We study the dynamics of the Internet topology based on empirical data on the level of the autonomous systems. It is found that the fluctuations occurring in the stochastic process of connecting and disconnecting edges are important features of the Internet dynamics. The network's overall growth can be described approximately by a single characteristic degree growth rate g eff ≈ 0.016 and the fluctuation strength σ eff ≈ 0.14, together with the vertex growth rate α ≈ 0.029. A stochastic model which incorporates these values and an adaptation rule newly introduced reproduces several features of the real Internet topology such as the correlations between the degrees of different vertices. PACS numbers: 89.70.+c, Recently many studies on complex systems [1,2] paid attention to complex networks [3,4]. An interesting feature emerging in such complex systems is a power-law behavior in the degree distribution, P D (k) ∼ k −γ [5,6], where the degree k is the number of edges incident upon a given vertex. Recently, Barabási and Albert (BA) [6] introduced an evolving network model to illustrate such networks, called the scale-free (SF) networks, in which the number of vertices N increases linearly with time, and a newly introduced vertex is connected to already existing vertices following the so-called preferential attachment (PA) rule.Huberman and Adamic (HA) [7] proposed another scenario for SF networks. They argued that the fluctuation effect arising in the process of connecting and disconnecting edges between vertices, is the essential feature to describe the dynamics of the Internet topology correctly. In this model, the total number of vertices N (t) increases exponentially with time asNext, it is assumed that the degree k i at a vertex i evolves through the multiplicative process [20],where ζ i (t) is the growth rate of the degree k i at time t, which fluctuates from time to time. Thus, one may divide the growth rate ζ i (t) into two parts,where g 0,i is the mean value over time, and ξ i (t) the rest part, representing fluctuations [21]. ξ i (t) is assumed to be a white noise satisfying ξ i (t) = 0 and ξ i (t)ξ j (t ′ ) = σ 2 0,i δ t,t ′ δ i,j , where σ 2 0,i is the variance. Here · · · is the sample average. For later convenience, we denote the logarithm of the growth factor as G i (t) ≡ ln(1 + ζ i (t)). Then a simple application of the central limit theorem ensures that k i (t)/k i (t 0 ), t 0 being a reference time, follows the log-normal distribution for sufficiently large t. To get the degree distribution, one needs to collect all contributions from different ages τ i , growth rates g 0,i , standard deviations σ 0,i and initial degree k i (t 0 ). HA first assumed that ζ i are identically distributed so that g 0,i = g 0 and σ 0,i = σ 0 for all i. Then the conditional probability for degree,where g eff ≡ G i (t) and σ 2 eff ≡ (G i (t) − G i (t) ) 2 . g eff and σ 2 eff are related to g 0 and σ 2 0 as g eff ≈ g 0 −σ 2 0 /2 and σ 2 eff ≈ σ 2 0 , respectively [9]. Since the density of vertices with age τ is proportio...

We study structural feature and evolution of the Internet at the autonomous systems level. Extracting relevant parameters for the growth dynamics of the Internet topology, we construct a toy model for the Internet evolution, which includes the ingredients of multiplicative stochastic evolution of nodes and edges and adaptive rewiring of edges. The model reproduces successfully structural features of the Internet at a fundamental level. We also introduce a quantity called the load as the capacity of node needed for handling the communication traffic and study its time-dependent behavior at the hubs across years. The load at hub increases with network size N as ~ N1.8. Finally, we study data packet traffic in the microscopic scale. The average delay time of data packets in a queueing system is calculated, in particular, when the number of arrival channels is scale-free. We show that when the number of arriving data packets follows a power law distribution, ~ n-λ, the queue length distribution decays as n1-λ and the average delay time at the hub diverges as ~ N(3-λ)/(γ-1) in the N → ∞ limit when 2 < λ < 3γ being the network degree exponent.

Abstract. We track the evolutionary history of the Internet at the autonomous systems (ASes) level and provide the evidence that it can be described in the framework of the multiplicative stochastic process. It is found that the fluctuations arising in the process of diversifying connections of each node play an essential role in forming the status quo of the Internet. Extracting relevant parameters for the growth dynamics of the Internet topology, we are able to predict the connectivity (degree) exponent γ of the Internet AS map successfully. We also introduce a quantity called the load as the capacity of node needed for handling the communication traffic and study its distribution over the Internet across years. The load distribution follows a power law with the exponent δ ≈ 2.0 and the load at the hub scales with the network size as h ∼ N 1.8 .