On the Definiteness of the Weighted Laplacian and its Connection to Effective Resistance

TL;DR

This paper analyzes when the weighted graph Laplacian with negative weights is positive semi-definite, linking it to effective resistance, and demonstrates how negative weights can induce clustering in consensus networks.

Contribution

It provides a novel characterization of the definiteness of the weighted Laplacian with negative weights using effective resistance and extends the analysis to multiple negative edges.

Findings

Laplacian becomes indefinite if negative weight exceeds inverse effective resistance

Single negative weight edge can induce clustering behavior in consensus networks

Results enable design of networks with desired clustering properties

Abstract

This work explores the definiteness of the weighted graph Laplacian matrix with negative edge weights. The definiteness of the weighted Laplacian is studied in terms of certain matrices that are related via congruent and similarity transformations. For a graph with a single negative weight edge, we show that the weighted Laplacian becomes indefinite if the magnitude of the negative weight is less than the inverse of the effective resistance between the two incident nodes. This result is extended to multiple negative weight edges. The utility of these results are demonstrated in a weighted consensus network where appropriately placed negative weight edges can induce a clustering behavior for the protocol.