Diagram: Degrees of Freedom

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The notion of 'degrees of freedom' pertains to the potential range of behaviors available within a given system. Without some freedom for a system to change its state, no complex adaptation can occur.

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The notion of degrees of freedom comes to us from physics and statistics, where it describes the number of possible states a system can occupy. For example, a swinging pendulum is constrained to a limited number of 'states' (positions in space) that the pendulum can occupy. If we were to map all of the potential states of the pendulum's behavior we would have what is called a 'phase portrait' of the pendulum (see also Phase Space).

Understanding the degrees of freedom available within a complex system is important because it helps us understand the overall scope of potential ways in which a system can unfold. We can imagine that a given complex system is subject to a variety of inputs (many of which we might not know), but then we must ask, what is the system's range of possible outputs? The degrees of freedom tells us something about what a system is capable of doing: its potential. The system cannot act outside of the boundaries of this action potential.

For example, if we wanted to offer the maximum capacity for motion for a three-dimensional object own space, this can be provided using the following six degrees of freedom, which allow for changes in orientation (rotation) to occur through the 'roll', 'yaw' and 'pitch' dimensions, and displacement in space to occur through the 'up/down', 'back/forward' and 'left/right' parameters. We can see that all potentials of movement are covered within this framework.

If we were to have fewer degrees of freedom, certain types of movement would no longer exist as possibilities. Thus Phase Space - which alludes to the sum total of all potential behaviors - is sometimes referred to as 'possibility space'.

(image courtesy of Wikimedia commons)

So far we have been speaking about physical degrees of freedom, but we might also imagine degrees of freedom in relation to behavioral possibilities.

Imagine we are wanting to stay at an airbnb. We could think of each airbnb as being an agent in a system, competing to win us over with its 'fitness' for our stay. Each Airbnb would be able to tune a number of parameters that one might consider as important in choosing a particular place to stay. These parameters could include cost, cleanliness, distance to center, size, and quietness. Different people might value (or weigh) these parameters differently, and choose their airbnb accordingly. At the same time, we can imagine that each airbnb has a capacity to adjust its 'offering' to different degrees in different dimensions. Location is clearly a limited parameter: a given airbnb unit has no capacity to adapt its location far from the city to center. But it does have the capacity to adjust its price point. Size is also difficult to alter. But cleanliness might have more flexibility. If airbnbs can be considered as agent in a complex system, competing to find patrons who wish to stay at their collection, then they have to operate within the boundary of certain degrees of freedom in terms of how they seek to align themselves with different user needs. Thus if they can't compete on the basis of location then they can attempt to compete on the basis of cost.

The airbnb case should also serve to illustrate that, in many scenarios, the degrees of freedom available to an agent in a complex system cannot be plotted in three dimensional space. We can have a multi-parameter space that the agent is located within.

Degrees of freedom is thus another way of thinking about the responsive capacity of a system, in light of a range of environmental changes or fluctuations. Another way to describe this is the idea of a system's {{requisite-variety}} HANDLEBAR FAIL HANDLEBAR FAIL HANDLEBAR FAIL: a phrase coined by Ross Ashby to explain the amount of variability a system could enact. According to Ashby, a system needs to have a variety of responses commensurate with the variety of inputs.

 


Photo Credit and Caption: Underwater image of fish in Moofushi Kandu, Maldives, by Bruno de Giusti (via Wikimedia Commons)

Cite this page:

Wohl, S. (2019, 31 October). Degrees of Freedom. Retrieved from https://kapalicarsi.wittmeyer.io/definition/degrees-of-freedom

Degrees of Freedom was updated October 31st, 2019.


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    Phase space is an abstract concept that refers to all possible behaviors available to an agent within a complex system.

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