Graph Filters and the Z-Laplacian

TL;DR

The paper introduces the Z-Laplacian, a unifying framework for graph filters that generalizes the traditional Laplacian, enabling modeling of diverse dynamical processes on networks such as information flow, epidemic spreading, and communication protocols.

Contribution

It proposes the Z-Laplacian as a versatile graph shift operator that encompasses all Z-matrices, unifying continuous and discrete-time dynamical processes in graph signal processing.

Findings

The Z-Laplacian models various dynamical processes on graphs.

Application to wireless networks demonstrates modeling of protocol effects.

Analysis of brain connectivity reveals insights across frequency bands.

Abstract

In network science, the interplay between dynamical processes and the underlying topologies of complex systems has led to a diverse family of models with different interpretations. In graph signal processing, this is manifested in the form of different graph shifts and their induced algebraic systems. In this paper, we propose the unifying Z-Laplacian framework, whose instances can act as graph shift operators. As a generalization of the traditional graph Laplacian, the Z-Laplacian spans the space of all possible Z-matrices, i.e., real square matrices with nonpositive off-diagonal entries. We show that the Z-Laplacian can model general continuous-time dynamical processes, including information flows and epidemic spreading on a given graph. It is also closely related to general nonnegative graph filters in the discrete time domain. We showcase its flexibility by considering two applications. First, we consider a wireless communications networking problem modeled with a graph, where the framework can be applied to model the effects of the underlying communications protocol and traffic. Second, we examine a structural brain network from the perspective of low- to high-frequency connectivity.