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We seek to understand Palm's definition of cell assembly in context. In particular, we seek to determine whether a brain-sized random graph can contain a realistic number of assemblies by Palm's definition.

In this report, we describe the process of k-assembly enumeration and explain some preliminary experimentation using that algorithm.

The report exists in two forms. The abridged version includes the material described above, while the full version goes on to include more trend examples, an extensive collection of visual examples of k-Assemblies, and implementation.

This is the abridged version. The full version is available here .

Finding k-assemblies

Finding k-assemblies takes place in two steps: k-core enumeration and k-assembly confirmation:

K-core enumeration

The process of k-Core enumeration follows the general form of a branch and bound algorithm. That is, it follows the process:

  1. If possible, solve the problem, otherwise:
  2. Break the problem into several smaller problems, for each of these smaller problems, go to step 1.

Applied to our problem, the general algorithm looks like this, given some input graph G ( [link] ):

An exhaustive 2-core enumeration: Ultimately, every induced subgraph is enumerated except those that are not 2-cores.
The maximum k-core algorithm for a 3-core: 1,2) A vertex is eliminated from the graph, leaving an induced subgraph which is not a minimal 3-core. 3,4) All vertices which have insufficient degree with respect to the new subgraph graph (less than 3 in this case) are eliminated. 5,6) Again, a vertex without sufficient degree is eliminated. The subgraph pictured in step 6 has no vertices of less than k degree, so the process is completed.
  1. If G is a minimal k-core, stop, otherwise:
  2. Dissect G as follows:
    1. Find the maximum k-core that is an induced subgraph of G , call it H ( [link] ).
    2. For each vertex v V ( H ) , go to step 1, using for the new G and induced subgraph of H such that V ( G ) = V ( H ) v .

This process finds all k-cores in the original G . To sketch a proof:

  • An algorithm that finds all induced subgraphs in G and then filters k-cores from the rest will trivially enumerate all induced k-cores in G .
  • The k-core enumeration algorithm described is equivalent to such an algorithm. By in turn eliminating every vertex from a graph, the algorithm find all induced subgraphs except for those it skips. The skipped subgraphs will never generate k-cores not already generated:
    • Skipped subgraphs have at least one vertex of degree less than k , with respect to that subgraph; call this vertex set W 1 . The subgraph may contain vertices that would have degrees less than k if excluding the vertices in W 1 ; call this set W 2 . There may also be vertices dependent on the vertices in W 1 and W 2 to maintain a degree of k ; call those W 3 . We can continue forming these sets until G ( W 1 W 2 . . . W n ) is a k-core for some positive integer n . Define W as ( W 1 V 2 . . . W n )
    • Trivially, take some induced subgraph of G called g . v V ( g ) , D ( v , g ) D ( v , G )
    • Consequently, no induced subgraph of G will contain a k-core including any vertices in W , since no vertex in W has gained degree upon finding an induced subgraph, and, consequently, the collective W still cannot meet the degree threshold to be included in a k-core.
    • Subgraphs are skipped iff they contain vertices in W .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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