Degree-dependent network growth: From preferential attachment to explosive percolation

Abstract

We present a simple model of network growth and solve it by writing down the dynamic equations for its macroscopic characteristics like the degree distribution and degree correlations. This allows us to study carefully the percolation transition using a generating functions theory. The model considers a network with a fixed number of nodes wherein links are introduced using degree-dependent linking probabilities $p_k$. To illustrate the techniques and support our findings using Monte-Carlo simulations, we introduce the exemplary linking rule $p_k$ proportional to $k^{-\alpha}$, with $\alpha$ between -1 and plus infinity. This parameter may be used to interpolate between different regimes. For negative $\alpha$, links are most likely attached to high-degree nodes. On the other hand, in case $\alpha>0$, nodes with low degrees are connected and the model asymptotically approaches a process undergoing explosive percolation.