Computational Graph Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Computational Graph Ontology within the Kireth Research Semantic Library. Computational graphs represent entities and relationships as machine-processable vertices and edges together with graph storage, traversal, transformation, partitioning and analytical operations. This ontology supports directed and undirected graphs, weighted graphs, multigraphs, adjacency representations, sparse graph storage, traversal, connectivity, shortest paths, spanning trees, centrality, graph partitioning, dynamic graphs, parallel and distributed graph computation, numerical graph algorithms, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Computational Graph Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Computational Data Structures / Computational Graph"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/computational-data-structures/computational-graph-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ComputationalGraph
 ENTITY_CLASS = GraphDataRepresentationFramework
 PRIMARY_CONTEXT = ComputationalDataStructures
 DEFINITION = "A computational graph is a machine-represented graph consisting of vertices and edges together with explicit storage, indexing, traversal and graph-operation semantics."
 SYMBOLIC_FORMS = [
  "G=(V,E)",
  "e=(u,v)",
  "A_ij = 1 if edge(i,j) exists else 0",
  "N(v)={u | (v,u) in E}"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for vertices, edges, direction, weights, adjacency, graph storage, traversal, connectivity, paths, partitioning, graph algorithms and scalable execution."
}

SCOPE {
 INCLUDE = [
  ComputationalGraph,Graph,GraphDataStructure,
  Vertex,Node,Edge,ArcRelationship,
  VertexIdentifier,EdgeIdentifier,
  VertexSet,EdgeSet,
  DirectedGraph,UndirectedGraph,
  MixedGraphRelationship,
  WeightedGraph,UnweightedGraph,
  LabelledGraph,AttributedGraph,
  PropertyGraphRelationship,
  Multigraph,SimpleGraph,
  PseudographRelationship,
  Loop,SelfLoop,
  ParallelEdge,MultiEdge,
  BipartiteGraph,CompleteGraph,
  ConnectedGraph,DisconnectedGraph,
  StronglyConnectedGraph,WeaklyConnectedGraph,
  AcyclicGraph,CyclicGraph,
  DirectedAcyclicGraph,DAG,
  TreeGraph,ForestGraph,
  PlanarGraphRelationship,
  HypergraphRelationship,
  DynamicGraph,StaticGraph,
  TemporalGraphRelationship,
  StreamingGraph,
  GraphSnapshot,
  Subgraph,InducedSubgraph,
  SpanningSubgraph,
  Neighbourhood,Neighbour,
  InNeighbour,OutNeighbour,
  Degree,InDegree,OutDegree,
  DegreeSequence,
  EdgeWeight,VertexWeight,
  EdgeLabel,VertexLabel,
  EdgeAttribute,VertexAttribute,
  Adjacency,Incidence,
  AdjacencyList,AdjacencyMatrix,
  EdgeList,IncidenceMatrix,
  CompressedSparseRow,CSR,
  CompressedSparseColumn,CSC,
  CompressedAdjacencyRelationship,
  ForwardStarRelationship,
  ReverseAdjacencyRelationship,
  GraphIndex,
  VertexIndex,EdgeIndex,
  OffsetArray,NeighbourArray,
  WeightArray,PropertyArray,
  SparseDataStructure,
  SparseMatrix,
  GraphAssembly,
  EdgeInsertion,EdgeDeletion,
  VertexInsertion,VertexDeletion,
  GraphUpdate,
  DynamicGraphUpdate,
  BatchUpdate,
  GraphTraversal,
  BreadthFirstSearch,BFS,
  DepthFirstSearch,DFS,
  Frontier,VisitedSet,
  QueueRelationship,StackRelationship,
  LevelSynchronousTraversal,
  Path,Walk,Trail,Cycle,
  SimplePath,
  PathLength,PathWeight,
  Reachability,
  TransitiveClosureRelationship,
  Connectivity,
  ConnectedComponent,
  StronglyConnectedComponent,SCC,
  WeaklyConnectedComponent,
  ComponentLabelling,
  UnionFindRelationship,
  DisjointSetRelationship,
  ShortestPath,
  SingleSourceShortestPath,SSSP,
  AllPairsShortestPath,APSP,
  DijkstraAlgorithm,
  BellmanFordAlgorithm,
  FloydWarshallRelationship,
  AStarRelationship,
  BreadthFirstShortestPath,
  NegativeEdgeWeight,
