Random Projections of Sparse Adjacency Matrices
This provides a method for dynamic graph representation that allows aggregation and manipulation of graphs of different sizes in a unified space, which is incremental over existing random projection techniques.
The paper tackles the problem of representing sparse graphs efficiently by analyzing a random projection method for adjacency matrices, showing that projections can scale linearly with the number of vertices while retaining first-order graph information.
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.