Distributed Linear Network Operators using Graph Filters

We study the optimal design of graph filters (GFs) to implement arbitrary linear transformations between graph signals. GFs can be represented by matrix polynomials of the graph-shift operator (GSO). Since this operator captures the local structure of the graph, GFs naturally give rise to distributed operators. In most setups the GSO is given, so that GF design consists fundamentally in choosing the (filter) coefficients of the matrix polynomial to resemble desired linear transformations. We determine spectral conditions under which a specific linear transformation can be implemented perfectly using GFs. For the cases where perfect implementation is infeasible, we address the optimization of the filter coefficients to approximate the desired transformation. Additionally, for settings where the GSO itself can be modified, we study its optimal design too. After this, we introduce the notion of a node-variant GF, which allows the simultaneous implementation of multiple (regular) GFs in different nodes of the graph. This additional flexibility enables the design of more general operators without undermining the locality in implementation. Perfect and approximate designs are also studied for this new type of GFs. To showcase the relevance of the results in the context of distributed linear network operators, the paper closes with the application of our framework to two particular distributed problems: finite-time consensus and analog network coding. Index terms — Graph signal processing, Graph filter design, Distributed linear transformation, Network operator, Finite-time consensus, Analog network coding.