fitness landscape

a fitness landscape is a way of visualizing a problem in optimization or adaptation. There are peaks and valleys, peaks are solutions and the highest peak is the best solution. How to get there? The movement of adaptation can be seen as always moving uphill, since at any time a more adapted condition is preferable to a less adapted one. Depending on the features of the landscape, a particular fitness walk may lead to local optima from which there are no paths leading higher. (see basins of attraction) The behaviour of actual populations may not always be movement upwards. They may cluster at a local optimum, move accross ridges somewhat below the fitness peaks, or drift down from the peaks and wander virtually anywhere accross the fitness landscape. See epigenetic landscape
The apparent inversion between climbing fitness landscapes and descending into basins of attraction, is an expression of the preference of Biologists to think of adaptation as "higher" evolution and Physicists to seek states of least energy. Problem solving is also thought of as an uphill climb.

In The Origins of Order, Stuart Kauffman studies some of the formal characteristics of fitness landscapes using the NK model ( N is the number of variables and K is the number of inputs per variable -- how many of the other variables affect any single variable-- genetics: epistatic interactions) and Boolean networks.

Fitness landscapes can be described as rugged or smooth, as correlated or uncorrelated.

In the NK model, when K=0, each variable N operates independently. The landscape corresponding to this model, comparable to Fujiyama, has one smooth peak, the global optimum. Any uphill route will get there, and values change smoothly. (they are correlated in that nearby points have similar values--they do not vary by more than 1/N)

As K increases, the landscape becomes increasingly rugged to the maximum ( chaotic) state at K=N-1(where every variable is affected by every other variable). Here the values of neighboring points are entirely uncorrelated (this is the chaos). The number of local optima is extremely large and they fall towards the mean fitness of the space of genotypes. This is because conflict and compromise increases as the variables become more interlinked, leading to what Kauffman calls a " complexity catastrophe" where the system becomes intractable to selection.

Christopher Alexander's Notes on the Synthesis of Form looks at design "fit" and tries to identify the subsets of a problem by an analysis similar to the NK model, at least insofar as he seeks variables and then for each one tries to identify the other variables that affect it. Alexander's intention is to break down a design problem into pieces that can be resolved independently. His goal is similar to that of the systematic taxonomists such as Linneaus, who sought to discover the natural orderings and groupings among all living things, the inherent logic in the "Plan of Creation." (see population / typological ) Alexander's fascination with the good fit of traditional design can be compared to Claude Levi-Strauss's analysis of the systems of sensible intuition that he calls "the science of the concrete." (see order)