Working definition
A graph consists of vertices connected by edges, optionally directed, weighted, temporal, or typed. Graph algorithms expose reachability, dependency, centrality, paths, and communities, but those structural properties inherit the meaning and quality of the encoded edges.
Notation
G = (V, E)Aᵢⱼ = edge weight from i to jAssumptions
- Node and edge semantics are explicit.
- Direction and time are represented where material.
- Missing edges are not automatically negative evidence.
Invariants
- Isomorphic graphs preserve topology.
- A directed path encodes reachability under edge semantics.
- Graph metrics depend on graph construction.
Reproducible procedure
- Define a typed node and edge schema.
- Build the graph from provenance-bearing records.
- Run algorithms appropriate to edge meaning and validate sensitivity.
Error and boundary controls
- Incomplete graphs bias centrality.
- Projection can erase edge types.
- Correlation edges must not be rendered as causal links.
What this does not establish
A knowledge graph connects claims and methods; visual proximity or centrality does not establish truth, causation, or predictive power.
Explicit applications
3 cross-domain bridges
Manufacturing dependency graph
Connect process steps, equipment, materials, defects, metrology, suppliers, and downstream failure modes as typed edges.
Inputs
- process nodes
- typed dependencies
- source records
Outputs
- dependency paths
- critical interfaces
- evidence gaps
Transformation: Build a directed provenance graph and compute paths only under declared edge semantics.
Limit: Graph centrality is not proof of physical causation or commercial importance.
Open connected system →Observation-to-model provenance graph
Connect detector products, calibrations, measured quantities, model assumptions, and inferred claims.
Inputs
- data products
- calibration records
- claims and citations
Outputs
- claim lineage
- missing dependencies
- model boundaries
Transformation: Build typed provenance paths that keep observation and inference edges distinct.
Limit: Complete lineage improves auditability but does not guarantee a model is correct.
Open connected system →Tradition and source lineage graph
Connect rules to passages, editions, reviewers, variants, techniques, and report modules.
Inputs
- source catalog
- rule records
- review records
Outputs
- lineage paths
- variant clusters
- coverage gaps
Transformation: Create typed provenance edges without merging disagreements.
Limit: Historical influence and graph proximity do not prove predictive efficacy.
Open connected system →Authoritative references
- [1]Dictionary of Algorithms and Data Structures · National Institute of Standards and Technology
Establishes: Reference vocabulary for graphs, optimization, search, data structures, complexity, and computational methods used to make algorithms explicit.
Boundary: A formal data structure can represent domain relationships without establishing that the represented causal or interpretive relationships are true.
Direct answer
- A graph consists of vertices connected by edges, optionally directed, weighted, temporal, or typed. Graph algorithms expose reachability, dependency, centrality, paths, and communities, but those structural properties inherit the meaning and quality of the encoded edges.
Mechanism and method
- Define a typed node and edge schema.
- Build the graph from provenance-bearing records.
- Run algorithms appropriate to edge meaning and validate sensitivity.
What is measured
- Isomorphic graphs preserve topology.
- A directed path encodes reachability under edge semantics.
- Graph metrics depend on graph construction.
Limitations
- Incomplete graphs bias centrality.
- Projection can erase edge types.
- Correlation edges must not be rendered as causal links.
- Node and edge semantics are explicit.
- Direction and time are represented where material.
- Missing edges are not automatically negative evidence.
What this does not establish
- A knowledge graph connects claims and methods; visual proximity or centrality does not establish truth, causation, or predictive power.
Bridge: Manufacturing dependency graph
- Connect process steps, equipment, materials, defects, metrology, suppliers, and downstream failure modes as typed edges.
- Input: process nodes
- Input: typed dependencies
- Input: source records
- Output: dependency paths
- Output: critical interfaces
- Output: evidence gaps
- Limit: Graph centrality is not proof of physical causation or commercial importance.
Bridge: Observation-to-model provenance graph
- Connect detector products, calibrations, measured quantities, model assumptions, and inferred claims.
- Input: data products
- Input: calibration records
- Input: claims and citations
- Output: claim lineage
- Output: missing dependencies
- Output: model boundaries
- Limit: Complete lineage improves auditability but does not guarantee a model is correct.
Bridge: Tradition and source lineage graph
- Connect rules to passages, editions, reviewers, variants, techniques, and report modules.
- Input: source catalog
- Input: rule records
- Input: review records
- Output: lineage paths
- Output: variant clusters
- Output: coverage gaps
- Limit: Historical influence and graph proximity do not prove predictive efficacy.