The effect of preferential node deletion on the structure of networks that evolve via preferential attachment

We present analytical results for the effect of preferential node deletion on the structure of networks that evolve via node addition and preferential attachment. To this end, we consider a preferential-attachment-preferential-deletion (PAPD) model, in which at each time step, with probability $P_{\mathrm{a}\mathrm{d}\mathrm{d}}$ there is a growth step where an isolated node is added to the network, followed by the addition of $m$ edges, where each edge connects a node selected uniformly at random to a node selected preferentially in proportion to its degree. Alternatively, with probability $P_{\mathrm{d}\mathrm{e}\mathrm{l}}=1-P_{\mathrm{a}\mathrm{d}\mathrm{d}}$ there is a contraction step, in which a preferentially selected node is deleted and its links are erased. The balance between the growth and contraction processes is captured by the growth/contraction rate $\eta =P_{\mathrm{a}\mathrm{d}\mathrm{d}}-P_{\mathrm{d}\mathrm{e}\mathrm{l}}$. For $0<\eta \le 1$ the overall process is of network growth, while for $-1\le \eta <0$ the overall process is of network contraction. Using the master equation and the generating function formalism, we study the time-dependent degree distribution $P_t(k)$. It is found that for each value of $m>0$ there is a critical value ${\eta }_c(m)=-(m-2)\mathrm{/}(m+2)$ such that for ${\eta }_c(m)<\eta \le 1$ the degree distribution $P_t(k)$ converges towards a stationary distribution $P_{\mathrm{s}\mathrm{t}}(k)$. In the special case of pure growth, where $\eta =1$, the model is reduced to a preferential attachment growth model and $P_{\mathrm{s}\mathrm{t}}(k)$ exhibits a power-law tail, which is a characteristic of scale-free networks. In contrast, for ${\eta }_c(m)<\eta <1$ the distribution $P_{\mathrm{s}\mathrm{t}}(k)$ exhibits an exponential tail, which has a well-defined scale.This implies a phase transition at $\eta =1$, in contrast with the preferential-attachment-random-deletion (PARD) model [B. Budnick, O. Biham and E. Katzav, J. Stat. Mech. 013401 (2025)], in which the power-law tail remains intact as long as $\eta >0$.