Determine whether the spectral degree exponent is submodular and useful for assortativity control

Investigate whether the spectral degree exponent is submodular and whether it can be used successfully in applications to enhance or reduce degree assortativity.

Background

The paper introduces the spectral degree exponent (SDE) as an implicitly defined graph metric related to the spectral radius and the degree sequence. Its empirical analysis finds a strong correlation between the SDE and degree assortativity across small graphs, random graph models, and real-world communication networks.

Despite this observed relationship, the paper does not establish structural properties such as submodularity, nor does it demonstrate whether the SDE can serve as a practical tool for increasing or decreasing degree assortativity in applications. The authors explicitly identify both questions as unresolved directions for future research.