Graph Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Graph Mathematics Ontology within the Kireth Research Semantic Library. Graph mathematics studies mathematical structures made from vertices, edges and relationships, together with connectivity, paths, neighbourhoods, spectral properties, graph operators and graph-based computation. In artificial intelligence this ontology supports graph representation learning, graph neural networks, message passing, knowledge graphs, relational reasoning, graph embeddings, spectral graph theory, random walks, graph attention, link prediction, node classification, graph clustering and relationships with discrete mathematics, linear algebra, probability, optimisation and AI mathematics.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Graph Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Artificial Intelligence Mathematics / Graph Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/artificial-intelligence-mathematics/graph-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = GraphMathematics
 ENTITY_CLASS = MathematicalRelationalStructureFramework
 PRIMARY_CONTEXT = AIMathematics
 DEFINITION = "Graph mathematics is the mathematical study of relational structures represented by vertices and edges, including their connectivity, geometry, spectra, dynamics and computational transformations."
 SYMBOLIC_FORMS = [
  "G=(V,E)",
  "A_ij=1 if (i,j) in E",
  "L=D-A",
  "h_v^(l+1)=UPDATE(h_v^(l),AGGREGATE({h_u^(l):u in N(v)}))"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for graph structure, adjacency, paths, connectivity, graph operators, spectral methods, message passing, graph embeddings, relational inference and graph-based AI computation."
}

SCOPE {
 INCLUDE = [
  GraphMathematics,
  Graph,
  Vertex,
  Node,
  Edge,
  Arc,
  DirectedGraph,
  UndirectedGraph,
  WeightedGraph,
  UnweightedGraph,
  SimpleGraph,
  Multigraph,
  Pseudograph,
  BipartiteGraph,
  CompleteGraph,
  RegularGraph,
  ConnectedGraph,
  DisconnectedGraph,
  Tree,
  Forest,
  DirectedAcyclicGraph,
  DAG,
  Hypergraph,
  HeterogeneousGraph,
  DynamicGraph,
  TemporalGraph,
  AttributedGraph,
  KnowledgeGraph,
  PropertyGraph,
  Adjacency,
  Neighbourhood,
  Degree,
  InDegree,
  OutDegree,
  DegreeMatrix,
  AdjacencyMatrix,
  IncidenceMatrix,
  GraphLaplacian,
  NormalisedLaplacian,
  TransitionMatrix,
  Path,
  Walk,
  Trail,
  Cycle,
  ShortestPath,
  GeodesicDistance,
  Connectivity,
  ConnectedComponent,
  StronglyConnectedComponent,
  WeaklyConnectedComponent,
  Cut,
  MinimumCut,
  Flow,
  MaximumFlow,
  Centrality,
  DegreeCentrality,
  BetweennessCentrality,
  ClosenessCentrality,
  EigenvectorCentrality,
  PageRank,
  RandomWalk,
  StationaryDistribution,
  Community,
  CommunityDetection,
  Modularity,
  GraphPartition,
  GraphClustering,
  SpectralGraphTheory,
  Eigenvalue,
  Eigenvector,
  GraphSpectrum,
  AlgebraicConnectivity,
  SpectralEmbedding,
  GraphSignal,
  GraphFourierTransform,
  GraphConvolution,
  GraphFilter,
  GraphLearning,
  GraphRepresentationLearning,
  GraphEmbedding,
  NodeEmbedding,
  EdgeEmbedding,
  GraphEmbeddingVector,
  GraphNeuralNetwork,
  GNN,
  MessagePassing,
  MessagePassingNeuralNetwork,
  MPNN,
  AggregationFunction,
  UpdateFunction,
  GraphConvolutionalNetwork,
  GCN,
  GraphAttentionNetwork,
  GAT,
  GraphSAGE,
  GraphIsomorphismNetwork,
  GIN,
  RelationalGraphConvolutionalNetwork,
  RGCN,
  GraphTransformer,
  AttentionOnGraphs,
  LinkPrediction,
  NodeClassification,
  EdgeClassification,
  GraphClassification,
  GraphRegression,
  GraphGeneration,
  KnowledgeGraphEmbedding,
  RelationEmbedding,
  EntityEmbedding,
  TransE,
  RotatE,
  DistMult,
  ComplEx,
  GraphAutoencoder,
  VariationalGraphAutoencoder,
  NegativeSampling,
  NeighbourSampling,
  SubgraphSampling,
  Oversmoothing,
  Oversquashing,
  Expressivity,
  WeisfeilerLehmanTest,
  PermutationInvariance,
  PermutationEquivariance,
  GraphPooling,
  ReadoutFunction,
  GraphCoarsening,
  GraphSparsification,
  GraphMatching,
  GraphIsomorphism,
  SubgraphIsomorphism,
  Homomorphism,
  GraphKernel,
  RandomGraph,
  ErdosRenyiGraph,
  ScaleFreeGraph,
  SmallWorldGraph,
  StochasticBlockModel,
  GraphDiffusion,
  HeatKernel,
  PersonalizedPageRank,
  LabelPropagation,
  GraphRegularisation
 ]
 EXCLUDE_AS_PRIMARY = [
  GeneralDiscreteMathematicsWithoutGraphStructure,
  TabularMachineLearningWithoutRelationalStructure,
  DatabaseGraphsWithoutMathematicalGraphSemantics,
  NetworkVisualisationWithoutGraphAnalysis
 ]
 DISAMBIGUATION_RULE = "When entities are connected by explicit mathematical relations represented as vertices and edges and the structure itself affects inference or computation, interpret the context as graph mathematics."
}

