We present analytical results for the emerging structure of networks that evolve via a combination of growth (by node addition and random attachment) and contraction (by random node deletion). To this end we consider a network model in which at each time step a node addition and random attachment step takes place with probability $P_{add}$ and a random node deletion step takes place with probability $P_{del}=1-P_{add}$. The balance between the growth and contraction processes is captured by the parameter $\eta=P_{add}-P_{del}$. The case of pure network growth is described by $\eta=1$. In case that $0<\eta<1$ the rate of node addition exceeds the rate of node deletion and the overall process is of network growth. In the opposite case, where $-1<\eta<0$, the overall process is of network contraction, while in the special case of $\eta=0$ the expected size of the network remains fixed, apart from fluctuations. Using the master equation we obtain a closed form expression for the time dependent degree distribution $P_t(k)$. The degree distribution $P_t(k)$ includes a term that depends on the initial degree distribution $P_0(k)$, which decays as time evolves, and an asymptotic distribution $P_{st}(k)$. In the case of pure network growth ($\eta=1$) the asymptotic distribution $P_{st}(k)$ follows an exponential distribution, while for $-1<\eta<1$ it consists of a sum of Poisson-like terms and exhibits a Poisson-like tail. In the case of overall network growth ($0 < \eta < 1$) the degree distribution $P_t(k)$ eventually converges to $P_{st}(k)$. In the case of overall network contraction ($-1 < \eta < 0$) we identify two different regimes. For $-1/3 < \eta < 0$ the degree distribution $P_t(k)$ quickly converges towards $P_{st}(k)$. In contrast, for $-1 < \eta < -1/3$ the convergence of $P_t(k)$ is initially very slow and it gets closer to $P_{st}(k)$ only shortly before the network vanishes.

中文翻译：

---

在增长（通过节点添加和随机附着）和收缩（通过随机节点删除）的组合下演化的网络结构

我们提出了新兴网络结构的分析结果，这些结构通过增长（通过节点添加和随机连接）和收缩（通过随机节点删除）的组合发展。为此，我们考虑一个网络模型，在该模型中，在每个时间步，以概率 $P_{add}$ 发生节点添加和随机附着步骤，以概率 $P_{del}=1-P_ 发生随机节点删除步骤{添加}$。增长和收缩过程之间的平衡由参数 $\eta=P_{add}-P_{del}$ 捕获。纯网络增长的情况用 $\eta=1$ 来描述。如果$0<\eta<1$，则节点添加率超过节点删除率，整个过程是网络增长。反之，$-1<\eta<0$，整个过程是网络收缩，而在 $\eta=0$ 的特殊情况下，除了波动之外，网络的预期大小保持不变。使用主方程，我们获得了时间相关度数分布 $P_t(k)$ 的封闭形式表达式。度分布 $P_t(k)$ 包括一个依赖于初始度分布 $P_0(k)$ 的项，它随着时间的推移而衰减，以及一个渐近分布 $P_{st}(k)$。在纯网络增长 ($\eta=1$) 的情况下，渐近分布 $P_{st}(k)$ 遵循指数分布，而对于 $-1<\eta<1$，它由泊松的总和组成-like 项，并表现出类似 Poisson 的尾巴。在整体网络增长的情况下（$0 < \eta < 1$），度分布 $P_t(k)$ 最终收敛到 $P_{st}(k)$。在整体网络收缩的情况下（$-1 < \eta < 0$) 我们确定了两种不同的制度。对于 $-1/3 < \eta < 0$，度分布 $P_t(k)$ 迅速收敛到 $P_{st}(k)$。相反，对于 $-1 < \eta < -1/3$，$P_t(k)$ 的收敛最初非常缓慢，并且仅在网络消失前不久才接近 $P_{st}(k)$。