Wolfram Function Repository
Instant-use add-on functions for the Wolfram Language
Function Repository Resource:
Adjacency
Compute the hypergraph with a specified adjacency tensor
Contributed by:
Jonathan Gorard
ResourceFunction["AdjacencyHypergraph"][atens] gives the (ordered or orderless) hypergraph with adjacency tensor atens. |
Details and Options
An adjacency tensor is a generalization of the concept of an adjacency matrix from graphs to hypergraphs, in which hyperedges may be of arbitrary arity.
The arity of the hyperedges in the hypergraph is equal to the rank of the adjacency tensor.
The number of vertices will be equal to n, where the dimensions of the adjacency tensor are n×n×…×n.
ResourceFunction["AdjacencyHypergraph"] accepts adjacency tensors specified both as SparseArray objects, and as ordinary nested lists.
The number of hyperedges of the form {vi,vj,…,vk} is specified by the entry aij…k of the adjacency tensor.
The number of self-loops for the vertex vi is given by the diagonal entry aii…i.
ResourceFunction["AdjacencyHypergraph"] has the following option:
| "OrderedHyperedges" | Automatic | whether to treat hyperedges as being ordered (directed) |
The following settings for "OrderedHyperedges" can be used in ResourceFunction["AdjacencyHypergraph"]:
| Automatic | construct an orderless hypergraph if and only if the adjacency tensor is symmetric across all indices |
| True | construct an ordered hypergraph |
| False | construct an orderless hypergraph |
When the adjacency tensor is of rank 2, the output of ResourceFunction["AdjacencyHypergraph"] is identical to that of AdjacencyGraph.
Examples
Basic Examples (3)
Construct an ordered hypergraph automatically from an asymmetric adjacency tensor:
| In[1]:= |
![ResourceFunction[
"AdjacencyHypergraph"][{{{0, 0, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {1, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}}]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/4e1023e93f1f721a.png)
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| Out[1]= |
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Construct an orderless hypergraph automatically from a symmetric adjacency tensor:
| In[2]:= |
![ResourceFunction[
"AdjacencyHypergraph"][{{{0, 0, 0, 0}, {0, 0, 1, 0}, {0, 1, 0, 1}, {0, 0, 1, 0}}, {{0, 0, 1, 0}, {0, 0, 0, 0}, {1, 0, 0, 1}, {0, 0, 1, 0}}, {{0, 1, 0, 1}, {1, 0, 0, 1}, {0, 0, 0, 0}, {1, 1, 0, 0}}, {{0, 0, 1, 0}, {0, 0, 1, 0}, {1, 1, 0, 0}, {0, 0, 0, 0}}}]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/4c78adf05092e90a.png)
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| Out[2]= |
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Treat the hypergraph as being ordered instead:
| In[3]:= |
![ResourceFunction[
"AdjacencyHypergraph"][{{{0, 0, 0, 0}, {0, 0, 1, 0}, {0, 1, 0, 1}, {0, 0, 1, 0}}, {{0, 0, 1, 0}, {0, 0, 0, 0}, {1, 0, 0, 1}, {0, 0, 1, 0}}, {{0, 1, 0, 1}, {1, 0, 0, 1}, {0, 0, 0, 0}, {1, 1, 0, 0}}, {{0, 0, 1, 0}, {0, 0, 1, 0}, {1, 1, 0, 0}, {0, 0, 0, 0}}}, "OrderedHyperedges" -> True]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/2f5a10ad98cae498.png)
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| Out[3]= |
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Construct an ordered hypergraph of arity 5 automatically from an asymmetric adjacency tensor specified as a SparseArray:
| In[4]:= |
![ResourceFunction["AdjacencyHypergraph"][SparseArray[
Automatic, {10, 10, 10, 10, 10}, 0, {
1, {{0, 24, 96, 168, 240, 288, 336, 408, 432, 456, 480}, CompressedData["
1:eJwllNGh3DAIBG1AYLuLtJQSXgNp+ZWSneXn7haGPQkJ/fn59/cnruv6va+r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"]}, CompressedData["
1:eJxTTMoPSmJkYGB4ACQYR8GwBgCZcgRp
"]}]]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/38142ad820f31a10.png){kind=small}
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Scope (7)
AdjacencyHypergraph accepts both SparseArray and nested list specifications of adjacency tensors:
| In[5]:= |
![ResourceFunction["AdjacencyHypergraph"][SparseArray[
Automatic, {4, 4, 4}, 0, {
