Scaling Breakdown in Flow Fluctuations on Complex Networks

We propose a model of random diffusion to investigate flow fluctuations in complex networks. We derive an analytical law showing that the dependence of fluctuations with the mean traffic in a network is ruled by the delicate interplay of three factors, respectively, of dynamical, topological and statistical nature. In particular, we demonstrate that the existence of a power-law scaling characterizing the flow fluctuations at every node in the network cannot be claimed. We show the validity of this scaling breakdown under quite general topological and dynamical situations by means of different traffic algorithms and by analyzing real data.

Figure 1

Flow fluctuation $\sigma$ as a function of $⟨f⟩$ for the RD model with various parameter values. In panel (a), $\delta ={10}^3$ and $W={10}^4$. In panel (b), $W$ has the same value while $M$ has been fixed to 10. In both figures, points correspond to the solution of Eq. (7) for different values of $k_i$ (1…18). The total number of links is $K=33500$. Dashed lines are guides to the eyes and correspond to $\sigma \sim ⟨f{⟩}^{\alpha }$, with $\alpha =1/2$ (lower curves) and $\alpha =1$ (upper curves). See the text for further details.

Figure 2

Flow fluctuation $\sigma$ as a function of $⟨f⟩$ from numerical simulations of the Internet traffic model (see text for details) on synthetic scale-free networks with $N={10}^4$ nodes, $K=37551$ links, and degree exponent $\gamma =2.2$. Different panels correspond to different values of $M$, respectively $M=1$, $5\times {10}^2$, $35\times {10}^3$, ${10}^5$. Color-coded values represent the logarithm of node degree. The continuous line is the curve $y=x^{0.5}$, while the dashed line is $y\sim x$.

Figure 3

Flow fluctuation $\sigma$ as a function of $⟨f⟩$ for the Abilene Interfaces. The values of $M$ used in each panel are: $M=5$ (a), $M=30$ (b), $M=60$ (c), and $M=720$ (d). Time is in minutes. The value of $\alpha$ for each $M$ is also reported. Averages are taken over one month of data corresponding to the period between January 11 to February 11 of 2006.