Filtering Random Graph Processes Over Random Time-Varying Graphs

Abstract

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 stochasticity 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 timevarying 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.

On the Transferability of Spectral Graph Filters

“…is a rational function, g(T) can be defined in two ways. First, by (2), and second by compositions, linear combinations, and inversions, as…”

“…Since the eigendecomposition is not stable to perturbations in the topology of the graph, this result does not prove robustness to such perturbations. [2] showed that the expected graph filter under random edge losses is equal to the accurate output. However, [2] did not bound the error in the output in terms of the error in the graph topology.…”

Graph Error Effect in Graph Signal Processing

“…. , λ N −1 of W appear as coefficient in (16). Thus, the difference between the largest and smallest coefficients in each equation in (16) can be quite significant.…”

“…, λ N −1 of W appear as coefficient in (16). Thus, the difference between the largest and smallest coefficients in each equation in (16) can be quite significant. Moreover, the difference between coefficients in (16) for the case of true A compared to the case of mismatched W can be also significant.…”

“…In [15], a new centrality measure was introduced and the robustness with respect to edge weight perturbation was considered. Autoregressive moving average graph filters and finite impulse response graph filters for time-varying graph signals were compared in [16]. Among other scenarios, the case when some of the edges are missing had been studied there.…”