Fluctuation of the Download Network

Abstract

The scaling behavior of fluctuation for a download network which we have investigated a few years ago based upon Zhang's Encophysics web page has been presented. A power law scaling, namely σ ∼ f α exists between the dispersion σ and average flux f of the download rates. The fluctuation exponent α is neither 1/2 nor 1 which was claimed as two universal fluctuation classes in previous publication, instead it varies from 1/2 to 1 with the time window in which the download data were accumulated. The crossover behavior of fluctuation exponents can be qualitatively understood by the external driving fluctuation model for a small-size system or a network traffic model which suggests congestion as the origin.PACS numbers: 89.75. Hc, 89.75.Da, 89.40.Dd Many phenomenological and statistical analysis have been made for the complex networks [1,2]. In those researches, most studies focused on the long-time behavior of a certain complex network. In this sense, the feature of the network corresponds to its static characteristic. However, time evolution of the network topology is also very important. During its evolution, the network nodes experience different traffic flux time by time and the fluctuation is unavoidable. Actually, fluctuation is a universal phenomenon which exists in many different fields, such as nuclear fragmentation or hadron production [3,4,5], which can also be related to the critical behavior or selforganized criticality. For instance, the dispersion (σ) of an order parameter, such as the charge of the largest fragments in nuclear fragmentation, shows a transition from σ ∝ f 1/2 (the ordered phase) to σ ∝ f (the disordered phase) when the multifragmentation phase transition takes place in hot nuclear system [3,4] (here f is the average of the order parameter). For network dynamics, recently, Menezes and Barabási investigated the fluctuation in a number of real world networks, which includes internet, river network, microchip, WWW and highway network dynamics and presented a model to understand the origin of fluctuation in traffic process [6]. They found that the fluctuation is dominantly driven by either internal or external dynamics of the complex system [6]. In their studies, they found there is a power-law scaling for the dispersion and the average flux, namely σ ∝ f α , and there are two classes of universality for real systems. In the Internet and the computer chip there is robust internal dynamics which leads to the fluctuation exponent α = 1/2, while highway and Web traffic are driven by external demand which leads to the fluctuation exponent α = 1. Authors use a stylized model of random walkers throughout network, they thought what is probably one of the most important factors in the traffic dynamics on networks is the limited capacity of nodes to handle packets simultaneously, which leads to packpack interaction and induce large fluctuations or even network congestion. However, a recent study on scaling of fluctuation in internet traffic shows that the fluctuation is different from 1/2 which ...

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