Numerousins

Numerousins are a hypothetical family of finite combinatorial objects used to encode multiplicity information in pairwise relations within a set. The term appears in theoretical discussions but has not gained widespread adoption in mainstream mathematics.

A numerousin consists of a finite vertex set V and a multiplicity function m defined on a subset E of unordered pairs from V. For each {u,v} in E, m(u,v) is a nonnegative integer representing how many independent relations hold between u and v. If E includes all pairs, it is a complete numerousin; otherwise it is a partial numerousin. A simple numerousin has m(u,v) in {0,1} for all pairs.

Relation to other structures: collapsing multiplicities greater than one to one yields a simple graph; allowing larger multiplicities links the concept to multigraphs and weighted graphs. The total multiplicity M = sum over {u,v} in E of m(u,v) measures overall relational density.

Examples: let V={a,b,c}, E={{a,b},{b,c},{a,c}}, with m(a,b)=2, m(a,c)=1, m(b,c)=3. Then M=6 and the structure encodes two parallel relations between a and b, one between a and c, and three between b and c.

Applications and status: Numerousins are discussed as models for multi-relational networks, motif counting with multiplicity, and extremal questions about maximizing total multiplicity under local constraints. They sit conceptually near multigraphs and hypergraphs and are used mainly in speculative or informal discussions.

Name and origin: the term emerged in late 2010s discussions, with several informal references and non-peer-reviewed notes. The name appears to blend “numerous” and “cousin,” signaling a multiplicity-based kin to simple graphs.

See also: multigraph, weighted graph, hypergraph, edge multiplicity. References: no established standard references; mentions exist only in informal notes and preprints in circulation.