1. Volumes of Bitnets http://omega.albany.edu:8008/bitnets
Carlos Rodriguez SUNY Albany.

2. Volumes of

3. Meet first bit
Adam 1

4. Meet second bit 2 1 or 2 1

5. Complete Bitnets Simplex P Cube T Sphere S t p t p t p x x x 3 32 p 2 t 1 1 2
Constant Ricci Scalar 1 x 3 x 2 x
Sphere S 3 + 1

6. The Line of n … Tough! Ricci scalar n=31 2 … n
Tough!
Ricci scalar n=3 1 2 1 2 3
Based on billions of Monte Carlo iterations!
Complete dag so constant curvature!

7. X is the probability that the center node is on.
Explode(n) Star
(Naïve Bayes)
n+1 1 2 3 n
X is the probability that the center node is on.
n > 1
Same vol. as Line of 3!
Volumes increase very fast, n=2,3,…8
, , , , , ,
Theorem:
Exactly one arrow can be reversed without changing the total Volume!

8. Collapse(n) Star ? n d a b < R > 2 6 10 3 11 54 4 20 14 16 1/2… n
n+1 1 n d a b
< R > 2 6 10
1/2 3 11 54 4 20
272 14 16 ?

9. Volume matters M compact, dimension d, volume V
Minimum Description Length
Generalization power of heat kernels

10. ( Line(n+1), Explode(n) ) v/s Collapse(n)
Explode > Line

11. B = symmetric beta function equal to a ratio of gammas
A Fast Approx. Alg.
The vol. of a dag of n nodes of dim. d is:
Where:
, p(j) is the prob that the parents of node i show jth pattern.
Jensen’s inequality applied twice in two different ways shows:
From where upper and lower bounds for the volume are obtained as:
ai and bi are very simple sums that depend on the local topology of the dag.
B = symmetric beta function equal to a ratio of gammas
The approximations are exact for total disconnected and for complete dags.
The sqrt(UL) is remarkably accurate as an estimate of Z… and it works!