I Under-appreciated the Kronecker Product...

Today, I came to the realization that I massively under-appreciated the Kronecker product of two matrices.  Until today, I thought that it wasn't all that useful...boy was I wrong!  Kronecker products are wonderfully useful, particularly if you are interesting in networks.  Here are links to a some papers on using Kronecker products as a method for generating network graphs that have power law degree distributions and small diameter.  I will write additional posts on this topic later, as I expect that it will be extremely useful in my own research...

Online Lecture Series on Differential Equations...

MIT OCW saves the day again!  A whole course on solving ODE's from MIT's Arthur Mattuck...

Excellent PhD Thesis Lecture on Dynamics of Large Networks...

I must say that I am a little bit jealous...if my PhD thesis and defense are half as good as the one presented here then I will be happy...

A major finding of his research is that networks tend to become more dense over time and that this densification process follows a power law.  My network version of Minsky's Financial Instability Hypothesis would postulate some type of limit cycle behavior (either stable of chaotic) of network density.  Now I am not convinced that his empirical work contradicts my idea for the following reason.  His data come from LinkedIn, MSN, and other large online data-sets.  It does not seem like there would be strong incentives to break links in these types of networks given that the cost of continuing to maintain the link is zero.  However, in my world where the links represent financial inter-dependencies of agents, there can be very strong incentives to break links as the cost of maintaining a link could conceivably be quite high...    

First Cooked Breakfast of the Year...

Today being Sunday, I took myself out to a nice full breakfast of sausage, bacon, haggis, eggs, potato pancakes, mushrooms, and crusty bread.  Sounds like a lot of food...and it is, but it will probably be the only meal I eat today (at least this is how I justify it!).

I ate my breakfast at a place called Always Sunday on High Street.  It was a bit expensive (10 GBP) for a full breakfast and OJ, and even though the food was good I am convinced that there are better breakfasts in Edinburgh for less money...

I will file another report next week.  If anyone has any suggestions on where to get a good cooked breakfast in Edinburgh please pass them along...

Modeling Financial Instability...

Most intriguing paper by Steve Keen on Financial and Economic Breakdown: Modeling Minsky's Financial Instability Hypothesis.  Using Goodwin's limit cycle model as his foundation, Keen extends the model to account for the four keys to Minsky's idea:

1. Tendency of capitalists to incur debt on the basis of euphoric expectations
2. The importance of long-term debt
3. The destabilizing impact of income inequality
4. The stabilizing role of government

Inclusion of Minsky's ideas converts Goodwin's stable but cyclical system into a chaotic one with the possibility of a divergent breakdown (i.e., depression)...

My interest in this paper is that it is the first attempt that I have come across so far that presents a mathematical model of Minsky's Financial Instability Hypothesis.  I would like to try to somehow incorporate these ideas into a network model...here is a link to a paper by Gallegati et al that uses an agent-based approach to model some of these ideas.  

Another Brilliant Linear Algebra Lecture...

Prof. Gilbert Strang at MIT delivers a beautiful lecture on how to solve systems of first order differential equations using linear algebra...an absolute must for anyone studying graduate economics...

The America I Know...

I typically avoid writing posts on politics (I hope!) but...

As I reflect today about the how the events of September 11th 2001 touched my life, and in a much more tragic way, the lives of my friends, I find myself distracted by current events: the embarrassing debate over the so-called "Ground Zero Mosque" and the dangerously insensitive Qu'ran burning event that has garnered massive news coverage...

These two events paint a picture of an America that is intolerant at best, and bigoted at worst. This is not the America that I know, and I suspect that this is not the America that those of you who have had the good fortune to travel in and around the U.S. know.

With this in mind, I hope that you will read this story on CNN about two Muslim friends who traveled around the U.S. on a 30 Mosques in 30 States in 30 days for Ramadan...the America that they experienced on their journey is the America that I know and love...