# Graph Filters and the Z-Laplacian

arXiv: https://arxiv.org/abs/1707.05851 · HTML: https://mlsanity.com/paper/1707.05851/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jul 2017.

## Built on by (most cited first, 1 of 1)

- [CenGCN: Centralized Convolutional Networks with Vertex Imbalance for Scale-Free Graphs](https://mlsanity.com/paper/2202.07826/) (2022)

## Abstract (arXiv)

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.