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

Abstract

In several real-world networks like the Internet, WWW etc., the number of links grow in time in a non-linear fashion. We consider growing networks in which the number of outgoing links is a non-linear 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: $P(k) \propto k^{-1-\frac{2} {1-\theta}}$, while the degree of the node introduced at time $t_i$ is given by $k(t_i,t) = t_i^{\theta}[ \frac {t}{t_i}]^{\frac {1+\theta}{2}}$ when the network is evolved till time $t$. Numerical results show a growth in the degree distribution for small $k$ values at any non-zero $\theta$. In the stochastic picture, $m(t)$ is a random variable. As long as $<m(t)>$ is independent of time, the network shows a behaviour similar to the Barab\'asi-Albert (BA) model. Different results are obtained when $<m(t) >$ is time-dependent, e.g., when $m(t)$ follows a distribution $P(m) \propto m^{-\lambda}$. The behaviour 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 behaviour. Characteristic features of the clustering coefficients in both models have also been discussed.