Complexity, Structure and Control: From Degrees of Freedom to Requisite Variety

In a previous paper (DOI 10.5281/zenodo.23588254) we defined the complexity of an open subsystem as the rate at which it raises the entropy of its quasi-isolated host. Here we connect that dynamic definition to two older ideas: the intuition that complexity is set by internal structure (number of elements, their degrees of freedom and the links between them), and Ashby's law of requisite variety. For structure, we review counting, graph-based and information-based measures, then compute the entropy of a system and of its environment directly in a linear transport network with N elements, f degrees of freedom per element and M links. The balance closes to machine precision, and the steady entropy production is exactly C = f·G_eff·X², where G_eff is the Kirchhoff effective conductance between the driving gradients. The product N·f·M is an upper bound on C, not its value: across 600 random architectures it predicts C with Spearman ρ = 0.59, while the size of the interface with the gradients gives ρ = 0.91. Hub architectures are efficient but lose all throughput when the hub is removed. For control, we combine Ashby's law in the information-theoretic form of Touchette and Lloyd with Landauer's bound. A regulator that holds outcome entropy at h_O against disturbances of entropy rate h_D must have C_R ≥ η_R k_B ln2 (h_D − h_O), where η_R ≥ 1 is its thermodynamic inefficiency. In a contest between two systems with equal efficiencies and allocations, the less complex system can never prevail; the more complex one prevails only with a margin max(θ, 1/θ). Resources do not multiply variety; they enter through separate conditions on power and endurance. A stylised application to political systems shows that the claim "democracies always defeat autocracies" does not follow; the model supports only a conditional advantage for distributed architectures in high-variety, long-horizon contests. Version 2 adds Appendix A, an application to Lotka–Volterra models. The competitive-exclusion conditions coincide exactly with the contest condition (coexistence ↔ mutual security, bistability ↔ mutual vulnerability). In energy-closed food chains the knockout index of the top species alternates in sign with chain length. In 300 random food webs entropy production follows the number of producers (ρ = 0.68), not the structural product N·M (ρ = 0.14). Files: English version (main.pdf), Russian version (main_ru.pdf), simulation script (model.py, lv.py) and generated data. .