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 model, in which at each time step, with probability $P_{\mathrm{add}}$ 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 $P_{\mathrm{del}}=1-P_{\mathrm{add}}$ 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 $\eta =P_{\mathrm{add}}-P_{\mathrm{del}}$. 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 $P_t\left(k\right)$. It is found that for each value of $m>0$ there is a critical value ${\eta }_c\left(m\right)=-(m-2)/(m+2)$ such that for ${\eta }_c\left(m\right)<\eta \le 1$ the degree distribution $P_t\left(k\right)$ converges toward a stationary distribution $P_{\mathrm{st}}\left(k\right)$. In the special case of pure growth, where $\eta =1$, the model is reduced to a preferential attachment growth model and $P_{\mathrm{st}}\left(k\right)$ exhibits a power-law tail, which is a characteristic of scale-free networks. In contrast, for ${\eta }_c\left(m\right)<\eta <1$ the distribution $P_{\mathrm{st}}\left(k\right)$ 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 model [Budnick et al., J. Stat. Mech. (2025) 013401], in which the power-law tail remains intact as long as $\eta >0$. These results illustrate the sensitivity of evolving networks to preferential node deletion, in contrast with their robustness to random node deletion. While for $\eta \ge max\{{\eta }_c\left(m\right),0\}$ the stationary degree distribution $P_{\mathrm{st}}\left(k\right)$ lasts indefinitely, for ${\eta }_c\left(m\right)<\eta <0$ (and $m>2$) it persists for a finite lifetime, until the network vanishes. It is also found that in the regime of $-1\le \eta \le {\eta }_c\left(m\right)$ the time-dependent degree distribution $P_t\left(k\right)$ does not converge toward a stationary form, but continues to evolve until the network is reduced to a set of isolated nodes. These results provide insight on the structure of transient social networks, such as dating networks and job-seeking platforms, in which user turnover is intrinsically high.

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• Degree distributions
• Evolving networks
• Network formation & growth
• Network phase transitions
• Network structure