Random Projections of Sparse Adjacency Matrices

TL;DR

This paper introduces a random projection technique for sparse adjacency matrices that preserves graph properties and enables unified, scalable representations of graphs of different sizes, with theoretical guarantees on accuracy.

Contribution

It presents a novel random projection method for adjacency matrices that retains graph functionality and scales linearly with vertices, extending Johnson-Lindenstrauss principles to graph representations.

Findings

Projections preserve first-order graph information.

Projection size scales linearly with vertices for accuracy.

Method acts as a distance-preserving map for adjacency matrices.

Abstract

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.