3.Fitness Functions And Rugged Landscapes

There exists a considerable literature that analyses the structure of these fitness functions. A frequent metaphor is that of a fitness landscape. The location in this landscape identifies a specific behavioral pattern whereas the elevation of a point in the landscape corresponds to its fitness value. Different authors use different conventions for the sign of this fitness function. In some publications optimal fitness corresponds to a peak in the landscape ("mountain summit"); in others increasing fitness values correspond to lower elevations in the landscape, i.e. the system evolves towards a local or global minimum in the landscape. Fitness landscapes in most complex systems have a number of complicating features:

• The dimension of the behavioral space can be much larger than the two dimensions of topographic landscapes. In a biological setting for instance the dimensions can be determined by the number of different genes that influence the specific behavior. "Moving" in that landscape then corresponds to mutations in the genetic composition of a species. In the context of a commercial enterprise a point the landscape can correspond to investment decisions and the fitness values to the profit rates.
• The shape of the landscape is not static and changes with the movement of each individual agent in that landscape like a poorly designed waterbed.
• The landscape can be rugged and contain cliffs as well as smooth hills.

This implies that small changes in the position can lead to dramatic changes in elevation. Those sudden transitions are sometimes also called "catastrophes" and can be categorized into a number of different classes. It is those sudden changes in fitness values that are associated with "innovation" in the context of complex systems.