CORE_CONCEPTS {
 GraphMathematics = "Mathematical study of relational structures represented by vertices and edges."
 Graph = "Ordered structure containing a vertex set and an edge or relation set."
 AdjacencyMatrix = "Matrix encoding pairwise graph adjacency."
 GraphLaplacian = "Matrix operator encoding graph connectivity and diffusion structure."
 Path = "Sequence of adjacent vertices or edges connecting locations in a graph."
 Connectivity = "Property describing whether and how graph vertices are mutually reachable."
 SpectralGraphTheory = "Study of graph properties through eigenvalues and eigenvectors of graph matrices."
 GraphRepresentationLearning = "Learning vector or tensor representations of graph entities and structures."
 MessagePassing = "Local graph computation that aggregates information from neighbouring nodes or edges."
 GraphNeuralNetwork = "Neural architecture whose operations respect graph structure and permutation symmetries."
}

RELATIONS {
 GraphMathematics --is_child_of--> AIMathematics
 GraphMathematics --uses--> DiscreteMathematics
 GraphMathematics --uses--> LinearAlgebra
 Graph --contains--> Vertex
 Graph --contains--> Edge
 Edge --connects--> Vertex
 DirectedGraph --has--> DirectedEdge
 WeightedGraph --assigns--> EdgeWeight
 AdjacencyMatrix --represents--> GraphAdjacency
 DegreeMatrix --contains--> VertexDegrees
 GraphLaplacian --is_defined_from--> DegreeMatrix
 GraphLaplacian --is_defined_from--> AdjacencyMatrix
 Path --connects--> Vertices
 Connectivity --depends_on--> Paths
 RandomWalk --uses--> TransitionMatrix
 PageRank --uses--> RandomWalk
 CommunityDetection --partitions--> Graph
 Modularity --evaluates--> CommunityPartition
 SpectralGraphTheory --uses--> GraphSpectrum
 SpectralEmbedding --uses--> GraphEigenvectors
 GraphConvolution --uses--> GraphLaplacianOrAdjacency
 GraphNeuralNetwork --uses--> MessagePassing
 MessagePassing --aggregates--> NeighbourInformation
 GraphAttentionNetwork --uses--> AttentionOnGraphs
 GraphSAGE --uses--> NeighbourSampling
 GraphPooling --produces--> GraphLevelRepresentation
 ReadoutFunction --aggregates--> NodeRepresentations
 KnowledgeGraphEmbedding --represents--> EntitiesAndRelations
 LinkPrediction --predicts--> MissingEdges
 NodeClassification --predicts--> NodeLabels
 GraphClassification --predicts--> GraphLabels
 PermutationEquivariance --preserves--> NodeReindexingStructure
 WeisfeilerLehmanTest --relates_to--> GNNExpressivity
}

