Master-equation analysis of accelerating networks

In many real-world networks, the rates of node and link addition are time dependent. This observation motivates the definition of accelerating networks. There has been relatively little investigation of accelerating networks and previous efforts at analyzing their degree distributions have employed mean-field techniques. By contrast, we show that it is possible to apply a master-equation approach to such network development. We provide full time-dependent expressions for the evolution of the degree distributions for the canonical situations of random and preferential attachment in networks undergoing constant acceleration. These results are in excellent agreement with results obtained from simulations. We note that a growing nonequilibrium network undergoing constant acceleration with random attachment is equivalent to a classical random graph, bridging the gap between nonequilibrium and classical equilibrium networks.

Figure 1

(Color online) Degree distributions for stochastic RA and PA networks undergoing constant acceleration. Each simulation comprises a network grown to $N_{tot}={10}^6$ nodes with mean degree of $\lambda =10\approx a{N}_{tot}$. Also illustrated is the Poisson distribution with the same mean and the analytic expression of Eq. (13) for the preferential attachment, constantly accelerating network derived in Sec. 4. The numerical implementation simply generates the degree distribution through iterating the master equation (10). The nonaccelerating degree distributions for random and preferential attachment [Eqs. (9, 15)] with the same mean ($m=5$ links added with each new node) are also illustrated to highlight the effect of accelerating behavior.