# Density Independent Algorithms for Sparsifying $k$-Step Random Walks

## TL;DR

This paper introduces faster algorithms for sparsifying transition matrices of k-step random walks on graphs, improving efficiency by leveraging density-independent methods and tighter sampling bounds.

## Contribution

It presents novel algorithms that produce sparse approximations of k-step random walk matrices with improved runtime and size bounds, based on better sampling analysis.

## Key findings

- Produces a graph with about n log n edges approximating the k-step walk
- Achieves runtime of m + n log^4 n for the sparsification process
- Revisits and refines density independent graph sparsification techniques

## Abstract

We give faster algorithms for producing sparse approximations of the transition matrices of $k$-step random walks on undirected, weighted graphs. These transition matrices also form graphs, and arise as intermediate objects in a variety of graph algorithms. Our improvements are based on a better understanding of processes that sample such walks, as well as tighter bounds on key weights underlying these sampling processes. On a graph with $n$ vertices and $m$ edges, our algorithm produces a graph with about $n\log{n}$ edges that approximates the $k$-step random walk graph in about $m + n \log^4{n}$ time. In order to obtain this runtime bound, we also revisit "density independent" algorithms for sparsifying graphs whose runtime overhead is expressed only in terms of the number of vertices.