OPERATORS {
 GRAPH(V,E) := G=(V,E)
 ADJACENCY_MATRIX(G) := A_ij=1_or_weight_if_edge(i,j)_exists_else_0
 DEGREE(v) := deg(v)=|N(v)|
 DEGREE_MATRIX(G) := D=diag(deg(v_1),...,deg(v_n))
 GRAPH_LAPLACIAN(G) := L=D-A
 NORMALISED_LAPLACIAN(G) := L_sym=I-D^(-1/2)*A*D^(-1/2)
 RANDOM_WALK_MATRIX(G) := P=D^(-1)*A
 PATH_LENGTH(path) := sum_edges weight(edge)
 SHORTEST_PATH(u,v) := min_path length(path)
 MODULARITY(A,k,m,c) := Q=(1/(2m))*sum_ij [A_ij-k_i*k_j/(2m)]*delta(c_i,c_j)
 GCN_LAYER(H,A,W) := H_next=sigma(D_hat^(-1/2)*A_hat*D_hat^(-1/2)*H*W)
 MESSAGE_PASS(v,l) := m_v=AGGREGATE({MESSAGE(h_v,h_u,e_uv):u in N(v)})
 NODE_UPDATE(v,l,m_v) := h_v_next=UPDATE(h_v,m_v)
 GAT_ATTENTION(i,j) := alpha_ij=softmax_j(LeakyReLU(a^T[W h_i || W h_j]))
 GRAPH_READOUT(H) := h_G=READOUT({h_v:v in V})
 TRANSE_SCORE(h,r,t) := score=-||e_h+e_r-e_t||
 DISTMULT_SCORE(h,r,t) := score=sum_k e_hk*e_rk*e_tk
}

AXIOMS {
 A1 = "Graph semantics require explicit vertex and edge definitions."
 A2 = "Directed and undirected edges have different adjacency semantics."
 A3 = "Graph algorithms must respect graph structure and edge direction where applicable."
 A4 = "Graph neural operations should preserve appropriate permutation invariance or equivariance."
 A5 = "Local message passing limits information propagation by graph distance and network depth."
 A6 = "Spectral graph quantities depend on the selected graph operator and normalisation."
 A7 = "Graph representation quality depends on structure, attributes, objective and sampling."
 A8 = "Graph-based conclusions require structural, numerical and task-level validation."
}

REASONING {
 R1_CONTEXT = identify(vertices,edges,direction,weights,attributes,task)
 R2_GRAPH_CLASS = classify(simple,directed,weighted,bipartite,heterogeneous,dynamic,knowledge_graph,hypergraph)
 R3_MATRIX = construct(adjacency,degree,incidence,Laplacian,transition_matrix)
 R4_STRUCTURE = analyse(degree,paths,components,cycles,centrality,communities)
 R5_SPECTRAL = compute(eigenvalues,eigenvectors,algebraic_connectivity,spectral_embedding)
 R6_LEARNING = classify(node,edge,link,graph,knowledge_graph_or_generation_task)
 R7_MESSAGE_PASSING = select(aggregation,update,attention,sampling,normalisation)
 R8_REPRESENTATION = compute(node_edge_graph_or_relation_embeddings)
 R9_COMPLEXITY = analyse(sparsity,neighbourhood_size,depth,oversmoothing,oversquashing)
 R10_VALIDATION = check(permutation_properties,split_integrity,negative_sampling,stability,task_metrics)
}

CONSTRAINTS {
 C1 = "Graph direction must not be discarded when it carries semantic meaning."
 C2 = "Disconnected components can alter spectral and diffusion behaviour."
 C3 = "High-degree nodes can dominate naïve aggregation."
 C4 = "Deep message-passing networks can suffer oversmoothing."
 C5 = "Long-range dependencies can suffer oversquashing through limited graph bottlenecks."
 C6 = "Neighbour sampling introduces stochastic approximation error."
 C7 = "Negative sampling changes the effective training distribution for link prediction."
 C8 = "Knowledge-graph embedding scores are model-specific and not universal probabilities."
 C9 = "Graph train-test splitting must avoid structural leakage."
 C10 = "Visual graph layouts do not themselves define mathematical distance or importance."
}

