Are the flows of complex-valued Laplacians and their pseudoinverses related?

TL;DR

This paper investigates the relationship between complex-valued Laplacian flows and their pseudoinverses, establishing conditions for consensus and demonstrating their interdependence in certain network types, with applications to power networks.

Contribution

It introduces a pseudoinverse Laplacian flow system and proves its consensus conditions, linking it to classical Laplacian flows in complex-valued networks.

Findings

Pseudoinverse Laplacian flows achieve consensus under specific conditions.

Consensus in pseudoinverse flows is equivalent to Laplacian flow consensus in certain networks.

The approach is validated with examples, especially in power network contexts.

Abstract

Laplacian flows model the rate of change of each node's state as being proportional to the difference between its value and that of its neighbors. Typically, these flows capture diffusion or synchronization dynamics and are well-studied. Expanding on these classical flows, we introduce a pseudoinverse Laplacian flow system, substituting the Laplacian with its pseudoinverse within complex-valued networks. Interestingly, for undirected graphs and unsigned weight-balanced digraphs, Laplacian and the pseudoinverse Laplacian flows exhibit an interdependence in terms of consensus. To show this relation, we first present the conditions for achieving consensus in the pseudoinverse Laplacian flow system using the property of real eventually exponentially positivity. Thereafter, we show that the pseudoinverse Laplacian flow system converges to consensus if and only if the Laplacian flow system achieves consensus in the above-mentioned networks. However, these are only the sufficient conditions for digraphs. Further, we illustrate the efficacy of the proposed approach through examples, focusing primarily on power networks.