Hypercubes and

Neural Networks

bill wolfe

9/21/2005

Modeling

neuron

wi

Net = Swi ai

ai

“activation level”

“Net Input”

0 <= ai <= 1

Saturation

daj/dt = Netj (1-aj)(aj)

Dynamics

a1 a2

a3

w12

w23w13

3 Neuron Example

000

111

100110

010

001 011

a1

a2

a3

Brain State: <a1, a2, a3>

000

111

100110

010

001 011

a1

a2

a3

“Thinking”

Binary Model

aj = 0 or 1

Neurons are either “on” or “off”

Binary Stability

aj = 1 and Netj >=0

Or

aj = 0 and Netj <=0

Hypercubes

n=0

n=1

n=2

4-Cube

4-Cube

5-Cube

5-Cube

5-Cube

http://www1.tip.nl/~t515027/hypercube.html

Hypercube Computer Game

00

01 11

100

1

2

3

2-Cube

0123

0 1 2 3

0110

0110

1001

1001

Q2 =Adjacency Matrix:

Hypercube Graph

=

I

QQ

nn

1

1nQ

I

Recursive Definition

Theorem 1: If v is an eigenvector of Qn-1 with eigenvalue x then the concatenated vectors [v,v] and [v,-v] are eigenvectors of Qn with eigenvalues x+1 and x-1 respectively.

Eigenvectors of the Adjacency Matrix

v

vQn =

I

Qn 1

1nQ

I

=

=

=

v

v

vv

vv

vQv

vvQ

v

v

n

n)1(

1

1

Proof

=

= +1

11

= -1

1-1

= +2

1111

= 0

11-1-1

= 0

1-11-1

= -2

1-1-11

= +3

11111111

= +1

1111-1-1-1-1

= +1

11-1-111-1-1

= -1

11-1-1-1-111

= +1

1-11-11-11-1

= -1

1-11-1-11-11

= -1

1-1-111-1-11

= -3

1-1-11-111-1

+1

+1

+1 +1 +1 +1

+1

-1

-1

-1-1

-1

-1 -1

n=0

n=1

n=2

n=3

Generating Eigenvectors and Eigenvalues

Walsh Functions for n=1, 2, 3

1

1

1

1

-1

-1

-1

-1

000

001

010

011

100

101

110

111

eigenvector binary number

000

100 110

010

111000

001 011

x

y

z

x

==

=-1k=1

=-1k=1

=+1k=2

=+1k=2

=+1k=2

=-3k=0

=-1k=1

y

z n=3

n=3

Theorem 3: Let k be the number of +1 choices in the recursive construction of the eigenvectors of the n-cube. Then for k not equal to n each Walsh state has 2n-k-1 non adjacent subcubes of dimension k that are labeled +1 on their vertices, and 2n-k-1 non adjacent subcubes of dimension k that are labeled -1 on their vertices. If k = n then all the vertices are labeled +1. (Note: Here, "non adjacent" means the subcubes do not share any edges or vertices and there are no edges between the subcubes).

n=5k=3

reduced graph

n=5k=2

reduced graph

Schamtice of the 5-cube Schamtice of the 5-cube

n=5, k= 3 n=5, k= 2

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1 -1

Inhibitory Hypercube

Theorem 5: Each Walsh state with positive, zero, or negative eigenvalue is an unstable, weakly stable, or strongly stable state of the inhibitory hypercube network, respectively.