SEMANTIC_EQUIVALENCE {
 "GraphMathematics" <=> "MathematicsOfGraphsForAI"
 "GraphNeuralNetwork" <=> "GNN"
 "MessagePassingNeuralNetwork" <=> "MPNN"
 "GraphConvolutionalNetwork" <=> "GCN"
 "GraphAttentionNetwork" <=> "GAT"
 "GraphIsomorphismNetwork" <=> "GIN"
 "RelationalGraphConvolutionalNetwork" <=> "RGCN"
 "DirectedAcyclicGraph" <=> "DAG"
}

VALIDATION {
 V1 = "Vertex and edge semantics are explicit."
 V2 = "Direction, weights and edge types are documented."
 V3 = "Adjacency and Laplacian construction matches graph conventions."
 V4 = "Graph splits avoid edge, node or temporal leakage."
 V5 = "Permutation invariance or equivariance is tested where required."
 V6 = "Sampling procedures are documented."
 V7 = "Oversmoothing and depth effects are assessed for deep GNNs."
 V8 = "Link-prediction negatives are generated consistently."
 V9 = "Graph metrics are evaluated against appropriate baselines."
 V10 = "Sparse computational representations preserve graph structure."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 GRAPH_MATHEMATICS_DEFINITION = 0.99
 STRUCTURAL_REASONING = 0.99
 MATRIX_REASONING = 0.99
 SPECTRAL_REASONING = 0.98
 MESSAGE_PASSING_REASONING = 0.99
 GNN_REASONING = 0.99
 KNOWLEDGE_GRAPH_REASONING = 0.98
 SAMPLING_REASONING = 0.98
 POLICY = "Reduce confidence when edge semantics, direction, weights, graph operator, sampling procedure, node attributes, task split or negative-sampling process are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  GraphMathematicsReasoning,
  GraphStructureReasoning,
  AdjacencyReasoning,
  ConnectivityReasoning,
  ShortestPathReasoning,
  CentralityReasoning,
  CommunityDetectionReasoning,
  SpectralGraphReasoning,
  GraphLaplacianReasoning,
  RandomWalkReasoning,
  GraphRepresentationLearningReasoning,
  GraphEmbeddingReasoning,
  GNNReasoning,
  MessagePassingReasoning,
  GCNReasoning,
  GATReasoning,
  GraphSAGEReasoning,
  KnowledgeGraphEmbeddingReasoning,
  LinkPredictionReasoning,
  NodeClassificationReasoning,
  GraphClassificationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "graph mathematics" -> GraphMathematics
  "graph neural network" -> GraphNeuralNetwork
  "GNN" -> GraphNeuralNetwork
  "message passing neural network" -> MessagePassingNeuralNetwork
  "MPNN" -> MessagePassingNeuralNetwork
  "GCN" -> GraphConvolutionalNetwork
  "GAT" -> GraphAttentionNetwork
  "GraphSAGE" -> GraphSAGE
  "GIN" -> GraphIsomorphismNetwork
  "DAG" -> DirectedAcyclicGraph
 }
}

EXECUTION_RULES {
 E1 = "If explicit entities and relations form a mathematical graph, activate GraphMathematicsOntology."
 E2 = "If edge direction matters, preserve directed-graph semantics."
 E3 = "If diffusion, smoothness or spectral structure is central, construct an appropriate graph Laplacian."
 E4 = "If local relational learning is central, route to message-passing graph neural networks."
 E5 = "If neighbour importance varies, consider graph attention."
 E6 = "If graph scale is large, consider sparse matrices and neighbour sampling."
 E7 = "If entities and typed relations are central, route to knowledge-graph embeddings or relational GNNs."
 E8 = "If link prediction is evaluated, define negative sampling and graph splits explicitly."
 E9 = "Any graph-learning system should be validated for permutation properties, leakage, sampling effects and structural robustness."
}

