Distributed Linear Network Operators Based on Successive Graph Shift Matrices

In the context of graph signal processing, the existing distributed approaches for implementing linear network operators rely on the notion of graph shift matrix, which captures the local structure of the graph. Most of the existing approaches consider only a restricted set of linear network operators. However, in this paper, we focus on approximating general linear network operators as fast as possible after a finite number of local exchanges, with a negligible error. We propose a new distributed successive method based on designing a sequence of different graph shift matrices, which are optimized to approximate the desired network operator in an approximately minimal number of iterations. We also consider the robustness of the distributed computation of linear operators against graph perturbations. For this, we first analyze the effect of graph perturbations on our successive method and then, we incorporate the effect of graph perturbations in our design by proposing an online kernel-based estimator, which enables the nodes of the network to estimate the missing values caused by graph perturbations across iterations via available information received from neighbor nodes. Our numerical results demonstrate the superior performance of our methods over the existing state-of-the-art approaches.