Implementing Huge Sparse Random Graphs

Consider a scenario where one desires to simulate the execution of some graph algorithm on huge random G(N,p) graphs, where N= 2nvertices are fixed and each edge independently appears with probability p= pn. Sampling and storing these graphs is infeasible, yet Goldreich et al. [7], and Naor et al. [12] considered emulating dense G(N,p) graphs by efficiently computable `random looking' graphs. We emulate sparseG(N,p) graphs - including the densities of the G(N,p) threshold for containing a giant component (p~1 / N), and for achieving connectivity (p茂戮驴 ~ln N/ N). The reasonable model for accessing sparse graphs is neighborhood queries where on query-vertex v, the entire neighbor-set Γ(v) is efficiently retrieved (without sequentially deciding adjacency for each vertex). Our emulation is faithful in the sense that our graphs are indistinguishable from G(N,p) graphs from the view of any efficient algorithm that inspects the graph by neighborhood queries of its choice. In particular, the G(N,p) degree sequence is sufficiently well approximated.