RECURSION {
 ENTRY = GraphMathematics
 EXPAND(GraphMathematics) -> [
  Graph, Vertex, Node, Edge, Arc, DirectedGraph, UndirectedGraph,
  WeightedGraph, UnweightedGraph, SimpleGraph, Multigraph, Pseudograph,
  BipartiteGraph, CompleteGraph, RegularGraph, ConnectedGraph,
  DisconnectedGraph, Tree, Forest, DirectedAcyclicGraph, Hypergraph,
  HeterogeneousGraph, DynamicGraph, TemporalGraph, AttributedGraph,
  KnowledgeGraph, PropertyGraph, Adjacency, Neighbourhood, Degree,
  InDegree, OutDegree, DegreeMatrix, AdjacencyMatrix, IncidenceMatrix,
  GraphLaplacian, NormalisedLaplacian, TransitionMatrix, Path, Walk,
  Trail, Cycle, ShortestPath, GeodesicDistance, Connectivity,
  ConnectedComponent, StronglyConnectedComponent, WeaklyConnectedComponent,
  Cut, MinimumCut, Flow, MaximumFlow, Centrality, DegreeCentrality,
  BetweennessCentrality, ClosenessCentrality, EigenvectorCentrality,
  PageRank, RandomWalk, StationaryDistribution, Community,
  CommunityDetection, Modularity, GraphPartition, GraphClustering,
  SpectralGraphTheory, Eigenvalue, Eigenvector, GraphSpectrum,
  AlgebraicConnectivity, SpectralEmbedding, GraphSignal,
  GraphFourierTransform, GraphConvolution, GraphFilter, GraphLearning,
  GraphRepresentationLearning, GraphEmbedding, NodeEmbedding,
  EdgeEmbedding, GraphEmbeddingVector, GraphNeuralNetwork, MessagePassing,
  MessagePassingNeuralNetwork, AggregationFunction, UpdateFunction,
  GraphConvolutionalNetwork, GraphAttentionNetwork, GraphSAGE,
  GraphIsomorphismNetwork, RelationalGraphConvolutionalNetwork,
  GraphTransformer, AttentionOnGraphs, LinkPrediction, NodeClassification,
  EdgeClassification, GraphClassification, GraphRegression,
  GraphGeneration, KnowledgeGraphEmbedding, RelationEmbedding,
  EntityEmbedding, TransE, RotatE, DistMult, ComplEx, GraphAutoencoder,
  VariationalGraphAutoencoder, NegativeSampling, NeighbourSampling,
  SubgraphSampling, Oversmoothing, Oversquashing, Expressivity,
  WeisfeilerLehmanTest, PermutationInvariance, PermutationEquivariance,
  GraphPooling, ReadoutFunction, GraphCoarsening, GraphSparsification,
  GraphMatching, GraphIsomorphism, SubgraphIsomorphism, Homomorphism,
  GraphKernel, RandomGraph, ErdosRenyiGraph, ScaleFreeGraph,
  SmallWorldGraph, StochasticBlockModel, GraphDiffusion, HeatKernel,
  PersonalizedPageRank, LabelPropagation, GraphRegularisation
 ]
 EXPAND(GraphLearningFamilies) -> [
  SpectralGraphMethods,
  MessagePassingNetworks,
  GraphConvolutionNetworks,
  GraphAttentionNetworks,
  KnowledgeGraphModels,
  GraphGenerativeModels
 ]
 STOP_CONDITION = "Stop when graph structure, operators, connectivity, representation method, learning task, sampling, complexity and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "AI Mathematics Ontology"
 IMMEDIATE_CONTEXTS = [
  "AI Mathematics Ontology",
  "Feature Space Ontology",
  "Embedding Space Ontology",
  "Neural Network Mathematics Ontology"
 ]
 RELATED_ONTOLOGIES = [
  "AI Mathematics Ontology",
  "Artificial Intelligence Mathematics",
  "Feature Space Ontology",
  "Embedding Space Ontology",
  "Neural Network Mathematics Ontology",
  "Optimisation for AI Ontology",
  "Discrete Mathematics Ontologies",
  "Graph Theory Ontologies",
  "Linear Algebra Ontologies",
  "Probability & Statistics Ontologies",
  "Spectral Theory Ontologies",
  "Knowledge Graph Ontologies",
  "Computational Complexity Ontology"
 ]
 ROUTE_RULES = [
  "Graph Mathematics -> AI Mathematics Ontology for parent semantics",
  "Graph Mathematics -> Feature Space for attributed graph representations",
  "Graph Mathematics -> Embedding Space for node, edge and graph embeddings",
  "Graph Mathematics -> Neural Network Mathematics for GNN architectures",
  "Graph Mathematics -> Optimisation for AI for graph-learning objectives",
  "Graph Mathematics -> Discrete Mathematics and Graph Theory for structural foundations",
  "Graph Mathematics -> Linear Algebra for adjacency, Laplacian and spectral operators",
  "Graph Mathematics -> Probability & Statistics for random graphs and stochastic walks",
  "Graph Mathematics -> Spectral Theory for graph eigenstructure",
  "Graph Mathematics -> Knowledge Graph Ontologies for entity-relation semantics",
  "Graph Mathematics -> Computational Complexity for graph algorithm cost"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible graph structures, graph-learning methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve vertex-edge semantics.",
  "Preserve directed-versus-undirected distinction.",
  "Preserve weighted-versus-unweighted distinction.",
  "Preserve structural-versus-attributed distinction.",
  "Preserve spectral-versus-spatial graph-method distinction.",
  "Preserve node-edge-graph task distinction.",
  "Preserve permutation symmetry semantics.",
  "Preserve machine-readable graph-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Graph mathematics represents and analyses relational structures through vertices, edges, graph operators and graph-based learning methods."
 STRUCTURE_SUMMARY = "Adjacency, paths, components, degree and centrality describe graph structure and connectivity."
 SPECTRAL_SUMMARY = "Graph Laplacians, eigenvalues and eigenvectors encode diffusion, partitioning and spectral geometry."
 LEARNING_SUMMARY = "Graph representation learning maps nodes, edges or whole graphs into task-relevant representations."
 GNN_SUMMARY = "Graph neural networks propagate and aggregate information across graph neighbourhoods while respecting permutation structure."
 KNOWLEDGE_GRAPH_SUMMARY = "Knowledge-graph models represent typed entities and relations through structured graph embeddings."
 COMPLEXITY_SUMMARY = "Large graphs require sparse operators, sampling, scalable aggregation and careful memory management."
 VALIDATION_SUMMARY = "Graph systems should be checked for structural leakage, permutation properties, sampling effects, oversmoothing, oversquashing and task-specific accuracy."
}

