This summer, I am in a reading group working through the book Complex Adaptive Systems. I don’t have the book yet but a PDF of the first two chapters and I thought I would post my thoughts here before I go to the talk this afternoon.

I was originally a big fan of complex systems. I took a few courses at Michigan on complex systems and I’m very familiar with statistical techniques for running simulations.  I even have a sample NetLogo widget on my website somewhere of a theoretical model from my adviser.  Running simulations is kind of exciting and sexy – you usually see graphics of how agents in a system interact and plot out the results.  Then, using stats and inference you can make awesome claims about how the system works and try to infer how this relates to things in the real universe.  This is cool and fun and what made me interested in going into academia.

But then I started my PhD program in the theory department at Northwestern. In theory, we try to make much grander claims: we want to show that something is true or false in all cases or for wide ranges of classes, not just a specific instance.  For example, we want to show certain classes of algorithms can be computed in a certain amount of time. As long as we can show an algorithm is in that class of problems, then we instantly know something about it.  Theoretical findings are powerful but hard to reach, they require extensive mathematical proofs and a great deal of intellectual and logical rigor.

Simulation is useful when you’re using it to compare a theoretical finding against some form of empirical data.   In other words, you’re given some actual real world data points.  You can simulate the theoretical models and show that the data fits your theory.   Verifying theory is important however I contend it is not as powerful of a finding as the theoretical result itself.

Where do complex systems come into this? The authors of the aforementioned book argue that some systems are too complex: we can’t break them down into their components. Therefore, this implies that they argue that theoretical results cannot be found for these systems! Perhaps the systems are too complex; there are too many of what accountants call “intangibles”, things that have value but cannot be counted as assets.  I am not yet convinced of this line of argument; theoreticians should seek to develop mathematical models and tools precisely to model these cases, such that we can develop a better understanding of how our world is classified.

Yesterday, I finished my semester project for my randomized algorithms course.  I started the project almost a month in advance since I knew there was going to be a significant amount of research and reading to do. The original plan was to create a mathematica module for processing Exponential Random Graph simulations.  The concept is this:

You have some observed social network, collected from data or in the field.  You want to know things about the relationships of people in the network, like, how likely are people to form connections randomly or do they form connections based on other sociological things. For example, if Alice is connected to both Bob and Eve, is there likely going to be a relationship between Bob and Eve? In other words, do they complete the triangle?  Standard random graph models can’t test for this but we can use exponential random graphs. The output of the algorithm is a set of values that indicate how strong various network structures are.

My presentation went alright. In the mathematical sciences (engineering, math, etc) proofs are the only method you can use to show something is true. In the applied sciences (communications, sociology, statistics) the only method you can use to prove something is statistical inference. So, explaining inference to engineers is difficult since they don’t encounter it (I think they should!). Explaining math to social scientists is challenging since they’re not familiar with proof techniques (what is the contrapositive again?)

I haven’t finished the paper yet (almost done) and I will post the results here shortly. In the mean time, I’ve compiled some of my thoughts about approaching this topic in the future. This is what I want to study (using computer science theory in other fields) so I am noting this for posterity.

• Use more visualizations to explain inference.  Mathematicians love proofs and it is ok to use math on your slides. However, when talking about statistical inference, you’re looking at how something observed fits something hypothesized.  The best way to do this, I suspect, is to plaster a N(0,1) curve on the slides and point to where things fit.
• For social networks stuff, use examples! I used a couple examples in my slides and people found it helpful and interesting.  There are so many great visualization tools for social networks, so I should use them more.
• Take a course on econometrics.  I’m doing this next year.  Econometrics is using statistical inference to reach economic conclusions. There has to be some good techniques they use.
• Write the slides AFTER you write the paper. In this case, I was so worried about the presentation, I did it before I wrote the paper and ended up rushing the paper.  Next time, I’ll flip it and spend time worrying more about visualizations and teaching people than plastering equations on slides.