On the topology of overlay-networks

Random-graph models are about to become an important tool in the study of wireless ad-hoc and sensor-networks, peer-to-peer networks, and, generally, overlay-networks. Such models provide a theoretical basis to assess the capabilities of certain networks, and guide the design of new protocols. Especially the recently proposed models for so-called small-world networks receive much attention from the networking community. This paper proposes the use of two more mathematical concepts for the analysis of network topologies, dimension and curvature. These concepts can intuitively be applied to, e.g., sensor-networks. But they can also be sensibly defined for certain other random-graph models. The latter is non-trivial since such models may describe purely virtual networks that do not inherit properties from an underlying physical world. Analysis of a random-graph model for Gnutella-like overlay-networks yields strong indications that such networks might be characterized as a sphere with fractal dimension.