A general framework for complex networks Mark Changizi Sloan-Swartz Center for Theoretical Neuroscience Caltech

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A general framework for complex networksMark
ChangiziSloan-Swartz Center for Theoretical
NeuroscienceCaltech
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Taxonomy
?
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Four parts to the talk

1. Behavioral complexity
2. Structural complexity
3. Connectivity
4. Parcellation

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Part 1 Building behaviors
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Behaviors are built out of combinations of
structures
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Examples
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How is behavioral repertoire size increased?
Universal language approach
Invariant-length approach
?
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structural repertoire versus behavioral repertoire
None are universal languages. I.e., none are
flat. Instead, behavior length is invariant.
Changizi, 2001, 2002, 2003
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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.
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Part 2 Building structures
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Structures are built out of combinations of nodes
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Examples
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How is structure repertoire size increased?
Universal language approach
Invariant-length approach
?
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node types versus network size
None are universal languages. Instead, structure
length is invariant. (Also true in
competitivenetworks)
Changiz et al., 2002
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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
Cortical modules, barrels... Number of neurons
across versus network size
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Computer software also has invariant length
structures
Invariant-length structures Lines of code
operator types increases in larger programs
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Part 3 Connectivity and network diameter for
behaviors
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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.
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Examples
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How is node-degree increased?
?2
? invariant ???
?N1 ??1
?N1/2 ??2
? 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).
Behavior not redundant, but wire too costly
Wire cost low, but diameter too high and
thus behavior increasingly redundant
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How electronic circuits and neocortex scale
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Some other consequences of a node-degree increase

• Node density decreases
• - neocortex ?Vgray-1/3.
• - circuits
• Wires (and somas) thicken
• - neocortex RVgray1/9
• - circuits
• White matter disproportionately increases
• - neocortex VwhVgray4/3 disproportionate due
  to wire thickening

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Also node-degree increases in larger software
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Part 4 Parcellation
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The partition problem Broadly expect that
partitions scales up disproportionately slowly
as network size increases
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Theory for neocortex
Well-connectedness
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Economical well-connectedness implies
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Parcellation also increases disproportionately
slowly in other behavioral networks
Probably electronic circuits too (partition
problem)
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Conclusions
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The long-term grand goal The ability to parse
complex networks so as to reveal their underlying
program.