Distributed implementation of linear network operators using graph filters

A signal in a network (graph) can be defined as a vector whose elements represent the value of a given magnitude at the different nodes. A linear network (graph) operator is then a linear transformation whose input and output are graph signals. This paper investigates how to use graph filters to implement generic network linear operators in a distributed manner, so that nodes only need to exchange a finite number of messages with their neighbors. The schemes are designed within the framework of shift-invariant graph filters, which are polynomials of the so-called graph-shift operator. The graph-shift operator is a matrix that accounts for the topology of the network, and the filter coefficients - which are the same across nodes - are the coefficients of that polynomial. First, we identify conditions under which the linear operator can be computed exactly. Then, we provide approximate designs for cases where perfect implementation is unfeasible. Setups where each node is allowed to use a different set of filter coefficients are briefly discussed. Finally, we apply this framework to the problem of finite-time consensus and analyze the graph-filter approximation performance for general linear graph operators.