Distributed Kronecker Graph Generation with Ground Truth of Many Graph Properties

Abstract

Computing various global and local topological graph features is an important facet of data analysis. To do so robustly and scalably requires efficient graph algorithms that either calculate topological features exactly or approximate topological features accurately. For this reason researchers developing distributed graph analytic algorithms desire generated graph benchmarks that share the challenging characteristics of real-world graphs (small-world, scale-free, heavy-tailed degree distribution) with efficiently calculated ground truth to the desired ouput. Given two small scale-free graphs with adjacency matrices A and B, their Kronecker product graph [1] has adjacency matrix C = A × B. Such Nonstochastic Kronecker graphs are highly compressible, and many expensive global graph calculations can be computed in sublinear time, with local graph statistics computed exactly in linear time, both from a sublinear amount of storage. Therefore, this class of graphs are likely of high interest to those pursuing data analysis tasks that incorporate diverse graph-based features. Here, we extend previous results regarding local triangle statistics and demonstrate that ground truth Kronecker formulas apply to: (i) some distance-based vertex centrality metrics (vertex eccentricity and closeness centrality), (ii) internal and external edge density of communities. Moreover, we demonstrate several scaling laws apply that allow researchers to have control over various ground truth quantities.