FLOW {
 INPUT
 -> detect_graph_mathematics_context
 -> identify_vertices_edges_direction_weights_attributes_and_graph_task
 -> classify_simple_directed_weighted_bipartite_heterogeneous_dynamic_knowledge_graph_or_hypergraph
 -> construct_adjacency_degree_incidence_laplacian_and_transition_operators
 -> analyse_paths_components_connectivity_centrality_communities_and_spectrum
 -> classify_node_edge_link_graph_generation_or_knowledge_graph_learning_task
 -> select_spectral_embedding_message_passing_GCN_GAT_GraphSAGE_GIN_relational_or_transformer_method
 -> configure_aggregation_update_attention_sampling_and_readout
 -> train_or_compute_graph_representations
 -> assess_oversmoothing_oversquashing_sampling_error_permutation_properties_and_scalability
 -> validate_graph_splits_negative_sampling_task_metrics_and_structural_robustness
 -> route_to_AI_mathematics_feature_space_embedding_space_neural_network_mathematics_optimisation_for_AI_discrete_mathematics_graph_theory_linear_algebra_probability_statistics_spectral_theory_knowledge_graph_or_computational_complexity_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Graph Mathematics Ontology defines relational mathematics for artificial intelligence. It supports directed, undirected, weighted, bipartite, heterogeneous, temporal and knowledge graphs, vertices, edges, adjacency, degree and incidence matrices, graph Laplacians and transition matrices, paths, connectivity, cuts and flows, centrality, PageRank and random walks, communities and modularity, spectral graph theory, graph signals and graph Fourier methods, graph representation learning, graph embeddings, graph neural networks and message passing, GCN, GAT, GraphSAGE, GIN, relational GCNs and graph transformers, link prediction, node, edge and graph prediction, knowledge-graph embeddings including TransE, RotatE, DistMult and ComplEx, graph autoencoders, negative and neighbour sampling, oversmoothing and oversquashing, permutation invariance and equivariance, graph pooling, coarsening and sparsification, graph matching and isomorphism, graph kernels, random-graph models, diffusion, Personalized PageRank, label propagation and graph regularisation."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-GRAPH-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW