Tuesday, January 8, 2008

A Simple Model for Understanding Emergence

The model is for understanding changes in complexity over scale in the context of cooperation, and also, with very few parts, offers valuable clarification of ideas like robustness, modularity and the 'plant-animal' spectrum (Sometimes you cut some living organism into two and both parts die, sometimes both parts survive and sometimes just one part dies. Does this have significance for a general understanding in interrelationship, and if so, how?). So:

There are 5 cells. Each cell has five parts and manufactures one unit of chemical in each part. Each cell needs to have 5 different chemicals at any moment to be legal. If a cell is not receiving a unit of all five different chemicals, it is not legal.

In the simplest case, each of the five cells is using each part of itself to produce each chemical it needs for itself. Five parts, five chemicals, easy. Each cell is simple and legal.

Now for the next step. Let them share chemical. This makes possible (but not practical) an important new configuration: We can make it so each cell is specializing in one chemical. Cell One makes five units of Chemical One. Cell Two makes five units of Chemical Two, etc. All cells are sharing, so each cell is receiving the five chemicals that it needs to be legal.

I noted that this is possible but not practical because if we separate one cell from the other four, than all five are illegal. The removed cell is receiving five units of only one chemical and the other four are receiving all but that chemical.

At this step of the model, this 'collective' arrangement offers no advantages over the 'individualist' arrangement above.

Now let us introduce economies of scale. Within one cell, if Chemical One is being produced by two of the cell's parts, it gets four units of that Chemical One. If it is produced by three parts (3/5 of its manufacturing capacity) it produces 9 units of Chemical One and so on. The benefit can be any kind of greater-than-linear growth, I chose n^2 for this model where n is number of parts devoted to manufacture of a particular chemical. Lets call the exponent of this function the "economy". In this case the economy is 2. In the starting case, it was 1.

Where the economy was 1, there was no important difference between the collectivist and individualist arrangements. With an economy greater than one, the collective arrangement produces more chemical than the individualist arrangement. Where five individualist cells collectively produce 25 units of chemical, the cooperating cells collectively produce 125 units of chemical.

With a small tweak to the model, we can call go a step further and call the 5 cooperating cells More Complex than the five individual cells. To make this possible I introduced efficiency, where, working at 100% efficiency, the cooperating cells produce 125 units of chemical and at 40% efficiency they produce 50 units. These five cells are all legal down to 20% efficiency, and so they have 80 legal states, while five individual cells have only one legal state (at 100% efficiency).

We are now looking at a tradeoff. Where, as individuals, the cells are robust to separation and are individually more complex, the cooperating cells are individually simpler and collectively more complex than individualized competitors. At intermediate levels of cooperation, we will see robustness and efficiency slide past each other.

The economy exponent provides the incentive to scale up. It provides an incentive for higher scale organization. Not only does this exponent control the extent to which cooperation gets rewarded, my preliminary poking around indicates that it has influence over how 'plantlike' or 'animallike' a collective of units is. Really! But I won't go into the work that shows it until you respond to this initial work.

So, in summary, this model provides a way to analyze changes in complexity of a simple system over changes in scale. It also provides a way to analyze plantlikeness and animallikeness. It also emphasizes that there are different kinds of robustness (robust to being separated or to being starved) and forces a rigorous definition of modular (are the specialized cells modules because they are specialized, or are the individual cells modules because they are self sufficient?). Both of these concepts are far too butchered in everyday discussion of ideas like this, and the model shows some of the nuance in them.

Also, this discussion only looked at the extreme ends of the spectrum, full collectiveness and individualness, but the intermediate points on the spectrum are ripe. By digging around I've seen how plantlikeness and animallikeness are meaningless distinctions at economies of one, in an intuitive way. It has also shown me how the more complex an individual cell is (the more parts it has), the more it will be animallike.

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