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 fluctua-tion is different from 1/2 which was claimed in the above papers. They developed a model where the arrival and departure of "packets" follow exponential distribution, and the processing capability of nodes is either unlimited or finite was proposed by Duch and Arenas [7]. This model presents a wide variety of exponents between 1/2 and 1, revealing their dependence on the few parameters considered, and questioning the existence of universality classes. Hence it seems that the universal classes of fluctuation scaling for network dynamics are far from reaching consensus and therefore it is worthy to further investigate what about the fluctuation behavior in other real networks. Neverthless, so far there are few analysis on the fluctuation behavior of other specific networks rather than the networks which have been investigated in Ref. [6,7]. In this work, we will investigate the network evolution and fluctuation based on our previous study of the download network.
In our previous work in 2004 [8], we reported, for the first time, that the download frequency of the papers in a web page is also a scale-free network. Its rank-ordered download distribution can be described by the Zipf law [9,10] or Tsallis’ non-extensive entropy [11]. The data set of the download rates comes from a well constructed web page in the field of economical physics (so-called Econophysics) by Zhang since 1998 [12]. Furthermore, the mechanism of network growth was explained by the preferential attachment network model of Barabasi and Albert. Since three years have passed after this network analysis, it is of interesting to see how this network evolves and how about the fluctuation of download rate.
Firstly let us see some plots of rank distributions of the download numbers from the data on 2004/08/31 to 2007/07/29 which is shown in Fig. 1. Roughly, rank distribution are almost linear in double logarithm plots and they can be described by the Zipf law [9] N ≃ rank -γ ,
where the γ is the Zipf law exponent. Zipf’s law or scale free networks is different from the predictions of pure random networks introduced by Erdos and Renyi [13]. Roughly, the shapes of these distributions keep similar in all times. However, quantitative analysis shows nonconstant behavior of the evolution of Zipf exponent (γ) which is shown in the inset of the Figure 1. Especially there is a bump during 2006, i.e. the rank-ordered distributions tend to be steeper, which reflects higher download frequency for higher rank papers. However, the exponent decreases in 2007, i.e. more flatter distribution, which is obviously seen in the Figure 1 (diamond points).
In this case, the web visitors prefer to download more papers listed in the web
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