Is the Internet Fractal? The Multicast Power Law Revisited

One of the main bene...ts of multicast communication is the overall reduction of network load. To quantify this reduction, when compared to traditional unicast, experimental studies by Chuang and Sirbu indicated the so called power law which asserts that the number of links L(m) in a multicast delivery tree connecting a source to m (distinct) sites satis...es L(m) 1⁄4 cm where c is a constant. In order to explain theoretically this behavior, Phillips, Shenker, and Tangmunarunkit examined approximately L(m) for a V -ary complete tree topology, and concluded that L(m) grows nearly linearly with m, thus not obeying the power law. We ...rst re-examine the analysis by Phillips et.al. and provide precise asymptotic expansion for L(m) that con...rms the nearly linear (with some wobbling) growth. Claiming that the essence of the problem lies in the modeling assumptions, we replace the V -ary complete tree topology by a V ary self-similar tree with similarity factor 0 · μ < 1. In such a tree a node at level k is replicated CV (D¡k)μ times, where D is the depth of the tree and C is a constant. Under this assumption, we analyze again L(m) and prove that L(m) » cm1¡μ where c is an explicitly computable constant. Hence self-similar trees provide a plausible explanation of the multicast power law. Next, we examine more general conditions for general trees, under which the power law still holds. We also discuss some experimental results in real networks that rea¢rm the power law and show that in these networks the general conditions hold. In particular, we experimentally establish that for the studied network μexp 1⁄4 0:12 , leading to the power law L(n) = £(n). Our theoretical ...ndings have been established by analytical techniques of the precise analysis of algorithms such as Mellin transform and complex asymptotics. These methods are interesting in their own right, and they potentially may shed some more light on the qualitative behavior of the Internet.