Phase transition in evolving networks that combine preferential attachment and random node deletion

Abstract

Analytical results are presented for the structure of networks that evolve via a preferential-attachment-random-deletion (PARD) model in the regime of overall network growth and in the regime of overall contraction. The phase transition between the two regimes is studied. At each time step a node addition and preferential attachment step takes place with probability $P_{\rm add}$, and a random node deletion step takes place with probability $P_{\rm del} = 1 - P_{\rm add}$. The balance between growth and contraction is captured by the parameter $\eta = P_{\rm add} - P_{\rm del}$, which in the regime of overall network growth satisfies $0 < \eta \le 1$ and in the regime of overall network contraction $-1 \le \eta < 0$. Using the master equation and computer simulations we show that for $-1 < \eta < 0$ the time-dependent degree distribution $P_t(k)$ converges towards a stationary form $P_{\rm st}(k)$ which exhibits an exponential tail. This is in contrast with the power-law tail of the stationary degree distribution obtained for $0 < \eta \le 1$. Thus, the PARD model has a phase transition at $\eta=0$, which separates between two structurally distinct phases. At the transition, for $\eta=0$, the degree distribution exhibits a stretched exponential tail. While the stationary degree distribution in the phase of overall growth represents an asymptotic state, in the phase of overall contraction $P_{\rm st}(k)$ represents an intermediate asymptotic state of a finite life span, which disappears when the network vanishes.