Network Topologies, Power Laws, and Hierarchy

It has long been thought that the Internet, and its constituent networks, are hierarchical in nature. Consequently, the network topology generators most widely used by the Internet research community, GT-ITM [7] and Tiers [11], create networks with a deliberately hierarchical structure. However, recent work by Faloutsos et al. [13] revealed that the Internet’s degree distribution — the distribution of the number of connections routers or Autonomous Systems (ASs) have — is a power-law. The degree distributions produced by the GT-ITM and Tiers generators are not power-laws. To rectify this problem, several new network generators have recently been proposed that produce more realistic degree distributions; these new generators do not attempt to create a hierarchical structure but instead focus solely on the degree distribution. There are thus two families of network generators, structural generators that treat hierarchy as fundamental and degree-based generators that treat the degree distribution as fundamental. In this paper we use several topology metrics to compare the networks produced by these two families of generators to current measurements of the Internet graph. We find that the degree-based generators produce better models, at least according to our topology metrics, of both the AS-level and router-level Internet graphs. We then seek to resolve the seeming paradox that while the Internet certainly has hierarchy, it appears that the Internet graphs are better modeled by generators that do not explicitly construct hierarchies. We conclude our paper with a brief study of other network structures, such as the pointer structure in the web and the set of airline routes, some of which turn out to have metric properties similar to that of the Internet.