Bitnets

1
Volumes of Bitnets
http//omega.albany.edu8008/bitnets
Carlos Rodriguez SUNY Albany.
2
Volumes of
3
Meet first bit

• Adam

4
Meet second bit
or
5
CompleteBitnets
t
p
3
3
t
p
2
2
p
Constant Ricci Scalar
1
x
3
x
2
x
1
6
The Line of n

1
2
n
Tough!
Ricci scalar n3
1
2
3
Based on billions of Monte Carlo iterations!
Complete dag so constant curvature!
7
Explode(n) Star
(Naïve Bayes)
X is the probability that the center node is on.
n gt 1
Same vol. as Line of 3!
Volumes increase very fast, n2,3,8
38.2522, 160.231, 698.646, 3121.57, 14178.0,
65157.6, 302107.
Exactly one arrow can be reversed without
changing the total Volume!
Theorem
8
Collapse(n) Star

2
n
n1
1
n d a b lt R gt
2 6 10 1/2 2
3 11 54 3 6
4 20 272 14 16
?
9
Volume matters
M compact, dimension d, volume V
Minimum Description Length
Generalization power of heat kernels
10
( Line(n1), Explode(n) ) v/s Collapse(n)
Explode gt Line
11
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.
Jensens 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!