Arithmetic of complexity

I received some interesting questions and comments from readers about my most recent column on complexity, including the following (paraphrased for convenience):

The examples of writing books and a journey of a thousand miles both seem to be examples in which the sum is equal to the parts. What do you think?

Although the previous discussion was more philosophical than mathematical, a discussion of summation (no matter how abstract) nevertheless must begin the definition of summation. Summation is both associative and commutative; the order of operations and the order in which the numbers are written do not matter. (1+2)+3 = 1+(2+3) = 3+2+1 = 6. The result is always the same.

Books are clearly neither associative nor commutative. Words must be written in a certain sequence in order to preserve the intended meaning. For example, I can give my puppy a squeaky toy and I will have a happy puppy. But if I give my squeaky toy a puppy - well, that`s just weird. What would my squeaky toy do with a puppy anyway?

Similarly, the order in which a journey is undertaken is also important. Each step along the path brings new experiences and insights. You meet new people and learn new things. You experience joy and pain and sunburn. If you change the order of your journey, you change the order in which you experience these things (if not the experiences themselves) which in turn changes you. As a result, the way you react to those experiences is based on the order in which you experience them. Too abstract? Have you ever gone on a ride at an amusement park where you got very wet and then went on a rollercoaster to dry off? If you had ridden the attractions in the opposite order you wouldn`t have dried as quickly or annoyed the next rollercoaster passenger as much with your wet seat. Order of operations is clearly important.

Consider it another way. In a simple system that is essentially the sum of its parts, removing a small part will only have a small effect. For example, if you take a pile of 100 rocks and remove one, you are left with a pile of 99 rocks. It is changed very little. If you take a pile of pocket change and remove 25 cents, you are still left with money. (Maybe you can buy an ice cream sandwich with it and maybe not.) If you take a book from a library, you still have a library. No one will mind much unless they are looking for that specific book.

However, if a complex system is more than the sum of its parts, then removing a single part may have an unexpected and potentially unpleasant effect. If you remove one bar from a four-bar linkage you don`t get a three bar linkage (a truss), you get a broken four bar linkage. If you randomly remove one component from your cell phone, it probably won`t work properly anymore. It may not work at all. What if you change the U.S. Federal Reserve interest rate? All sorts of interesting things can happen to the economy - not all good. The more complex the system, the less easily you can predict the impact that a change will have.

How many words would you have to change or remove from a great poem before it loses that certain je ne sais quoi that makes it a great literary work? What if Tennyson`s "Valley of Death" were re-dubbed "the valley between Fedyukhin Heights and the Causeway Heights" in the "Charge of the Light Brigade?" Would we still remember the noble six hundred in rhyme? (Would we still be able to memorize it? Would we want to?) What if we called it the "Balaklava Plain Ravine" instead? The word count in the poem would be the same, but something would be lost.

What if the next step you take on your journey determines if you choose the road more or less traveled? Like Robert Frost, would it make all the difference? What if a small incident on your path started a larger conflict? Would it change history?

Yes, the laws of probability agree that an infinite number of monkeys typing on an infinite number of typewriters will eventually produce the complete works of Shakespeare. (Although an experiment in 2003 showed that monkeys are actually very poor random number-generators. Several keyboards were destroyed in the experiment.) And, indeed, you can walk a thousand miles on a treadmill and experience very little except sore knees. But both activities will be really boring to watch and to do. Most of the interesting things in life are complex.





Mary Kathryn Thompson, Ph.D., is an assistant professor in the Department of Civil and Environmental Engineering, the Korea Advanced Institute of Science and Technology. She can be reached at mkthompson@an.kaist.ac.kr. - Ed.