Random evolution in massive graphs

Many massive graphs (such as the WWW graph and Call graphs) share certain universal characteristics which can be described by the so-called "power law." In this paper, we, examine three important aspects of power law graphs, (1) the evolution of power law graphs, (2) the asymmetry of in-degrees and out-degrees, (3) the "scale invariance" of power law graphs. In particular, we give three increasingly general directed graph models and one general undirected graph model for generating power law graphs by adding at most one node and possibly one or more edges at a time. We show that for any given edge density and desired power laws for in-degrees and out-degrees, not necessarily the same, the resulting graph will almost surely have the desired edge density and the power laws for the in-degrees and out-degrees. Our most general directed and undirected models include nearly all known power law evolution models as special cases. Finally, we show that our evolution models generate "scale invariant" graphs. We describe a method for scaling the time in our evolution model such that the power law of the degree sequences remains invariant.