A week from now, we will be commemorating the fifth anniversary of

9/11. I can still remember that morning. As I was groggily waking up in my apartment in Cambridge, Massachusetts, I was shocked to hear on my alarm clock radio that the World Trade Center had been hit by two airliners. (It shook me up even more because I had seriously thought about flying to California around that time ... so it was possible that I might have found myself on one of those planes from Boston's Logan Airport to the West Coast.) So, I think it's appropriate to write this brief blog posting on how

math is being used to help in the fight against

terrorism.

The best news article I've read on how

mathematics is being employed in the war on terrorism is

Wired News: Can Math Help in Terror War?. The article mentions the two ways that mathematicians (and I will include computer scientists, economists, mathematical psychologists, et al., in that label for the purposes of this blog post) take in applying mathematics to counter-terrorism.

The first path mathematicians take in analyzing terrorism is

game theory and/or similar but alternative approaches that stem from applications in the social sciences (economics, psychology, etc.). One 'similar but alternative' approach is

reflexive theory (a topic I have wroted about previously). The basic theme common to this branch of thinking is the application of mathematics to the strategic thinking of terrorists.

The second road taken by applied mathematicians in the

War on Terrorism is based more on the organizational structure of terrorism (obviously, there is some overlap between strategy and organization). The methodologies approached to getting a clearer picture of the organizational structure of terrorists groups, like Al Qaeda, include

Social Network Analysis and

Formal Concept Analysis. These and other approaches basically boils down to drawing a 'map' of of the structure of terrorist group and/or its cells based on available data.

The assumption that underlies the attempts to uncover the organizational structure of terrorist groups is that terrorist groups are organized as, what mathematicians call, a

partially ordered set (or a

poset) as opposed to a

totally ordered set (without getting into the technical details, think trees and/or lattices when thinking of partially ordered sets (or posets), and think of a straight line when thinking of totally ordered sets). The advantages of trying to map the partially ordered structure of terrorists groups is that: (a) it is more realistic to think there are overlapping ('horizontal') links between terrorists interspersed within any top-down ('vertical') hierarchy, and (b) this realism would, hopefully, help us to pinpoint terrorists (and/or their cells) and what importance they have within the overall organizational structure.

There are problems with taking either of the two paths -- analyzing terrorist strategy or their organization -- to battling terrorism via math. [Some of these problems are highlighted in a New York Times op-ed piece by mathematician, Jonathan Farley,

The N.S.A.'s Math Problem (May 16, 2006).]

One of the problems with looking at terrorist strategy is that terrorists adapt their strategies as conditions change. For example, when it became more difficult to bring in knives and box cutters on airplanes, Al Qaeda came up with the strategy to bring hard-to-detect liquid explosives onto aircraft. Another problem with trying to unravel terrorist strategy is that strategy almost always (except for highly idealized games like chess) involves a random (or stochastic) component. In trying to unmask the strategy of terrorists, think poker instead of chess.

Trying to uncover the structure of terrorist organizations also has limitations. For example, using the methods that are currently available to us, one person who might be mis-identified (due to all of the empirical 'links' between this person and known terrorists) as a 'terrorist' would be Secretary of State Condeleeza Rice!

To be fair, the

Center for Computational Analysis of Social and Organizational Systems (CASOS) at Carnegie Mellon has succeeded in identifying the changing leadership of Hamas in the past. So, better data and better methods would probably lead to a more robust method of mapping terrorist networks, and hopefully, preventing another tragedy that woke all of us up to the dangers of terrorism on that fateful morning five years ago.