1, {{0, 4, 8, 14, 18}, {{2, 3}, {3, 2}, {3, 4}, {4, 3}, {1, 3}, {3, 1}, {3, 4}, {4, 3}, {1, 2}, {2, 1}, {1, 4}, {4, 1}, {2, 4}, {4, 2}, {1, 3}, {3, 1}, {2, 3}, {3, 2}}}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}]]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/1157b52e1a5aa70a.png){kind=small}
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| In[6]:= |
![ResourceFunction[
"AdjacencyHypergraph"][{{{0, 0, 0, 0}, {0, 0, 1, 0}, {0, 1, 0, 1}, {0, 0, 1, 0}}, {{0, 0, 1, 0}, {0, 0, 0, 0}, {1, 0, 0, 1}, {0, 0, 1, 0}}, {{0, 1, 0, 1}, {1, 0, 0, 1}, {0, 0, 0, 0}, {1, 1, 0, 0}}, {{0, 0, 1, 0}, {0, 0, 1, 0}, {1, 1, 0, 0}, {0, 0, 0, 0}}}]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/1fac74d7bf1562da.png)
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AdjacencyHypergraph supports multihypergraphs, with adjacency tensor entries representing hyperedge multiplicities:
| In[7]:= |
![ResourceFunction[
"AdjacencyHypergraph"][{{{0, 0, 0, 0}, {0, 0, 3, 2}, {0, 3, 0, 1}, {0, 2, 1, 0}}, {{0, 0, 3, 2}, {0, 0, 0, 0}, {3, 0, 0, 0}, {2, 0, 0, 0}}, {{0, 3, 0, 1}, {3, 0, 0, 0}, {0, 0, 0, 0}, {1, 0, 0, 0}}, {{0, 2, 1, 0}, {2, 0, 0, 0}, {1, 0, 0, 0}, {0, 0, 0, 0}}}]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/5256431db60a7600.png)
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| Out[7]= |
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When the rank of the adjacency tensor is equal to 2, the output of AdjacencyHypergraph is identical to the output of AdjacencyGraph:
| In[8]:= |
![hypergraph = ResourceFunction[
"AdjacencyHypergraph"][{{0, 1, 1, 0}, {1, 0, 1, 1}, {1, 1, 0, 1}, {0, 1, 1, 0}}]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/1198404d29968bfc.png){kind=small}
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| Out[8]= |
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| In[9]:= |
![graph = List @@@ EdgeList[AdjacencyGraph[{{0, 1, 1, 0}, {1, 0, 1, 1}, {1, 1, 0, 1}, {0, 1, 1, 0}}]]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/69868637301f1e41.png){kind=small}
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| Out[9]= |
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| In[10]:= |
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| Out[10]= |
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When the adjacency tensor is symmetric across all indices, the hypergraph is automatically orderless:
| In[11]:= |
![hypergraph1 = ResourceFunction[
"AdjacencyHypergraph"][{{{0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0,
1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0}}, {{0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {1, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0,
0}}, {{0, 1, 0, 0, 0, 0}, {1, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 1, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 1, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 1, 0}}, {{0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 0, 0}, {1, 0, 0, 1, 0, 0}}, {{0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0}, {1, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0}}}]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/57a121a234a0fa80.png)
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| Out[11]= |
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| In[12]:= |
![hypergraph2 = ResourceFunction[
"AdjacencyHypergraph"][{{{0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0,
1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0}}, {{0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {1, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0,
0}}, {{0, 1, 0, 0, 0, 0}, {1, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 1, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 1, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 1, 0}}, {{0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 0, 0}, {1, 0, 0, 1, 0, 0}}, {{0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0}, {1, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0}}},