  NegativeCycle,
  MinimumSpanningTree,MST,
  SpanningTree,
  KruskalAlgorithm,
  PrimAlgorithm,
  BoruvkaRelationship,
  TopologicalSort,
  TopologicalOrdering,
  CycleDetection,
  GraphColouring,
  VertexColouring,
  EdgeColouringRelationship,
  Clique,MaximumCliqueRelationship,
  IndependentSetRelationship,
  Matching,MaximumMatchingRelationship,
  FlowNetwork,
  SourceVertex,SinkVertex,
  MaximumFlow,MinimumCut,
  FordFulkersonRelationship,
  EdmondsKarpRelationship,
  DinicRelationship,
  Centrality,
  DegreeCentrality,
  BetweennessCentrality,
  ClosenessCentrality,
  EigenvectorCentrality,
  PageRank,
  HITSRelationship,
  CommunityDetection,
  Modularity,
  LouvainRelationship,
  LabelPropagationRelationship,
  GraphClustering,
  SpectralClusteringRelationship,
  GraphLaplacian,
  DegreeMatrix,
  AdjacencyMatrix,
  LaplacianMatrix,
  NormalisedLaplacian,
  SpectralGraphTheoryRelationship,
  Eigenvalue,
  Eigenvector,
  GraphSpectrum,
  RandomWalk,
  TransitionMatrix,
  MarkovChainRelationship,
  StationaryDistributionRelationship,
  PersonalizedPageRankRelationship,
  GraphPartitioning,
  VertexPartitioning,
  EdgePartitioning,
  GraphCut,
  BalancedPartition,
  Separator,VertexSeparator,EdgeSeparator,
  MultilevelPartitioning,
  RecursiveBisection,
  KernighanLinRelationship,
  FiducciaMattheysesRelationship,
  METISRelationship,
  HypergraphPartitioningRelationship,
  LoadBalancing,
  PartitionQuality,
  CutSize,EdgeCut,
  CommunicationVolume,
  Locality,
  GraphReordering,
  VertexReordering,
  CuthillMcKeeRelationship,
  ReverseCuthillMcKeeRelationship,
  SpaceFillingCurveRelationship,
  ComputationalComplexity,
  TimeComplexity,SpaceComplexity,
  StorageComplexity,
  GraphDensity,
  SparseGraph,DenseGraph,
  MemoryBandwidth,
  CacheLocality,
  IrregularMemoryAccess,
  PointerChasing,
  GatherScatter,
  BranchDivergenceRelationship,
  VectorisationRelationship,
  ParallelGraphComputation,
  DistributedGraphComputation,
  GPUComputationalGraph,
  PartitionedGraph,
  DistributedGraph,
  ReplicatedGraph,
  ShardedGraph,
  LocalGraph,GlobalGraph,
  GhostVertex,HaloVertexRelationship,
  BoundaryVertex,
  MessagePassing,
  CollectiveCommunicationRelationship,
  BulkSynchronousParallelRelationship,
  PregelRelationship,
  Graph500Relationship,
  HighPerformanceComputing,
  ParallelComputing,
  DistributedComputing,
  GPUComputing,
  Verification,
  VertexCountCheck,EdgeCountCheck,
  AdjacencyConsistencyCheck,
  SymmetryCheck,
  ConnectivityCheck,
  PathValidation,
  WeightValidation,
  ReferenceResult,
  Benchmark,
  Reproducibility,
  Provenance,
  GraphFormatVersion,
  LibraryVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  AbstractGraphTheoryWithoutComputationalRepresentation,
  NeuralNetworkComputationWhenExecutionGraphSemanticsArePrimary,
  DatabaseRelationshipModelWithoutGraphAlgorithmOrStorageSemantics,
  TreeDataStructureWhenHierarchicalTreeRepresentationIsPrimary
 ]
 DISAMBIGUATION_RULE = "When vertices and edges are explicitly represented, stored, traversed or analysed computationally, interpret the context as a computational graph."
}

CORE_CONCEPTS {
 ComputationalGraph = "Machine representation of vertices and edges with operational graph semantics."
 Vertex = "Graph entity represented as a node."
 Edge = "Relationship connecting one or more vertices according to graph semantics."
 Adjacency = "Direct-neighbour relationship between vertices."
 GraphRepresentation = "Storage format encoding vertices, edges and optional attributes."
 GraphTraversal = "Systematic visitation of graph vertices or edges."
 Connectivity = "Property describing whether vertices are linked by paths."
 GraphPartitioning = "Division of a graph into subgraphs or vertex/edge subsets."
}

