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The algorithm described above runs in approximately exponential ( O ( n ! ) ) time, which is to be expected since it solves an NP-hard problem.

A clique for which k = 4

The enumeration of k-cores reduces to the NP complete clique problem [link] as follows:

  • A clique is a k-core for which the number of vertices in the core is equal to k + 1 ( [link] ).
  • The clique problem is: determine whether an arbitrary graph contains a clique of at least size n .
  • If we could accomplish k-core enumeration in polynomial time, say O ( n k ) , where n is the number of nodes in the graph, then:
    • We could find cores of all k in, at most, O ( n × n k ) = O ( n k + 1 ) time, because there is no k core in a given graph for which k exceeds n .
    • Without increasing fundamental run-time, we could flag each of those k-cores which is a clique. In doing so, we find all cliques.
    • By simply finding the largest of this group of cliques, we have solved the clique problem in polynomial time.

Consequently, k-core enumeration belongs to the class of NP-hard problems, meaning that it is not clear whether an algorithm that runs significantly faster than exponential time can be devised.

That is not to say, however, that the algorithm cannot be improved upon at all. For instance, one of our implementations takes advantage of the fact that, if we form an induced subgraph H by removing some vertex v from a graph G , if v is not in a k-core, then the graph I resulting from removing any of v 's neighbors that are not in a k-core from G will have the same maximum k-core as H . Also, our algorithm takes care not to revisit previously enumerated branches. Different approaches can certainly speed up an algorithm to solve the problem, but it is not clear that any approach will make the algorithm run in sub-exponential time.

Assembly enumeration

k-Assembly enumeration on a reduced set of k-cores.

Upon enumerating all k-cores, every unique closure of a k-core is analyzed to determine whether any one of the k-cores that closes to that closure is tight ( [link] ). Checking tightness involves nothing more clever than simply verifying the above definition of tight. If at least one of the cores that closes to a given closure is tight, then that closure is a collected into the set of k-assemblies, otherwise, it is ignored.


After devising an algorithm to find every k-assembly, we attempted to discover attributes about k-assemblies through, first, generating random graphs, and then, enumerating the assemblies contained in those graphs.

Random graph generation

We employed two primary types of random graphs.

A Bernoulli random graph
A scale-free Cooper-Frieze random graph
  1. The classical Bernoulli random graph ( [link] ):
    • Pick some probability p , and a number of vertices, n . Create a graph G , for which V ( G ) = 1 , 2 , . . . n .
    • For all pairs of vertices, ( i , j ) , where i j and 0 < i , j n , an edge i j E ( G ) with probability p .
  2. The scale-free Cooper-Frieze random graph ( [link] ) [link] . As we implement it:
    • pick some positive integer T ; α , β , γ , δ [ 0 , 1 ] ; P , Q , which are 1-indexed lists. P = ( p 1 , p 2 , . . . , p n ) , where i = 1 n p = 1 , and p i [ 0 , 1 ] i { 1 , 2 , . . . , n } . Q = ( q 1 , q 2 , . . . , q m ) , where i = 1 m q = 1 , and p i [ 0 , 1 ] i { 1 , 2 , . . . , m } .
    • Begin with a graph G , where V ( G ) = { 1 } , and E ( G ) = { 11 } . (That is, G is a graph with a single vertex which has a single edge connected to itself).
    • t { 0 , 1 , 2 , . . . , T } :
      • Do the "Old" procedure with probability α , otherwise, do the procedure "New."
      • Do the procedure "Add Edges."
    • Old:
      • with probability δ , choose the vertex start from among the set of vertices in V ( G ) randomly, giving each vertex an even chance. Otherwise, choose start with the probability for each vertex proportional to the the degree of that vertex with respect to G.
      • with probability γ set the boolean variable terminateUniformly to true . Otherwise, set the variable to false .
      • choose an index of Q so that the index i has a probability q i of being chosen. Set the integer variable numberOfEdges to this chosen index.
    • New:
      • Add a new vertex v to V ( G ) . Call that vertex start .
      • With probability β set the boolean variable terminateUniformly to true . Otherwise, set the variable to false .
      • choose an index of P so that the index i has a probability p i of being chosen. Set the integer variable numberOfEdges to this chosen index.
    • Add Edges:
      • Create the set END by choosing numberOfEdges vertices from G . The vertices are chosen randomly, either with uniform probability if terminateUniformly is true , or in proportion to degree otherwise.
      • vertices e END , add to G an edge directed from start to e .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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.
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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