Distributed graph filters have recently found applications in wireless sensor networks (WSNs) to solve distributed tasks such as reaching consensus, signal denoising, and reconstruction. However, when implemented over WSNs, the graph filters should deal with network limited energy constraints as well as processing and communication capabilities. Quantization plays a fundamental role to improve the latter but its effects on distributed graph filtering are little understood. WSNs are also prone to random link losses due to noise and interference. In this instance, the filter output is affected by both the quantization error and the topological randomness error, which, if it is not properly accounted in the filter design phase, may lead to an accumulated error through the filtering iterations and significantly degrade the performance. In this paper, we analyze how quantization affects distributed graph filtering over both time-invariant and time-varying graphs. We bring insights on the quantization effects for the two most common graph filters: the finite impulse response (FIR) and autoregressive moving average (ARMA) graph filter. Besides providing a comprehensive analysis, we devise theoretical performance guarantees on the filter performance when the quantization stepsize is fixed or changes dynamically over the filtering iterations. For FIR filters, we show that a dynamic quantization stepsize leads to more reduction of the quantization noise than in the fixed-stepsize quantization. For ARMA graph filters, we show that decreasing the quantization stepsize over the iterations reduces the quantization noise to zero at the steady-state. In addition, we propose robust filter design strategies that minimize the quantization noise for both time-invariant and time-varying networks. Numerical experiments on synthetic and two real data sets corroborate our findings and show the different trade-offs between quantization bits, filter order, and robustness to topological randomness.

Distributed graph filters can be implemented over wireless sensor networks by means of cooperation and exchanges among nodes. However, in practice, the performance of such graph filters is deeply affected by the quantization errors that are accumulated when the messages are transmitted. The latter is paramount to overcome the limitations in terms of bandwidth and computation capabilities in sensor nodes. In addition to quantization errors, distributed graph filters are also affected by random packet losses due to interferences and background noise, leading to the degradation of the performance in terms of the filtering accuracy. In this work, we consider the problem of designing graph filters that are robust to quantized data and time-varying topologies. We propose an optimized method that minimizes the quantization error, while ensuring an accurate filtering over time-varying graph topologies. The efficiency of the proposed theoretical findings is validated by numerical results in random wireless sensor networks.