Hypergraphs offer flexible and robust data representations for many applications, but methods that work directly on hypergraphs are not readily available and tend to be prohibitively expensive. Much of the current analysis of hypergraphs relies on first performing a graph expansion -- either based on the nodes (clique expansion), or on the edges (line graph) -- and then running standard graph analytics on the resulting representative graph. However, this approach suffers from massive space complexity and high computational cost with increasing hypergraph size. Here, we present efficient, parallel algorithms to accelerate and reduce the memory footprint of higher-order graph expansions of hypergraphs. Our results focus on the edge-based $s$-line graph expansion, but the methods we develop work for higher-order clique expansions as well. To the best of our knowledge, ours is the first framework to enable hypergraph spectral analysis of a large dataset on a single shared-memory machine. Our methods enable the analysis of datasets from many domains that previous graph-expansion-based models are unable to provide. The proposed $s$-line graph computation algorithms are orders of magnitude faster than state-of-the-art sparse general matrix-matrix multiplication methods, and obtain approximately $5-31{\times}$ speedup over a prior state-of-the-art heuristic-based algorithm for $s$-line graph computation.
翻译:测深仪为许多应用提供了灵活而有力的数据表示,但直接在高测图上工作的方法并非现成的,而且往往费用高得令人望而却步。目前对高测图进行的大部分分析依赖于首先进行图表扩展 -- -- 要么基于节点(Clique 扩展),要么基于边缘(线形图) -- -- 然后在由此产生的代表图上进行标准图形分析。然而,这一方法存在巨大的空间复杂性和高超光谱尺寸的高计算成本问题。在这里,我们提出了高效、平行的算法,以加速和减少高测图扩张的记忆足迹。我们的成果侧重于基于边缘的美元线图扩展,但我们为更高测距扩展而开发的方法。据我们所知,我们的第一个框架是能够对单一共享的机器上的大型数据集进行超测光谱分析。我们的方法使得能够从许多领域分析以往基于图形的模型扩展模型无法提供的基于数据的数据集。拟议的以美元线为单位的图表图表缩略图计算,而我们开发的更高级的曲线扩展方法也是我们所开发的更高水平的州平面平面平面的直径矩阵算方法。