hypergraphHypergraph2010-05-09 23:51:282010-05-10 00:44:12Definitionfinite hypergraphhypergraphsimple incidence structureincidence structurefinite hypergraphhypergraphsimple incidence structureincidence structur% there are many more packages, add them here as you need
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A \emph{hypergraph} or \emph{metagraph} $\mathcal{H}$ is an ordered pair, or couple, $(V, \mathcal{E})$ where $V$ is the class of \emph{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 also considered to be a class.
\begin{remark}
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.
\end{remark}
\begin{remark}
A \emph{finite hypergraph} can also be considered as an example of a \emph{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.
\end{remark}
\begin{remark}
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 $n \times m$ \emph{incidence matrix} $A = (a_{ij})$ where
\[a_{ij} =
\begin{cases}
1 &\text{ if } ~ v_i \in e_j \\
0 &\text{ otherwise }
\end{cases}\]
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
\begin{center}$A =
\begin{pmatrix}
1 & 1 & 1 & 1 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1
\end{pmatrix}$
\end{center}
\end{remark}