hypergraph

A hypergraph or metagraph $ \mathcal{H}$ is an ordered pair, or couple, $ (V, \mathcal{E})$ where $ V$ is the class of vertices of the hypergraph and $ \mathcal{E}$ is the class of edges such that $ \mathcal{E} \subseteq \mathcal{P}(V)$ , where $ \mathcal{P}(V)$ is the powerset of $ V$ (the set of subsets of $ V$ ) and is also considered to be a class.

Remark 0.1   A hypergraph is as an extension of the concepts of a graph, colored graph and multi-graph. A finite hypergraph, with both $ V$ and $ \mathcal{E}$ being sets, is also related to a metacategory; therefore, it can also be considered as a special case of a supercategory, and can be thus defined as a mathematical interpretation of ETAS axioms.

Remark 0.2   A finite hypergraph can also be considered as an example of a simple incidence structure. Note also that the more general definition of a hypergraph given above avoids well known antimonies of set theory involving `sets' of sets in the general case.

Remark 0.3   Many specific graph definitions (but not all) can be extended to similar specific hypergraph, or multigraph, definitions. For example, let $ V = \{v_1, v_2, ~\ldots, ~ v_n\}$ and $ \mathcal{E} = \{e_1, e_2, ~ \ldots, ~ e_m\}$ . Associated to any finite hypergraph is the finite $ n \times m$ incidence matrix $ A = (a_{ij})$ where

\begin{displaymath}a_{ij} =
\begin{cases}
1 &\text{ if } ~ v_i \in e_j \\
0 &\text{ otherwise }
\end{cases}\end{displaymath}

For example, let $ \mathcal{H}=(V,\mathcal{E})$ , where $ V=\lbrace a,b,c\rbrace$ and $ \mathcal{E}=\lbrace \lbrace a\rbrace, \lbrace a,b\rbrace, \lbrace a,c\rbrace, \lbrace a,b,c\rbrace\rbrace$ . Defining $ v_i$ and $ e_j$ in the obvious manner (as they are listed in the sets), we have
$ A =
\begin{pmatrix}
1 & 1 & 1 & 1 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1
\end{pmatrix}$



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