Calculations with random simplicial complexes -- homological calculations on random simplicial complexes

In what follows we illustrate a collection of homological calculations that can be performed on random simplicial complexes.

Create a random abstract simplicial complex with vertices supported on a subset of $[n] = \{1,...,n\}$.

i1 : K = randomAbstractSimplicialComplex(4)

o1 = AbstractSimplicialComplex{-1 => {{}}                                           }
                               0 => {{1}, {2}, {3}, {4}}
                               1 => {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}
                               2 => {{1, 2, 3}, {1, 2, 4}}

o1 : AbstractSimplicialComplex

i2 : prune HH simplicialChainComplex K

       1       1
o2 = ZZ  <-- ZZ
              
     0       1

o2 : Complex

Create a random simplicial complex on $[n]$ with dimension at most equal to $r$.

i3 : L = randomAbstractSimplicialComplex(6,3)

o3 = AbstractSimplicialComplex{-1 => {{}}                                                                                                                    }
                               0 => {{1}, {2}, {3}, {4}, {5}, {6}}
                               1 => {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}}
                               2 => {{1, 2, 3}, {1, 2, 5}, {1, 2, 6}, {1, 3, 4}, {1, 4, 6}, {2, 3, 6}, {2, 4, 6}, {2, 5, 6}, {3, 4, 5}, {3, 5, 6}, {4, 5, 6}}

o3 : AbstractSimplicialComplex

i4 : prune HH simplicialChainComplex L

       1             1
o4 = ZZ  <-- 0 <-- ZZ
                    
     0       1     2

o4 : Complex

Create the random simplicial complex $Y_d(n,m)$ which has vertex set $[n]$ and complete $(d − 1)$-skeleton, and has exactly m dimension d faces, chosen at random from all $\binom{\binom{n}{d+1}}{m}$ possibilities.

i5 : M = randomAbstractSimplicialComplex(6,3,2)

o5 = AbstractSimplicialComplex{-1 => {{}}                                           }
                               0 => {{2}, {4}, {5}, {6}}
                               1 => {{2, 4}, {2, 5}, {2, 6}, {4, 5}, {4, 6}, {5, 6}}
                               2 => {{2, 4, 5}, {2, 4, 6}, {4, 5, 6}}

o5 : AbstractSimplicialComplex

i6 : prune HH simplicialChainComplex M

       1
o6 = ZZ
      
     0

o6 : Complex

Creates a random subsimplicial complex of a given simplicial complex.

i7 : K = randomAbstractSimplicialComplex(4)

o7 = AbstractSimplicialComplex{-1 => {{}}     }
                               0 => {{1}, {2}}
                               1 => {{1, 2}}

o7 : AbstractSimplicialComplex

i8 : J = randomSubSimplicialComplex(K)

o8 = AbstractSimplicialComplex{-1 => {{}}}
                               0 => {{1}}

o8 : AbstractSimplicialComplex

i9 : inducedSimplicialChainComplexMap(K,J)

           2              1
o9 = 0 : ZZ  <--------- ZZ  : 0
                | 1 |
                | 0 |

o9 : ComplexMap

Calculations with random simplicial complexes -- homological calculations on random simplicial complexes

See also