# [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

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

```