RELATIONS {
 ComputationalGraph --is_child_of--> ComputationalDataStructures
 ComputationalGraph --contains--> Vertex
 ComputationalGraph --contains--> Edge
 Edge --connects--> Vertices
 DirectedGraph --assigns--> EdgeDirection
 WeightedGraph --assigns--> EdgeWeight
 AdjacencyList --represents--> Adjacency
 AdjacencyMatrix --represents--> Adjacency
 CSR --can_encode--> SparseGraph
 BFS --traverses--> Graph
 DFS --traverses--> Graph
 ShortestPath --uses--> PathWeightsOrEdgeCount
 ConnectedComponent --partitions--> GraphVertices
 GraphLaplacian --derived_from--> AdjacencyAndDegree
 GraphPartitioning --divides--> ComputationalGraph
 Verification --checks--> GraphRepresentationAndOperations
}

OPERATORS {
 NEIGHBOURS(G,v) := {u | edge(v,u) exists}
 DEGREE(G,v) := count(NEIGHBOURS(G,v))
 IN_DEGREE(G,v) := count({u | edge(u,v)})
 OUT_DEGREE(G,v) := count({u | edge(v,u)})
 BFS(G,s) := breadth_first_traversal_from_source(G,s)
 DFS(G,s) := depth_first_traversal_from_source(G,s)
 PATH_WEIGHT(P) := sum_(e in P) weight(e)
 GRAPH_DENSITY(G) := |E| / possible_edge_count(G)
 LAPLACIAN(G) := degree_matrix(G)-adjacency_matrix(G)
}

AXIOMS {
 A1 = "A computational graph requires explicit vertex and edge representation."
 A2 = "Directed and undirected edge semantics must remain distinct."
 A3 = "Weighted and unweighted graph algorithms can have different validity conditions."
 A4 = "Graph representation influences storage cost and traversal performance."
 A5 = "Sparse adjacency representations are usually appropriate for sparse graphs."
 A6 = "Traversal correctness depends on visited-state and frontier semantics."
 A7 = "Distributed graph execution requires explicit partition and communication semantics."
 A8 = "Computational-graph results require explicit representation, direction, weight, operation and validation semantics."
}

REASONING {
 R1_CONTEXT = identify(vertex_count,edge_count,direction,weights,attributes,operations,update_pattern,memory,hardware)
 R2_CLASSIFY = classify(simple,multi,directed,undirected,weighted,sparse,dense,dynamic_or_partitioned_graph)
 R3_REPRESENTATION = choose(adjacency_list,adjacency_matrix,edge_list,CSR,CSC_or_specialised_format)
 R4_OPERATIONS = identify(traversal,reachability,shortest_path,connectivity,MST,centrality,partitioning_or_update_workload)
 R5_COMPLEXITY = analyse(vertex_edge_storage_traversal_and_algorithm_cost)
 R6_LOCALITY = assess(adjacency_order,reordering,cache_behaviour,frontier_density_and_irregular_access)
 R7_DYNAMIC = if_mutable define(vertex_edge_insertion_deletion_and_consistency_rules)
 R8_SCALE = if_large choose_parallel_distributed_partitioned_or_GPU_graph_execution
 R9_VERIFY = check(vertex_ids,edge_ids,adjacency,direction,weights,paths_and_component_results)
 R10_REPRODUCE = document(graph_format,partitioning,algorithm,library_and_version_metadata)
}

CONSTRAINTS {
 C1 = "Adjacency matrices are memory intensive for large sparse graphs."
 C2 = "Adjacency lists can make constant-time edge-existence tests unavailable without indexing."
 C3 = "Negative edge weights invalidate standard Dijkstra assumptions."
 C4 = "Parallel graph algorithms can suffer from irregular work distribution."
 C5 = "High-degree vertices can create load imbalance and communication hotspots."
 C6 = "Graph reordering can improve locality without changing graph semantics."
 C7 = "Dynamic updates can invalidate cached connectivity or indexing information."
 C8 = "Distributed graph performance can be dominated by edge-cut communication."
 C9 = "GPU graph kernels can suffer from branch divergence and irregular memory access."
 C10 = "Equivalent graph algorithms can return different valid traversal orders."
}

