Sp&Hrg 803
Complex Systems in Neurobiology, Language, and Speech

Day 7
19 March 1998
Hierarchical Organization

Summary of this Day's Class
We mostly finished up our discussion on Power Laws.
We were introduced with the idea of hierarchical organization in complex systems and noted the difficulties that exist in designating hierarchies.
The greatest problem is the question of whether the hierarchical organization is in a property of the system itself or is in the eye of the beholder.
We briefly discussed H.A. Simon's idea of near-decomposability in the case of dynamical hierarchies. In these cases one finds that the dynamical behavior at the lower levels of the hierarchy take place at faster rates than at the higher levels. Thus when studying the dynamics of any given level and considers the interactions due to the nearby levels, one finds that the behavior of the lower level can often be approximated as an average (since it is changing at a much faster rate). The effect of a higher level on a lower level can likewise be approximated as a constant since the higher level is changing at a much slower rate.

We will conclude this class by discussing Eduardo Caianiello's Hierarchical Modular System model.
Although the terminology is a bit outdated, he considers hierarchically organized systems that are self-similar in that each successive level is a constant factor (or modulus) higher than the previous. One finds that the organization of levels in this case is self-similar, and as expected, one obtains a power law for the distribution of elements in the system.
He applied this simple model to the study of monetary systems with great success.

Homework Assignment 7 - Due 26 March 98
The Game of Life is a set of rules played out on a grid of squares or cells.
Cells can be in two states: on or off (alive or dead).
The states of the cells in the next time step depend on the current state of the cell as well as the current states of the cells 8 neighbors (East, West, North, South, and the four corners).

The rules governing whether a square will turn on, stay on, or turn off are described by:
1. A cell will turn on (comes to life) if it is surrounded by exactly three live neighbors.
2. A cell will turn off (die) if fewer than two neighbors are alive (its lonely).
3. A cell will turn off (die) if more than three neighbors are alive (its overcrowded).
The last two rules can be expressed as one rule:
A cell will stay on if it is surrounded by exactly two or three live neighbors.

For homework, try these rules out by hand for these 7 initial patterns.



Notice that these rules generate a map, not so unlike the Logistic map. By this I mean that the Logistic Map takes an interval of real numbers and maps it onto a new interval of numbers. This 'game' maps a set of states onto a new set of states.
These rules just tell you how to get the next state. Can you see stable patterns, and periodic patterns like we did for the Logistic map?

Please check out this web site by Alan Hensel and have some fun.
It has a fast Java applet that allows one to load in premade patterns (or draw your own):
http://www.mindspring.com/~alanh/life/