Dynamics and changing environments in highly optimized tolerance

Highly optimized tolerance (HOT) is a mechanism for power laws in complex systems based on the robust design of systems in uncertain environments. Once the system, the environment, and the optimization scheme have been specified, the HOT state is fixed and corresponds to the set of measure zero (typically a single point) in the configuration space which minimizes a cost function U. Here we explore the U-dependent structures in configuration space which are associated with departures from the optimal state. We introduce dynamics, quantified by an effective temperature T, such that $T=0\mathrm{}$ corresponds to the original HOT state, while $\overrightarrow{T}\infty$ corresponds to completely random configurations. More generally, T defines the range in state space over which fluctuations are likely to be observed. In a fixed environment fluctuations always raise the average cost. However, in a time-dependent environment, mobile configurations can lower the average U because they adjust more efficiently to changes.