the number of outgoing links is a nonlinear function of time but new links between older nodes are forbidden. The attachments are made using a preferential attachment scheme. In the deterministic picture, the number of outgoing links m (t) at any time t is taken as N (t)(theta) where N (t) is the number of nodes present at that time. The continuum theory predicts a power-law decay of the degree distribution: P (k) proportional to k-(1-2/ (1-theta ) ), while the degree of the node introduced at time t(i) is given by k(t(i),t)=t(theta)(i) [t/t(i) ]((1+theta)/2) when the network is evolved till time t. Numerical results show a growth in the degree distribution for small k values at any nonzero theta. In the stochastic picture, m (t) is a random variable. As long as <m (t) > is independent of time, the network shows a behavior similar to the Barabási-Albert (BA) model. Different results are obtained when <m (t) > is time dependent, e.g., when m (t) follows a distribution P (m) proportional to m(-lambda). The behavior of P (k) changes significantly as lambda is varied: for lambda>3, the network has a scale-free distribution belonging to the BA class as predicted by the mean field theory; for smaller values of lambda it shows different behavior. Characteristic features of the clustering coefficients in both models have also been discussed.