Download Invariant-length structures

A general framework for complex networks
Mark Changizi
Sloan-Swartz Center for Theoretical Neuroscience
Caltech
Taxonomy
Selected
Behavioral
Non-behavioral
Nervous systems
Ant colonies
Organisms (cell-networks)
Businesses
Electronic circuits
Computer software
Legos
Furniture
Buildings
Non-selected
?
Competitive:
Ecosystems
World-wide web
Economies
Acquaintances
Non-competitive:
Crystals/molecules
Galaxies
Four parts to the talk
(1) Behavioral complexity
(2) Structural complexity
(3) Connectivity
(4) Parcellation
Part 1
Building behaviors
Behaviors are built out of combinations of structures
Structures in the network
Behaviors of the network
1
2
Nodes and edges not shown
3
etc
Examples
structures
behaviors
electronic devices
basic actions
device functions
computer software
instructions
runs
bird vocalization
syllables
songs
mammalian behavior muscle actions
cell
genes
behaviors
cell types
Legos
ecosystems
-----------------
connections
food chains
How is behavioral repertoire
size increased?
1
2
1
1
3
2
2
3
Universal language
approach
Invariant-length
3
approach
4
4
4
5
5
6
?
6
7
7
8
8
structural repertoire versus behavioral repertoire
# muscles types C (n=8)
y = 0.2668x + 2.6997
R2 = 0.5911
Log #
2.5
0.5
-2.0
-1.5
Cetacea
Primates
Insectivora
Rodentia
Chiroptera
Carnivora
Lagomorpha
Artiodactyla
1.5
-1.0
Mean #
ethobehavior types E (n=8)
Log number of syllable types
Primates
Rodentia
Proboscidea
Carnivora
Lagomorpha
Perissodactyla
Didelphimorphia
Artiodactyla
Mammalian behavior
3.5
y = 0.8014x + 2.7017
R2 = 0.8419
Bird vocalization
2.5
1.5
0.5
y = 0.8076x + 0.4157
R2 = 0.7273
-0.5
-0.5
-0.5
0.5
1.5
2.5
Log number of song types
Log index encephalization
Electronic user-interface languages
CD players
Log number of buttons
1.5
1
1
2
1.5
1
1.35
2
1.5
y = 0.4835x + 0.7947
2
R = 0.4896
1
1.25
1.5
1.75 1
Log length of manual
VCRs
2
Genes and cell types
5
Televisions
Human
y = 0.631x + 0.6669
R2 = 0.842
1.25
1.5
1.75
Log length of manual
Calculators
C. Elegans
log # genes
2
4
1.6
1.85
2.1 0
Log length of manual
Drosophila
E. coli
1.5
y = 0.2529x + 1.1603
R2 = 0.5078 1
Yeast
y = 0.114x + 1.4105
R2 = 0.8752
1
2
3
Log length of manual
3
-0.25
0.25
y = 0.3423x + 3.6712
R2 = 0.8408
0.75
1.25
1.75
2.25
log # cell types
None are universal languages. I.e., none are flat.
Instead, behavior length is invariant.
Changizi, 2001, 2002, 2003
Computer software too
Computer software also tends to have invariant length
behaviors, since programs must run within a feasible amount
of time.
Instead of allowing running time to increase, programmers
increase the number of instructions, or lines of
code, in the program. [This is why, for example,
quicksort has more lines of code than bubblesort.]
Part 2
Building structures
Structures are built out of combinations of nodes
...is actually...
Nodes and edges not shown
Edges not shown
Examples
nodes
structures
behaviors
electronic circuits
components
basic functional circuit
computer software
operators
instructions (or lines of code)
businesses
employees
basic functional groups
organisms
cells
basic functional cell combinations
ant colonies
ants
basic functional ant combinations
nervous systems
neurons
basic functional neuron combinations
universities
Legos
ecosystems
faculty
piece
organism
teaching combinations
connections
food chains
-------------------------------------
How is structure repertoire
size increased?
Universal language
approach
Invariant-length
approach
?
# node types versus network size
Legos: Networks of connectable pieces
2
1
y = 0.7089x + 0.2707
R2 = 0.9043
2
3
1
y = 0.0564x + 0.6062
R2 = 0.2489
0
4
Log # Lego pieces
3
log # of physical castes
1
y = 0.5512x - 0.6548
R2 = 0.6952
3
4
0.5
y = 0.1225x - 0.1926
R2 = 0.4827
-0.5
0
5
2
4
2
Log # of component types
1
y = 0.4262x + 0.1379
R2 = 0.7488
2
Log # of components
8
Neocortex: Networks of neurons
1
y = 0.2191x + 1.081
R2 = 0.5204
0
1
6
log colony size
Circuits: Networks of electronic components
0
15
log # cells
log # students (~ log # faculty)
2
10
Ant colonies: Networks of ants
2
0
5
1.5
Universities: Networks of faculty
2
(Also true in
competitive
networks)
1
Log # neuron types
Instead,
structure
length is
invariant.
0
2
0
0
log # departments
(log # faculty types)
None are
universal
languages.
Organisms: Networks of cells
3
log # cell types
Log # Lego piece types
3
3
0
-1.5
-1
-0.5
0
Log index of neuron encephalization
Changiz et al., 2002
Brains thus appear to have invariant length structures
Invariant-length structures:
Minicolumns and modules (below)
# neuron types increases in larger
nervous networks: neocortex and retina
2.5
Neocortex:
# neuron types versus brain size
1
y = 0 .0 8 4 1 x + 0 .8 3 0 5
R 2 = 0 .5 3 3 4
0 .5
0
0 .5
1
1 .5
2
2 .5
Log N
2
y = 0.0137x + 2.3338
R2 = 0.0024
1.5
-0.5
0
0.5
1
1.5
~ number of neurons
2
2.5
Retina:
# neuron types versus brain size
1.5
log # retinal neuron types
~ mean # neurons across
3
Log # neuron types
1 .5
Cortical modules, barrels...:
Number of neurons across versus network size
human
cat
goldfish
1
y = 0.1592x + 0.2311
R2 = 0.8964
0.5
4.5
5
5.5
log # optic fibers
6
6.5
Computer software also has invariant length structures
Invariant-length structures:
Lines of code
5
Lines of code versus program size
# operator types versus program size
4
5
3
y = 1.021x - 0.5733
R2 = 0.9368
1
n=144
1
2
3
4
log N
5
6
3
2
y = 0.3937x + 0.4352
R2 = 0.8415
1
n=185
0
-1
0
log # operators
log # lines of code
7
# operator types increases in
larger programs:
7
0
1
2
3
4
log N
5
6
7
Part 3
Connectivity and network diameter
for behaviors
Keeping structures “close” with edges
Behavior is combinatorial, and
thus the structures must all be “close”.
And this can only be accomplished via edges,
and edges are between nodes.
...is actually...
Edges not shown
Examples
nodes
electronic circuits
edges
components wires
computer software operators
program flow edges
businesses
employees
communication
nervous systems
neurons
axons/dendrites
Legos
ecosystems
piece
organism
links
trophic edges
structures behaviors
How is node-degree
increased?
 invariant

