Stochastic model for scale-free networks with cutoffs

We propose and analyze a stochastic model which explains, analytically, the cutoff behavior of real scale-free networks previously modeled computationally by Amaral et al. [Proc. Natl. Acad. Sci. U.S.A. 97, 11149 (2000)] and others. We present a mathematical model that can explain several existing computational scale-free network generation models. This yields a theoretical basis to understand cutoff behavior in complex networks, previously treated only with simulations using distinct models. Therefore, ours is an integrative approach that unifies the existing literature on cutoff behavior in scale-free networks. Furthermore, our mathematical model allows us to reach conclusions not hitherto possible with computational models: the ability to predict the equilibrium point of active vertices and to relate the growth of networks with the probability of aging. We also discuss how our model introduces a useful way to classify scale free behavior of complex networks.

Figure 1

$P(Z=z)$ distribution.

Figure 2

(Color online) $P(x⩽0.5)$ versus $p$ (probability of a vertex getting inactive) for $\alpha =1$.

Figure 3

(Color online) Evolution of the standard deviation of $x_t$ for $p=0.1$ and $\alpha =1$ with time.

Figure 4

(Color online) The simulation results for $P(Z=z)$ and curve fitting with an exponential $a{q}^{z-1}$.

Figure 5

(Color online) Cumulative degree distribution for network sizes 1000, 10 000, and 100 000, with $\alpha =1$ and $p=0.05$. Also in solid we plot the BAM.

Figure 6

(Color online) Cumulative degree distribution for probability of inactiveness $p=0.1$, $p=0.05$, and $p=0.01$, with $\alpha =1$ and size of the $\text{network}=10000$. Also in solid we plot the BAM.