# Filtering Random Graph Processes Over Random Time-Varying Graphs

## TL;DR

This paper analyzes the behavior of graph filters on random time-varying graphs and signals, providing statistical insights and novel methods for noise reduction and computational efficiency.

## Contribution

It introduces a stochastic analysis of FIR and ARMA graph filters on random graphs and signals, and proposes new methods leveraging randomness for noise cancellation and complexity reduction.

## Key findings

- Filters behave as deterministic filters on the expected graph.
- Upper bounds for the variance of filter outputs are established.
- Proposed methods outperform existing algorithms in noise reduction and complexity.

## Abstract

Graph filters play a key role in processing the graph spectra of signals supported on the vertices of a graph. However, despite their widespread use, graph filters have been analyzed only in the deterministic setting, ignoring the impact of stochastic- ity in both the graph topology as well as the signal itself. To bridge this gap, we examine the statistical behavior of the two key filter types, finite impulse response (FIR) and autoregressive moving average (ARMA) graph filters, when operating on random time- varying graph signals (or random graph processes) over random time-varying graphs. Our analysis shows that (i) in expectation, the filters behave as the same deterministic filters operating on a deterministic graph, being the expected graph, having as input signal a deterministic signal, being the expected signal, and (ii) there are meaningful upper bounds for the variance of the filter output. We conclude the paper by proposing two novel ways of exploiting randomness to improve (joint graph-time) noise cancellation, as well as to reduce the computational complexity of graph filtering. As demonstrated by numerical results, these methods outperform the disjoint average and denoise algorithm, and yield a (up to) four times complexity redution, with very little difference from the optimal solution.