Abstract
How does a ‘normal’ computer (or social) network look like? How can we spot ‘abnormal’ sub-networks in the Internet, or web graph? The answer to such questions is vital for outlier detection (terrorist networks, or illegal money-laundering rings), forecasting, and simulations (“how will a computer virus spread?”). The heart of the problem is finding the properties of real graphs that seem to persist over multiple disciplines. We list such “laws” and, more importantly, we propose a simple, parsimonious model, the “recursive matrix” (R-MAT) model, which can quickly generate realistic graphs, capturing the essence of each graph in only a few parameters. Contrary to existing generators, our model can trivially generate weighted, directed and bipartite graphs; it subsumes the celebrated Erdős-Renyi model as a special case; it can match the power law behaviors, as well as the deviations from them (like the “winner does not take it all” model of Pennock et al. [20]). We present results on multiple, large real graphs, where we show that our parameter fitting algorithm (AutoMAT-fast) fits them very well.
Highlights
Graphs, networks and their surprising regularities/laws have been attracting significant interest recently
3.3 Parameter fitting with AutoMAT-fast: The Recursive Matrix (R-MAT) model can be considered as a binomial cascade in two dimensions
Our R-MAT model is exactly a step in this direction: we illustrate experimentally that several, diverse real graphs can be well approximated by an R-MAT model with the appropriate choice of parameters
Summary
Networks and their surprising regularities/laws have been attracting significant interest recently. We propose the Recursive Matrix (R-MAT) model, which naturally generates power-law (or “DGX” [4] ) degree distributions. Bipartite graphs have edges between two sets of nodes, like, for example, the graph of the movie-actor database (www.imdb.com). Power-laws have been observed for the degree distributions of the Internet, the WWW and the citation graph, the distribution of “bipartite cores” (≈ communities), the eigenvalues of the adjacency matrix and others [10, 13, 2]. Pennock et al [21] observed deviations from power-laws for the Web graph, which are well-modeled by the truncated, discretized lognormal (“DGX”) distribution of Bi et al [4]. Broder et al [5] show that the WWW has a “bow-tie” structure, while Tauro et al [24] find that the Internet topology is organized as a set of concentric circles around a small core, like a “Jellyfish”
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