Conjectures on near-balanced biological networks

Formalize and prove the conjectures that (1) interaction graphs that are nearly balanced (i.e., close to sign-consistent/orthant-monotone structure) tend to exhibit more regular dynamical behavior and confer biological advantage compared to graphs that are far from monotone, and (2) real biological networks are substantially closer to being balanced than random networks of comparable size and with the same distribution of positive and negative edges. Provide precise mathematical definitions of "nearly balanced" and appropriate comparison baselines, and establish the conjectures rigorously.

Background

The paper develops a connection between signed interaction graphs and monotone dynamics, showing that balanced (sign-consistent) graphs correspond to orthant-monotone systems with strong qualitative predictability and robustness. It then presents empirical evidence that intracellular regulatory networks (e.g., yeast and E. coli) are far more balanced than random graphs, motivating conjectures about the prevalence and dynamical benefits of near-monotone structure.

The authors explicitly state two conjectural claims: that near-balanced networks may be biologically advantageous by yielding more regular dynamics, and that real networks are closer to balanced than matched random networks. They note the absence of precise mathematical formulation or proofs and call for work to make these ideas rigorous, including definitions of "distance to monotonicity" and statistical baselines.