PlanetPhysics/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.

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.

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.

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 $$a_{ij} = \begin{cases} 1 &= if = ~ v_i \in e_j \\ 0 &= otherwise = \end{cases}$$ 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 $$A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix}$$