Domenets

Domenets are a theoretical construct in discrete mathematics and computer science used to model domain-annotated network structures that generalize ordinary directed graphs. They focus on the notion of domains, which label nodes and edges to reflect their provenance, function, or security level.

A domenet consists of a finite node set N, a finite edge set E, a domain set Dom, a source function src: E -> N, a target function tgt: E -> N, and a domain labeling function domN: N -> Dom along with domE: E -> Dom that assigns a domain to each edge. Compatibility conditions may require that the domain of an edge aligns with the domains of its endpoints according to a user-defined rule. In practice, domenets can encode constraints such as permissible interactions across domains and domain-specific routing or processing requirements.

A domenet morphism between D1 and D2 consists of functions fN: N1 -> N2 and fE: E1 -> E2 that preserve endpoints and domain labels: domN2(fN(n)) = domN1(n) and domE2(fE(e)) = domE1(e); src and tgt are preserved as well. This makes Domenets amenable to categorical-style reasoning and composition of mappings between structures.

Variants and properties include layered domenets, where nodes and edges are associated with multiple domain levels, and acyclic domenets, analogous to directed acyclic graphs. Domenets can be finite or infinite, with additional constraints such as connectivity or resilience metrics, depending on the modeling goals.

Example: A simple domenet has N = {a, b}, Dom = {outside, internal}, with a in internal, b in outside, and a single edge e from a to b carrying domain label internal. This encodes an internal communication step from an internal node to an outside node.

Applications include modeling cross-domain data flows, modular system architectures, and provenance tracking in distributed networks. Domenets provide a framework for reasoning about interactions with domain-consistent constraints.

See also: Graph theory, labeled graphs, multi-layer networks, category theory.