On time-scale designs for networks

Abstract

We motivate the problem of designing a subset of the edge weights in a graph, to shape the spectrum of an associated linear time-invariant dynamics. We address a canonical design problem of this form by applying time-scale assignment methods, and give graph-theoretic characterisations of the designed dynamics.

Keywords:

• graph design
• eigenvalue placement
• time-scale assignment
• special coordinate basis

Acknowledgements

This work was supported by National Science Foundation grants ECS-0528882 and ECCS-0725589, NAVY grants ONR KKK777SB001 and ONR KKK760SB0012, and National Aeronautics and Space Administration grant NNA06CN26A. We thank Dr B. Lesieutre for an illumi-nating discussion on the subject.

Notes

Notes

1. We note that the following definitions are for arbitrary designable-edge graphs, not only forests.

2. We again note that the definition is in force for any partial edge design problem, not only one with designable-edge graph that is a z-forest.

3. We note that the results in this section hold for arbitrary designable-edge graphs, not only forests.