Local/Global Phenomena in Geometrically Generated Graphs

In recent years, a wealth of random graph models have arisen to explain the power-law degree distribution observed in the Internet [1] and other complex networks (see, for example, [6] and [8]). While power-law behavior often drives these models, less is known about other very natural properties found in complex networks. One such notion is the local versus global structure of a network. In [12] for example, the authors use notions from local flow to discover near regular and highly connected subgraphs in a hybrid model. In practice, however, networks are usually embedded in some ambient space, and the notions of local and global structure require less delicate definitions. Consider the problem of providing telephone service to some central hub. Each customer has some given coordinate in the plane, and thus a distance from the hub. For any connection in our telephone network, this distance provides a natural length scale to discuss whether the connection is local. Motivated by this example, let us form a random tree in the unit ball of R as follows. Place the first point, v1, uniformly at random. For each vi, i = 2, . . . , n thereafter, place this point uniformly in the ball and add an edge to the point vj , j < i which minimizes