We analyze a random projection method for adjacency matrices, studying its utility in representing sparse graphs. We show that these random projections retain the functionality of their underlying adjacency matrices while having extra properties that make them attractive as dynamic graph representations. In particular, they can represent graphs of different sizes and vertex sets in the same space, allowing for the aggregation and manipulation of graphs in a unified manner. We also provide results on how the size of the projections need to scale in order to preserve accurate graph operations, showing that the size of the projections can scale linearly with the number of vertices while accurately retaining first-order graph information. We conclude by characterizing our random projection as a distance-preserving map of adjacency matrices analogous to the usual Johnson-Lindenstrauss map.

翻译：我们分析了一种针对邻接矩阵的随机投影方法，并研究了其在表示稀疏图方面的效用。研究表明，这些随机投影不仅能保持原始邻接矩阵的功能，还具有额外特性，使其成为有吸引力的动态图表示。特别地，它们能够在同一空间中表示不同规模和顶点集合的图，从而允许以统一方式对图进行聚合与操作。我们还给出了投影规模需要如何扩展以保持精确图操作的结果，表明当投影规模与顶点数量呈线性关系时，能够准确保留一阶图信息。最后，我们将所提出的随机投影方法描述为一种保持距离的邻接矩阵映射，类似于经典的Johnson-Lindenstrauss映射。