Model

Mathematical model

In pypergraph, a hypergraph \(\mathcal{H}\) consists of a pair \((\mathcal{V}, \mathcal{E})\). The set \(\mathcal{V}\) contains the nodes of the hypergraph. The set \(\mathcal{E} = \{e_i\}_{i \in I}\) has an index set \(I\) and contains the hyperedges.

Each hyperedge may be undirected or directed.

• An undirected hyperedge is some subset of \(\mathcal{V}\).

  • That is, if \(e_i\) is an undirected hyperedge then \(e_i = (i, N_i)\) where \(N_i \subseteq \mathcal{V}\).

  • The symbol \(N_i\) represents the nodes incident with \(e_i\).

  • We write \(N_i = N(e_i)\).

• A directed hyperedge consists of a tail and a head, each of which is a subset of \(\mathcal{V}\).

  • That is, if \(e_i\) is a directed hyperedge then \(e_i = (i, T_i, H_i)\) where \(T_i, H_i \subseteq \mathcal{V}\).

  • The symbol \(T_i\) represents the tail of \(e_i\), while \(H_i\) represents the head of \(e_i\).

  • We write \(T_i = T(e_i)\) and \(H_i = H(e_i)\).

For avoidance of doubt, the following are possible within a hypergraph in pypergraph:

Situation	Description
Parallel undirected hyperedges	It may be that \(N(e_i) = N(e_j)\) for distinct \(i, j \in I\).
Parallel directed hyperedges	It may be that \((T(e_i), H(e_i)) = (T(e_j), H(e_j))\) for distinct \(i, j \in I\).
Undirected loops	It may be that \(N(e_i) = \{v\}\) for some \(i \in I\) and \(v \in \mathcal{V}\).
Directed loops	It may be that \(T(e_i) \cap H(e_i) \neq \emptyset\) for some \(i \in I\).
Degenerate undirected hyperedges	It may be that \(N(e_i) = \emptyset\) for some \(i \in I\).
Degenerate directed hyperedges	It may be that either \(T(e_i) = \emptyset\) or \(H(e_i) = \emptyset\) for some \(i \in I\).

It is not possible to have \(T(e_i) = H(e_i) = \emptyset\) for some \(i \in I\).

Logical data model

In pypergraph, Hypergraph is a container of nodes and hyperedges.

Within a Hypergraph instance, the nodes have unique identifiers known as names, and the hyperedges have unique identifiers known as labels. Node names and hyperedge labels must be hashable.

A hyperedge may be undirected, whereby it refers to one subset of the nodes. Alternatively, a hyperedge may be directed, whereby it refers to two subsets of the nodes, where one is known as the tail and the other as the head.

Each node or hyperedge has a single underlying object, which can be any Python object.

Each node or hyperedge has an associated mapping representing data attributes. A data attribute has a unique name which should be a str value as well as a value which can be any Python object. A given data attribute need not be uniformly specified across nodes or hyperedges.

The "inner" data attribute represents the underlying object for the node or hyperedge. If this is not specified, the underlying object is inferred as the node name or hyperedge label.