Fluctuations and Pseudo Long Range Dependence in Network Flows: A Non-Stationary Poisson Process Model

In the study of complex networks (systems), the scaling phenomenon of flow fluctuations refers to a certain power-law between the mean flux (activity) $<F_i>$ of the $i$th node and its variance $\sigma_i$ as $\sigma_i \propto < F_{i} > ^{\alpha}$. Such scaling laws are found to be prevalent both in natural and man-made network systems, but our understanding of their origins still remains limited. In this paper, a non-stationary Poisson process model is proposed to give an analytical explanation of the non-universal scaling phenomenon: the exponent $\alpha$ varies between 1/2 and 1 depending on the size of sampling time window and the relative strength of the external/internal driven forces of the systems. The crossover behavior and the relation of fluctuation scaling with pseudo long range dependence are also accounted for by the model. Numerical experiments show that the proposed model can recover the multi-scaling phenomenon.