A hypergraph or metagraph ℋ is an ordered pair, or couple, (V,ℰ) where V is the class of vertices
of the hypergraph and ℰ is the class of edges such that ℰ⊆𝒫(V ), where 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 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 = {v1,v2,…,vn} and
ℰ = {e1,e2,…,em}. Associated to any finite hypergraph is the finite n × m incidence matrix
A = (aij) where
For example, let ℋ = (V,ℰ), where V = {a,b,c} and ℰ = {{a},{a,b},{a,c},{a,b,c}}.
Defining vi and ej in the obvious manner (as they are listed in the sets), we have
