Complex Adaptive Systems: Computational Models of Social Life (Ch3: pp. 35-43)ModelingModels as MapsA More Formal Approach to ModelingBreaking Down the Formal ModelModeling Complex SystemsModeling Modeling
For every Complex problem, there is a solution that is simple, neat and wrong.
— H. L. Mencken
Things should be made as simple as possible — but no simpler.
— Albert Einstein
Nothing is built on stone; all is built in sand. But we must build as if the sand were stone.
— Jorge Luis Borges
One of the best models that we encounter in our daily experience is the road map.
Maps are valuable for a variety of reasons
They leave out a lot of unnecessary details
This minimizes distractions and allow us to focus on the questions that are most important to the user
Adding more and more details moves us closer to reality and can obfuscate the benefit of ever creating the map
They are easily understandable to others
This often requires that there is some sort of implicit understanding on the part of the user
They allow us to predict key features of the world, but they also enable us to discover new phenomena
A map of the world may give rise to the theory of continent drift
Plotting earthquakes locations could be used to find hidden fault systems
Snow's 1885 map of cholera cases in the Soho district of London revealed both the mode of transmission and source (a pump on Broad Street) of the disease
The top half of the picture represents the real world, the bottom represents the model
We assume that the real world consists of various states
A transition function exists which maps a given state at time into a new state at time
is unknown but is what the modeler hopes to uncover
To reduce the state space a modeler generates equivalence classes (designated )
Once equivalence classes have been generated, we then need to find a transition function for our model, , that can predict how patterns of pressure and humidity are transformed over time
The success of a particular mode is tied to its ability to capture the behavior of the real world.
If we end up in at the same model state regardless of whether we...
… the model coincides with the world.
Put another way (equations)...
If we start with real-world state
The model transforms this state into
While the real-world state becomes state in the next time period
The model "coincides" with the real world if
The requirement that the maps between the model and the real world must be commutative in this way is known as a homomorphism. Thus, the goal of modeling under this view is to find a set of equivalence classes and a transition function that results in a useful homomorphism.
A model requires choices of both the equivalence classes and the transition function, and the art of modeling lies in judicious choices of both.
There are likely many difference equivalence mappings (and associated transition functions) the value of any particular set of choices depends on the current need(s) of the modeler
Choosing a broad set of equivalence classes makes finding the transition function simpler, but loses resolution and value
Finding useful homomorphisms is often difficult in practice
This continual chasing of the "ideal" model results in a Schumperterian cycle of scientific creative destruction. Modelers attempt to reduce the world to a fundamental set of elements (equivalence classes) and laws (transition functions), and on this basis they hope to better understand and predict key aspects of the world. The ever present quest for refining old, and discovering new, ways to represent the world drives the process of scientific creative destruction.
Complex systems give rise to emergent behavior or properties
To model this, we take the model from the previous section (Figure 3.2) and add an addition section, to represent new emerging entities and/or behavior (brought on by transformation function )
In the social sciences, the two systems might be psychology and economics
In the higher level world, higher-level entities are governed by a new transition function
This idea that the higher level is completely unknown to us coincides with the idea of "hierarchical reductionism" put forth by Dawkins (1976)
Each time we move to a new level, we are confronted with a new world that requires new models. Moreover, creating a theory about how these new levels arise from existing ones, namely understanding the function , becomes important. We would like to be able to develop a theory that helps us understand how states of the world (composed of lower-level entities and interaction rules) are transformed into higher-level entities.
Regardless of the system or methodology, our goal is to employ high quality models. Thus, we apply the same standards of simplicity and elegance to our computational models that we do to our mathematical ones. Models need to be judged by what they eliminate as much as by what they include—like stone carving, the art is in removing what you do not need. Even though a computational model may require thousands of lines of code, if done well it can still embody the simplicity and elegance that is demonstrated in a mathematical model existing in only a few equations.
Having an explicit awareness of the issues surrounding quality modeling is important if we want to work on the frontiers of science. This awareness disciplines our efforts as we explore new problems and employ novel techniques. Creating a model is much like trying to solve a brain teaser. Finding such solutions is often an extremely difficult task involving a combination of theory, practice, and a bit of art. Yet, once discovered, the answer has strong intuitive appeal and appears all too obvious.
Notes by Matthew R. DeVerna