Algorithms for finding dense regions in an input graph have proved to be effective tools in graph mining and data analysis. Recently, Tatti and Gionis [WWW 2015] presented a novel graph decomposition (known as the locally-dense decomposition) that is similar to the well-known k-core decomposition, with the additional property that its components are arranged in order of their densities. Such a decomposition provides a valuable tool in graph mining. Unfortunately, their algorithm for computing the exact decomposition is based on a maximum-flow algorithm which cannot scale to massive graphs, while the approximate decomposition defined by the same authors misses several interesting properties. This calls for scalable algorithms for computing such a decomposition. In our work, we devise an efficient algorithm which is able to compute exact locally-dense decompositions in real-world graphs containing up to billions of edges. Moreover, we provide a new definition of approximate locally-dense decomposition which retains most of the properties of an exact decomposition, for which we devise an algorithm that can scale to real-world graphs containing up to tens of billions of edges. Our algorithm is based on the classic Frank-Wolfe algorithm which is similar to gradient descent and can be efficiently implemented in most of the modern architectures dealing with massive graphs. We provide a rigorous study of our algorithms and their convergence rates. We conduct an extensive experimental evaluation on multi-core architectures showing that our algorithms con- † This research was partially supported by the Hong Kong  RGC verge much faster in practice than their worst-case analysis. Our algorithm is even more efficient for the more specialized problem of computing a densest subgraph.
Danisch et al. REF introduced an alternative iterative technique for computing locally dense decomposition that scales well in practice.
Large Scale Density-friendly Graph Decomposition via Convex Programming
WWW
Computer Science