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We present analytical results for the emerging structure of networks thatevolve via a combination of growth (by node addition and random attachment) andcontraction (by random node deletion). To this end we consider a network modelin which at each time step a node addition and random attachment step takesplace with probability $P_{add}$ and a random node deletion step takes placewith probability $P_{del}=1-P_{add}$. The balance between the growth andcontraction processes is captured by the parameter $\eta=P_{add}-P_{del}$. Thecase 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 overallprocess is of network growth. In the opposite case, where $-1<\eta<0$, theoverall process is of network contraction, while in the special case of$\eta=0$ the expected size of the network remains fixed, apart fromfluctuations. Using the master equation we obtain a closed form expression forthe 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 asymptoticdistribution $P_{st}(k)$ follows an exponential distribution, while for$-1<\eta<1$ it consists of a sum of Poisson-like terms and exhibits aPoisson-like tail. In the case of overall network growth ($0 < \eta < 1$) thedegree distribution $P_t(k)$ eventually converges to $P_{st}(k)$. In the caseof overall network contraction ($-1 < \eta < 0$) we identify two differentregimes. For $-1/3 < \eta < 0$ the degree distribution $P_t(k)$ quicklyconverges towards $P_{st}(k)$. In contrast, for $-1 < \eta < -1/3$ theconvergence of $P_t(k)$ is initially very slow and it gets closer to$P_{st}(k)$ only shortly before the network vanishes.

physical review. e

The structure of networks that evolve under a combination of growth, via  node addition and random attachment, and contraction, via random node  deletion