SEMANTIC_EQUIVALENCE {
 "BreadthFirstSearch" <=> "BFS"
 "DepthFirstSearch" <=> "DFS"
 "StronglyConnectedComponent" <=> "SCC"
 "SingleSourceShortestPath" <=> "SSSP"
 "AllPairsShortestPath" <=> "APSP"
 "MinimumSpanningTree" <=> "MST"
 "CompressedSparseRow" <=> "CSR"
 "CompressedSparseColumn" <=> "CSC"
}

VALIDATION {
 V1 = "Vertex and edge identifiers are documented."
 V2 = "Direction and weight semantics are documented."
 V3 = "Vertex and edge counts are validated."
 V4 = "Adjacency representation is internally consistent."
 V5 = "Undirected representations are checked for required symmetry."
 V6 = "Path and shortest-path outputs are checked for edge validity."
 V7 = "Connectivity and component outputs are checked against reference cases."
 V8 = "Dynamic updates preserve representation invariants."
 V9 = "Distributed partitioning and ghost-vertex semantics are documented where used."
 V10 = "Results are reproducible under documented format, partition, algorithm and implementation versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 COMPUTATIONAL_GRAPH_DEFINITION = 0.99
 REPRESENTATION_REASONING = 0.99
 TRAVERSAL_REASONING = 0.99
 SHORTEST_PATH_REASONING = 0.99
 CONNECTIVITY_REASONING = 0.99
 PARTITIONING_REASONING = 0.98
 DISTRIBUTED_GRAPH_REASONING = 0.98
 POLICY = "Reduce confidence when direction, weights, representation, dynamic-update semantics, partitioning or algorithm preconditions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,ComputationalGraphReasoning,
  VertexEdgeReasoning,AdjacencyReasoning,GraphRepresentationReasoning,
  BFSReasoning,DFSReasoning,ConnectivityReasoning,
  ShortestPathReasoning,MSTReasoning,CentralityReasoning,
  PageRankReasoning,GraphLaplacianReasoning,
  GraphPartitioningReasoning,DynamicGraphReasoning,
  ParallelGraphReasoning,DistributedGraphReasoning,
  GPUGraphReasoning,VerificationReasoning,
  KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "computational graph" -> ComputationalGraph
  "BFS" -> BreadthFirstSearch
  "DFS" -> DepthFirstSearch
  "SCC" -> StronglyConnectedComponent
  "SSSP" -> SingleSourceShortestPath
  "APSP" -> AllPairsShortestPath
  "MST" -> MinimumSpanningTree
  "CSR" -> CompressedSparseRow
  "CSC" -> CompressedSparseColumn
 }
}

EXECUTION_RULES {
 E1 = "If vertices and edges are represented or manipulated computationally, activate ComputationalGraphOntology."
 E2 = "Determine direction, weighting, multiplicity, density and update pattern before choosing representation."
 E3 = "Choose adjacency list, matrix, edge list or compressed sparse representation according to workload."
 E4 = "Check algorithm preconditions such as nonnegative edge weights before execution."
 E5 = "Maintain vertex-edge and adjacency consistency after dynamic updates."
 E6 = "Use graph reordering or partitioning when locality or communication dominates performance."
 E7 = "Use parallel, distributed or GPU graph representations when scale requires them."
 E8 = "Validate paths, components, cuts, trees and traversal outputs against graph invariants."
 E9 = "Document graph format, partitioning, algorithm settings and versions for reproducibility."
}