"OrderedHyperedges" -> False]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/635f295a01e71253.png)
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| Out[12]= |
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| In[13]:= |
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| Out[13]= |
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When the adjacency tensor is asymmetric across any pair of indices, the hypergraph is automatically ordered:
| In[14]:= |
![hypergraph1 = ResourceFunction[
"AdjacencyHypergraph"][{{{0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0,
0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0,
0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}}]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/0bbfb8e41921b88e.png)
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| Out[14]= |
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| In[15]:= |
![hypergraph2 = ResourceFunction[
"AdjacencyHypergraph"][{{{0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0,
0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0,
0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}},
"OrderedHyperedges" -> True]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/049d855d38c1f3d3.png)
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| Out[15]= |
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| In[16]:= |
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| Out[16]= |
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Diagonal entries of the adjacency tensor specify self-loops in the hypergraph:
| In[17]:= |
![ResourceFunction[
"AdjacencyHypergraph"][{{{1, 0, 0, 0, 0}, {0, 0, 1, 0, 1}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}}, {{0, 0, 1, 0, 1}, {0, 1,
0, 0, 0}, {1, 0, 0, 1, 0}, {0, 0, 1, 0, 0}, {1, 0, 0, 0, 0}}, {{0, 1, 0, 0, 0}, {1, 0, 0, 1, 0}, {0, 0, 0, 0, 0}, {0, 1, 0,
0, 1}, {0, 0, 0, 1, 0}}, {{0, 0, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 1, 0, 0, 1}, {0, 0, 0, 0, 0}, {0, 0, 1, 0, 0}}, {{0, 1, 0, 0, 0}, {1, 0, 0, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 0, 0}}}]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/2f7d955f07f1cf93.png)
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| Out[17]= |
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Adjacency tensors can be of arbitrary rank:
| In[18]:= |
![ResourceFunction["AdjacencyHypergraph"][SparseArray[
Automatic, {12, 12, 12, 12, 12, 12}, 0, {
1, {{0, 240, 600, 960, 1440, 1800, 2040, 2160, 2280, 2400, 2640, 2760, 2880}, CompressedData["
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1:eJxTTMoPSmJkYGBw4GZgYBwFo2AUjIJRMApGwSgYBaNgFBANAFNBDTM=
"]}]]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/6dbcb7285618d192.png){kind=small}
|
| Out[18]= |
|
Options (3)
By default ("OrderedHyperedges"→Automatic), all hyperedges are treated as orderless (i.e. undirected) if the adjacency tensor is symmetric across all indices:
| In[19]:= |
![ResourceFunction[
"AdjacencyHypergraph"][{{0, 1, 1, 0}, {1, 0, 1, 1}, {1, 1, 0, 1}, {0,
1, 1, 0}}]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/218387b76cfa3457.png){kind=small}
|
| Out[19]= |
|
Use "OrderedHyperedges"→True to treat hyperedges as ordered (i.e. directed):
| In[20]:= |
![ResourceFunction[
"AdjacencyHypergraph"][{{0, 1, 1, 0}, {1, 0, 1, 1}, {1, 1, 0, 1}, {0,
1, 1, 0}}, "OrderedHyperedges" -> True]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/690736bb7587c034.png){kind=small}
|
| Out[20]= |
|
Conversely, all hyperedges are treated as ordered (i.e. directed) if the adjacency tensor is asymmetric across any pair of indices:
| In[21]:= |
![ResourceFunction[
"AdjacencyHypergraph"][{{0, 1, 1, 0}, {0, 0, 1, 1}, {0, 0, 0, 1}, {0,
0, 0, 0}}]](https://www.wolframcloud.com/obj/resourcesystem/images/df2/df2ac06a-307e-458c-8806-5237e6e212af/38bd86b88532ff19.png){kind=small}
|
| Out[21]= |
|
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- 1.0.0 – 17 March 2021
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This work is licensed under a Creative Commons Attribution 4.0 International License