| The subject of this work is the Internet, on both the microscopic and the macroscopic level: by the former we mean just a link within the vast network, and we focus on the study of the mathematical properties of the traffic through it; by the latter we mean the entire network, or at least a large part of it, which the “composite eye” of a group of BGP routers is able to reconstruct for us, and we focus on its connectivity properties.; Regarding the microscopic level, Internet traffic on a network link can be modeled as a stochastic process. After detecting and quantifying the properties of this process, using statistical tools, we develop a series of mathematical models, culminating in one that is able to generate “traffic” that exhibits—as a key feature—the same difference in behavior for different time scales, as observed in real traffic, and is moreover indistinguishable from real traffic by other statistical tests as well. Tools inspired from the models are then used to determine and calibrate the type of activity taking place in each of the time scales. Surprisingly, the above procedure does not require any detailed information originating from either the network dynamics, or the decomposition of the total traffic into its constituent user connections, but rather only the compliance of these connections to very weak conditions.; Regarding the macroscopic level, the Internet, or any part of it, can be modeled as a time evolving random graph, whose nodes represent ASs (or routers), and whose edges represent direct connections between them; its properties are fundamentally different from the ones of random graphs traditionally studied. After using real BGP data to determine these properties, we use them as a guide to suggest a model that can simulate the evolution of such a graph and match its key features.; In both of the aforementioned studies, we take care to keep the models simple enough to be mathematically tractable and to avoid introducing in them parameters that do not correspond to actual network quantities. Our ambition is not to produce models that “just work”, but that capture, in addition, the mechanisms of the underlying phenomenon. |
