Isomorphic Discovery — cross-domain structural equivalence detection

[anormang1992]

Summary

When VRE agents in a network query peers for cross-domain knowledge, the overlapping subgraphs between two domains can reveal structural isomorphisms — concepts that play analogous roles across different domains. These isomorphisms are a distinct epistemic category: they are not learning candidates to be persisted into the local graph, but structural bridges that enable reasoning by analogy.

An agent that discovers an isomorphism between its own domain and a peer's domain gains the ability to suggest novel approaches by drawing on patterns from the structurally similar domain.

Relationship to VRE Networks (#5)

The VRE Networks protocol sends all extracted concepts (not just gapped ones) when querying a peer. The peer's response includes the full subgraph. This design ensures the overlapping subgraph — the raw material for isomorphic discovery — is available without protocol changes.

Isomorphic discovery is a consumer of network peering data, not a modification to the peering protocol itself.

What Constitutes an Isomorphism

The system must be extremely strict about classification. False positives are epistemically devastating — they would cause the agent to reason by analogy from a structural relationship that doesn't actually hold, leading to confidently wrong conclusions.

Tentative criteria (all must hold):

1. Relata-type equivalence — The concepts share the same relationship types to their neighbors. If A's Write --[APPLIES_TO]--> Socket and B's Write --[APPLIES_TO]--> File, the relata type (APPLIES_TO) matches. A mere name match on the concept without relata-type alignment is NOT an isomorphism.

2. Depth-level parity — The concepts are grounded to comparable depth levels in their respective graphs. If A's Write is at D3 and B's Write is at D0, the knowledge asymmetry is too large for structural comparison. Both should be at D2+ minimum.

3. Neighborhood structure similarity — The local graph topology around the concept should be structurally similar, not just the immediate edges. If A's Write has three CONSTRAINED_BY edges and B's Write has one, the constraint profiles diverge and the analogy weakens.

4. Constraint pattern alignment — D3 (constraints) properties on the isomorphic concepts should express analogous restrictions. If A's Write is constrained by "connection timeout" and B's Write is constrained by "file lock", those are structurally analogous (both are temporal resource constraints). If B's Write is constrained by "file format validation", that's a different constraint category — weakening the isomorphism.

5. Provenance independence — The structural similarity should not be an artifact of shared authorship. If the same person seeded both graphs from the same mental model, the similarity is expected, not discovered. True isomorphisms emerge across independently authored domains.

Additional signals (supporting but not sufficient alone):

• Shared transitive dependencies — Both concepts REQUIRES or DEPENDS_ON a common concept (e.g., both Write operations REQUIRES Permission)
• Analogous gap patterns — When grounding fails in both domains, it fails at the same depth or for the same structural reason
• Cross-domain relata targets playing equivalent roles — The targets of matching relata play analogous roles in their respective domains (File in filesystem ≈ Socket in network ≈ Table in database)

What is NOT an isomorphism:

• Name match alone — Two concepts called "Read" in different domains are not isomorphic just because they share a name
• Partial structural overlap — Sharing one relata type but diverging on others is correlation, not isomorphism
• Inherited similarity — Concepts that are similar because they share a common abstraction node (Abstraction Nodes #3) are related by abstraction, not isomorphism

Epistemic Designation

Isomorphisms should be a first-class epistemic category in VRE, distinct from:

• Provenance (how knowledge was obtained)
• Learning candidates (gaps to be filled)
• Knowledge gaps (missing depth or existence)

An isomorphism is a structural claim about the relationship between two subgraphs in different domains. It should carry:

• The two concepts being compared
• The source agents/graphs
• The structural evidence (matching relata, depth parity, constraint alignment)
• A confidence indicator reflecting how many criteria were satisfied
• Timestamp of discovery

Behavioral Impact

When an agent has access to validated isomorphisms, it can:

• Suggest approaches from one domain that may apply to another ("filesystem write uses atomic rename for safety — the analogous pattern for network write might be...")
• Anticipate constraints by analogy ("filesystem write requires permission; network write likely has an analogous access control constraint")
• Surface potential knowledge gaps by analogy ("filesystem write has a D4 implication about data loss — does network write have an analogous implication?")

Open Questions

• Should isomorphism validation require human confirmation (like learning), or can it be fully automated with strict enough criteria?
• How should isomorphisms decay or be invalidated when the underlying graphs evolve?
• Should there be a threshold number of criteria before an isomorphism is even proposed?
• How does this interact with Abstraction Nodes (Abstraction Nodes #3) — are isomorphisms a precursor to discovering abstractions?

Dependencies

• VRE Agentic Networks — Cross-Graph Epistemic Grounding #5 (VRE Networks) — the peering protocol provides the cross-domain subgraphs
• Abstraction Nodes #3 (Abstraction Nodes) — related but distinct; isomorphisms are between instances, abstractions are generalizations