=2
~N1
1
~N1/2
2
Wire cost low,
but diameter too high and thus
behavior increasingly redundant
*****

Behavior not redundant,
but wire too costly
For behavioral networks, expect...
network diameter1/v, for N.
Payoff:  scales up very slowly, saving wire.
Cost: Behaviors are longer (roughly v times longer).
How electronic circuits and neocortex scale
electronic circuits
neocortex
node-degree
network diameter
 ~N0.7
 ~N0.5
 1.3
 2
Electronic circuits
2
N ~ Vgray2/3
Nsyn ~ Vgray1
log N
Neocortex
1
y = 0.5906x + 0.0593
R2 = 0.9382
0
0
0.5
1
1.5
log # edges
2
2.5
3
Some other consequences
of a node-degree increase
• Node density decreases
- neocortex: ~Vgray-1/3.
- circuits
• Wires (and somas) thicken
- neocortex: R~Vgray1/9
- circuits
• White matter disproportionately increases
- neocortex: Vwh~Vgray4/3 [disproportionate due to wire thickening]
Also…
node-degree increases in larger
software
Part 4
Parcellation
The partition problem
Broadly expect that # partitions scales up
disproportionately slowly as network size increases
Theory for neocortex
Well-connectedness
total # area-edges ~ A2
Economical well-connectedness
implies …
# areas ~ N1/2
area degree ~ N1/2
Parcellation also increases disproportionately slowly
in other behavioral networks
Computer software
4
Businesses
1.5
# program modules vs program size
# divisions vs # employees
2
1
y = 0.7028x - 1.3932
R2 = 0.9097
n=77
0
-1
Log # divisions
log # modules
3
1
0.5
y = 0.2122x + 0.1896
R2 = 0.3013
0
0
1
2
3
4
log N
5
6
7
1
2
3
Log # employees
Probably electronic circuits too (partition problem)
n=53
4
Conclusions
Summary
Selected
Behavioral
Non-behavioral
Nervous systems
1. Invariant-length
Ant colonies
structures
Organisms (cell-networks) 2. Invariant-length
Businesses
behaviors
Electronic circuits
3. Invariant network
Computer software
diameter (via slow
Legos
Furniture
Buildings
1. Invariant-length
structures
increase in degree)
4. Parcellation
increases
Non-selected
?
Competitive:
1. Invariant-length
Ecosystems
structures
World-wide web
Economies
Acquaintances
Non-competitive:
Crystals/molecules
Galaxies
short partition(MATRIXELREC edgelist[MATNUM],
int p,
int r)
{
float pivot;
short i, j;
MATRIXELREC temp;
pivot=edgelist[p].weight;
i=p-1;
j=r+1;
The long-term grand goal:
The ability to parse complex networks so
as to reveal their underlying program.
while (i<j)
{
do j--;
while (edgelist[j].weight > pivot);
do i++;
while (edgelist[i].weight < pivot);
if (i<j)
{
temp=edgelist[j];
edgelist[j]=edgelist[i];
edgelist[i]=temp;
}
else return j;
}
}
void quickedgesort(MATRIXELREC edgelist[MATNUM],
int p,
int r)
{
int q;
if (p<r)
{
q=partition(edgelist,p,r);
quickedgesort(edgelist,p,q);
quickedgesort(edgelist,q+1,r);
}
}
Software code,
carved at its joints
?
temp; r} else r return j; while partition } q } quickedgesort(
MATRIXELREC (edgelist[MATNUM], p, int r) { float;
short i, j; temp; pivot=edgelist[p].weight; j i=p-1; j=r+1;
int = (i<j) { do j--; quickedgesort edgelist pivot i while
MATRIXELREC (edgelist[j].weight >); void do; while
(edgelist[i].weight < pivot); great if (i<) pivot r {temp=
edgelist[j]; =edgelist[i]; ++ edgelist[i]=<) { q partition(,
p,r); short (edgelist,p,); (edgelist,q+,); }} quickedgesort 1
edgelist[MATNUM], int p, MATRIXELREC edgelist[j] int
r) { int q; if (p
Same software code,
but with nodes scrambled