[FoRK] Correction / expansion, was Re: Scary study on how lack of IPOs is harming US economy

Jeff Bone jbone at place.org
Wed Nov 11 04:36:26 PST 2009


On Nov 10, 2009, at 6:16 PM, Jeff Bone wrote:

> To see the above merely observe that the number of interconnections  
> in a fully-connected network is n^2 in the number of nodes in the  
> network.

Before somebody (rightfully) beats me up for either sloppy language or  
math (or both) let me correct this statement.

We're looking for something like the big-O complexity of such a  
network of interacting economic entities / agents.  The number of  
bidirectional edges in an arbitrary fully-connected network is of  
course quadratic, i.e. (n^2 - n) / 2.  In this case, though, we want  
the number of unidirectional edges, i.e., we're interested in the  
number of potential  interactions (i.e. transactions, messages or  
"flows" between entities in the graph, i.e. initiated by any entity or  
agent) hence we want the number of possible unidirectional edges ---  
so that is n^2 - n, or ~n^2.  Close enough for horseshoes and hand  
grenades, as they say.

You could argue, though I won't, that n^2 is *actually* precisely  
correct in the case of the specific example we're talking about, as it  
contemplates node self-connecting edges, and this models self-dealing  
over time --- which is precisely what e.g. saving in a depository  
institute is, more or less.  It actually involves a third party, so to  
model this we actually need at least two kinds of edges, one for the  
initial deposit, one for the withdrawal, and potentially others for  
both the interest-payment activity and the subsequent reinvestment or  
loaning activity.  Things get more complex still with fractional  
reserve banking, and still more complicated through a layered central  
bank and hub system, and at this point we lose the original spirit of  
the back-of-the-envelope in minutia that aren't really that interesting.

The more pertinent criticism of the back of the envelope SWAG here is  
an argument about whether fully-connected networks are a realistic  
model of any real economy.  In fact, for the various reasons I  
mentioned earlier (i.e., asymmetries, etc. in realistic, non-efficient  
markets) this is clearly not the case for a real economy, and indeed  
the actual internal topology is likely to be layered and absolutely  
essential to any high-fidelity model of the functioning of the  
economy.  And in fact, the very rules that Ken and Bill were talking  
about are one such mechanism of layering or forcing various kinds of  
internal topology onto the network in question.

The point, though, remains valid:  it's dangerous to extrapolate  
between phenomena involving same sets and experimental setups with  
substantial log-scale differences in various relevant and interesting  
dimensions.

jb



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