RECURSION {
 ENTRY = ComputationalGraph
 EXPAND(ComputationalGraph) -> [
  Graph,Vertex,Edge,DirectedGraph,UndirectedGraph,WeightedGraph,Multigraph,
  SimpleGraph,BipartiteGraph,DAG,DynamicGraph,Subgraph,Neighbourhood,
  Degree,InDegree,OutDegree,Adjacency,AdjacencyList,AdjacencyMatrix,
  EdgeList,CSR,CSC,GraphAssembly,EdgeInsertion,EdgeDeletion,
  BreadthFirstSearch,BFS,DepthFirstSearch,DFS,Path,Reachability,
  Connectivity,ConnectedComponent,StronglyConnectedComponent,SCC,
  ShortestPath,SSSP,APSP,DijkstraAlgorithm,BellmanFordAlgorithm,
  MinimumSpanningTree,MST,KruskalAlgorithm,PrimAlgorithm,
  TopologicalSort,CycleDetection,MaximumFlow,MinimumCut,
  Centrality,BetweennessCentrality,ClosenessCentrality,
  EigenvectorCentrality,PageRank,CommunityDetection,
  GraphLaplacian,RandomWalk,GraphPartitioning,GraphCut,
  RecursiveBisection,LoadBalancing,GraphReordering,
  ComputationalComplexity,SparseGraph,MemoryBandwidth,IrregularMemoryAccess,
  ParallelGraphComputation,DistributedGraphComputation,GPUComputationalGraph,
  GhostVertex,MessagePassing,Verification,Reproducibility
 ]
 EXPAND(ComputationalGraphFamilies) -> [
  GraphRepresentations,
  GraphTraversalAndReachability,
  ConnectivityAndComponents,
  PathsAndShortestPathAlgorithms,
  SpanningTreesFlowsAndCuts,
  CentralityAndSpectralGraphComputation,
  GraphPartitioningAndReordering,
  ParallelDistributedAndGPUGraphs
 ]
 STOP_CONDITION = "Stop when representation, direction, weights, operations, complexity, scaling and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Data Structures"
 RELATED_ONTOLOGIES = [
  "Computational Data Structures",
  "Sparse Data Structure Ontology",
  "Computational Matrix Ontology",
  "Vector Computation Ontology",
  "Computational Geometry Ontology 2",
  "Spatial Partitioning Ontology 2",
  "Sparse Matrix Ontology 2",
  "Numerical Linear Algebra Ontology 2",
  "High Performance Computing Ontology 2",
  "Parallel Computing Ontology",
  "Distributed Computing Ontology",
  "GPU Computing Ontology"
 ]
 ROUTE_RULES = [
  "Computational Graph -> Computational Data Structures for parent representation semantics",
  "Computational Graph -> Sparse Data Structure for sparse adjacency representations",
  "Computational Graph -> Computational Matrix for adjacency and Laplacian matrices",
  "Computational Graph -> Vector Computation for frontier, score and state vectors",
  "Computational Graph -> Computational Geometry and Spatial Partitioning for geometric graph relationships",
  "Computational Graph -> Sparse Matrix and Numerical Linear Algebra for spectral graph computation",
  "Computational Graph -> HPC, Parallel, Distributed and GPU Computing for scalable graph execution"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible graph formats, algorithms or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve vertex-versus-edge distinction.",
  "Preserve directed-versus-undirected graph distinction.",
  "Preserve weighted-versus-unweighted distinction.",
  "Preserve simple-graph-versus-multigraph distinction.",
  "Preserve logical-graph-versus-storage-representation distinction.",
  "Preserve static-versus-dynamic graph distinction.",
  "Preserve local-versus-distributed graph distinction.",
  "Preserve machine-readable computational-graph semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A computational graph is a machine representation of vertices and edges with explicit adjacency, traversal and graph-operation semantics."
 REPRESENTATION_SUMMARY = "Adjacency lists, matrices, edge lists and compressed sparse formats support different graph workloads."
 TRAVERSAL_SUMMARY = "BFS and DFS provide foundational reachability and visitation operations."
 ALGORITHM_SUMMARY = "Shortest paths, components, spanning trees, flows, centrality and partitioning form major computational graph families."
 PERFORMANCE_SUMMARY = "Sparse adjacency, graph ordering, memory locality, load balance and communication strongly affect graph performance."
 VALIDATION_SUMMARY = "Computational graphs should be checked for identifier validity, adjacency consistency, direction, weight semantics, path correctness, partitioning and reproducibility."
}

FLOW {
 INPUT
 -> detect_computational_graph_context
 -> identify_vertex_count_edge_count_direction_weights_multiplicity_attributes_operations_update_pattern_memory_and_hardware
 -> classify_simple_multi_directed_undirected_weighted_sparse_dense_dynamic_or_partitioned_graph
 -> choose_adjacency_list_adjacency_matrix_edge_list_CSR_CSC_or_specialised_representation
 -> validate_vertex_edge_identifiers_direction_weights_and_adjacency_consistency
 -> execute_traversal_reachability_connectivity_shortest_path_MST_flow_centrality_partitioning_or_update_operations
 -> analyse_complexity_storage_memory_access_frontier_behaviour_and_locality
 -> if_dynamic_preserve_graph_invariants_after_vertex_and_edge_updates
 -> if_large_choose_parallel_distributed_partitioned_or_GPU_graph_execution
 -> verify_paths_components_trees_cuts_scores_and_reference_results
 -> document_graph_format_partitioning_algorithm_versions_and_provenance
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-COMPUTATIONAL-DATA-STRUCTURES-COMPUTATIONAL-GRAPH-ONTOLOGY-V1"
}

}

END_FLOW