Analyzing and Characterizing Small-World Graphs

We study variants of Kleinberg's small-world model where we start with a k-dimensional grid and add a random directed edge front each node. The probability u's random edge is to v is proportional to d(u, v)(-r) where d(u, v) is the lattice distance and r is a parameter of the model. For a k-dimensional grid, we show that these graphs have poly-log expected diameter when k < r < 2k, but have polynomial expected diameter when r > 2k. This shows an interesting phase-transition between small-world and "large-world" graphs. We also present; a general framework to construct classes of small-world graphs with circle minus(log n) expected diameter, which includes several existing settings such as Kleinberg's grid-based and tree-based settings [15]. We also generalize the idea of 'adding links with probability alpha the inverse distance' to design small-world graphs. We use semi-metric and metric functions to abstract distance to create a class of random graphs where almost all pairs of nodes are connected by a path of length O(log u), and using only local information we can find paths of poly-log length.