Density-friendly Graph Decomposition

Decomposing a graph into a hierarchical structure via k-core analysis is a standard operation in any modern graph-mining toolkit. k-core decomposition is a simple and efficient method that allows to analyze a graph beyond its mere de-gree distribution. More specifically, it is used to identify areas in the graph of increasing centrality and connected-ness, and it allows to reveal the structural organization of the graph. Despite the fact that k-core analysis relies on vertex de-grees, k-cores do not satisfy a certain, rather natural, density property. Simply put, the most central k-core is not nec-essarily the densest subgraph. This inconsistency between k-cores and graph density provides the basis of our study. We start by defining what it means for a subgraph to be locally-dense, and we show that our definition entails a nested chain decomposition of the graph, similar to the one given by k-cores, but in this case the components are ar-ranged in order of increasing density. We show that such a locally-dense decomposition for a graph G = (V,E) can be computed in polynomial time. The running time of the exact decomposition algorithm is O(|V |2|E|) but is signifi-cantly faster in practice. In addition, we develop a linear-time algorithm that provides a factor-2 approximation to the optimal locally-dense decomposition. Furthermore, we show that the k-core decomposition is also a factor-2 ap-proximation, however, as demonstrated by our experimental evaluation, in practice k-cores have different structure than locally-dense subgraphs, and as predicted by the theory, k-cores are not always well-aligned with graph density

HyperPRAW : architecture-aware hypergraph restreaming partition to improve performance of parallel applications running on high performance computing systems

High Performance Computing (HPC) demand is on the rise, particularly for large distributed computing. HPC systems have, by design, very heterogeneous architectures, both in computation and in communication bandwidth, resulting in wide variations in the cost of communications between compute units. If large distributed applications are to take full advantage of HPC, the physical communication capabilities must be taken into consideration when allocating workload. Hypergraphs are good at modelling total volume of communication in parallel and distributed applications. To the best of our knowledge, there are no hypergraph partitioning algorithms to date that are architecture-aware. We propose a novel restreaming hypergraph partitioning algorithm (HyperPRAW) that takes advantage of peer to peer physical bandwidth profiling data to improve distributed applications performance in HPC systems. Our results show that not only the quality of the partitions achieved by our algorithm is comparable with state-of-the-art multilevel partitioning, but that the runtime performance in a synthetic benchmark is significantly reduced in 10 hypergraph models tested, with speedup factors of up to 14x