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WolframInstitute`Hypergraph`
HypergraphTransitionMatrix  |
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The entry for vertices and is proportional to (𝑒-1), summed over the hyperedges containing both and , where is the number of distinct vertices of ; each row is normalized to total 1.
𝑣
𝑤
∑
𝑒
𝑒
𝑣
𝑤
𝑒
𝑒
▪
Equivalently, at each step the walker at picks an incident hyperedge with probability proportional to , then moves to a vertex of chosen uniformly at random, possibly staying at ; diagonal entries are therefore generally nonzero.
𝑣
𝑒
𝑒(𝑒-1)
𝑒
𝑣
▪
Vertex membership is determined from the 0-1 incidence pattern, so repeated occurrences of a vertex within a hyperedge do not increase its weight, while repeated hyperedges count separately.
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The order and symmetry of vertices within hyperedges do not affect the result.
Examples
(5)
Basic Examples
(1)
Compute the transition matrix of a hypergraph:
In[1]:=
tm=
[{{1,2},{2,3},{1,2,3}}]
 | HypergraphTransitionMatrix |
 | Hypergraph |
Out[1]=
SparseArray
 |  |
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Display it as a matrix:
In[2]:=
MatrixForm[tm]
Out[2]=
3 8 | 3 8 | 1 4 |
3 10 | 2 5 | 3 10 |
1 4 | 3 8 | 3 8 |
Every row sums to 1:
In[3]:=
Total[tm,{2}]
Out[3]=
{1,1,1}
Scope
(2)
Applications
(1)
Possible Issues
(1)
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