The Lie algebraic structure of colored networks

Fahimeh Mokhtari and Jan Sanders from the Mathematics Department  of Vrije Universiteit Amsterdam explore the structural properties of colored network dynamical systems in their latest study preprint available on arXiv). This work investigates systems where components, grouped by color, share identical functions but differ in inputs, to investigate the inherent linear structure of these networks.

The paper introduces a novel approach for computing the normal form of colored network vector fields, crucial for local dynamics analysis, particularly when the system's organizing center is nonsemisimple. Utilizing the semigroupoid formalism introduced by Bob Rink and Jan Sanders, the study presents an algorithm for deriving the Levi decomposition of the Lie algebra associated with such networks, revealing valuable insights into their solvable and semisimple parts.

In addition to theoretical advancements, the paper demonstrates how this framework aids in determining multipliers for spectral analysis, with applications to bifurcation studies and critical system control.
This preprint marks a significant step towards a deeper understanding of high-dimensional dynamical systems and their complex behaviors.