Relational and Graph Behavioral Representations

Relational and Graph Behavioral Representations are structured scientific representations in which identified entities and declared relations among them are explicit parts of the scientific object. These representations embed node or entity identity, edge or relation identity, topology, directionality, weights or other edge values, relation types, attributes, temporal or support semantics, and graph-level schema as integral components. It is critical to distinguish that terms such as relation value, relation matrix, edge, graph, network, topology, and interaction are not synonyms. A graph representation organizes relational evidence by explicitly encoding entities as nodes and relations as edges or higher-order linkages; however, the existence of a graph does not by itself establish behavioral interaction, social connection, physical coupling, communication, influence, leadership, or causality.

Meaning and Relational Object Identity

A Relational Representation is an explicit organization of declared relations among identified entities. That is, entities are specified scientific objects, and relations between them are scientifically defined and declared. A Graph Representation is a special case of a relational representation in which entities are represented as nodes and declared binary or higher-order relations are represented through edges, hyperedges, incidence structures, layers, or other explicit relational constructs governed by a declared graph schema.

The identity of such a representation depends on the node set (the entities represented), the relation set (the declared relations included), endpoint roles (how entities participate in relations), relation-presence semantics (what it means for a relation to be present or absent), attributes (properties of nodes, edges, or the graph), support or validity conditions (e.g., temporal or measurement contexts), and the overall schema that governs allowed structures and types.

The following distinctions clarify core terminology:

A generic attributed graph abstraction can be formalized as:

where:

• G is one declared Graph Representation Instance,
• V is the set of nodes representing entities,
• E is the declared set of edges or relation instances,
• XV are optional node-level attributes,
• XE are optional edge-level attributes,
• XG are optional graph-level attributes.

Additional schema elements such as time, layer, relation type, uncertainty, hyperedge membership, or other domain-specific structures may require an enriched or more complex definition. The tuple above is a compact abstraction, not a universal ontology.

The boundary between relational descriptors and relational representations is fundamental. Relational descriptors such as correlation, coherence, mutual information, transfer entropy, event coincidence, recurrence, or geometric proximity can supply edge evidence or attributes when scientifically justified. However, constructing a graph from such descriptors does not redefine the mathematics or interpretation of the underlying relation. Conversely, the mere existence of a pairwise relation value between two entities does not mandate that it become a graph edge; inclusion depends on declared representation policies and scientific context.

Node, Edge, and Schema Semantics

Stable entity identity refers to the persistent scientific object recognized across contexts and representations. In contrast, graph-local node identity is a representational artifact such as a node key or index that may vary across serializations, reorderings, or implementations. Entities can be participants, sensors, channels, joints, body parts, events, regions, modalities, behavioral states, descriptor components, or abstract objects. Because node indices or positions (e.g., adjacency matrix rows/columns) can change, these must not be used as sole durable identifiers of the underlying entity.

Node membership states extend beyond simple presence or absence and include:

• Present-connected: Node is present and has at least one incident edge.
• Present-isolated: Node is present but has no incident edges.
• Absent by definition: Node not included in the graph instance by schema or policy.
• Unobserved: Node existence or participation is unknown or missing.
• Unavailable: Node cannot be observed or measured in the current context.
• Conditionally present: Node presence depends on external conditions or filters.
• Structurally inapplicable: Node cannot participate in any edge under the schema.

Absence of incident edges does not imply absence of the represented entity. Removing isolated nodes changes representation semantics when node presence itself encodes scientific information.

Edge identity is defined by:

• Endpoint identities and their roles (e.g., source, target, participant).
• Relation type or definition.
• Directionality (directed or undirected).
• Edge key or index when parallel relations between the same endpoints are allowed.
• Support dimensions such as time, layer, or context.
• Provenance or versioning information.

Endpoint pairs alone are insufficient to uniquely identify edges when multiple relation types, supports, frequency bands, contexts, or measurement sources coexist between the same entities.

A Graph Schema defines:

• Allowed node/entity types.
• Allowed relation/edge types.
• Endpoint-type constraints (which node types can connect via which edge types).
• Directionality policies.
• Self-loop permissions.
• Parallel-edge policies.
• Attribute schemas for nodes, edges, and graph.
• Temporal or layer semantics.
• Higher-order relation policies.
• Missingness and default-value semantics.
• Graph-level support and validity.

Schema is distinct from any single observed Graph Instance; one instance cannot reveal every legal or prohibited structure.

Edge Presence, Direction, Weight, and Uncertainty

Edges can exist in several distinct states:

• Present: Edge is declared and included in the graph.
• Absent: Edge is declared not present (explicitly excluded).
• Unobserved: Edge presence is unknown due to missing data or measurement failure.
• Below inclusion threshold: Edge relation value exists but does not meet inclusion criteria.
• Censored: Edge value is partially or noisily measured and adjusted.
• Invalid: Edge estimate is known to be erroneous or unreliable.
• Not applicable: Edge is prohibited by schema or domain rules.
• Structurally prohibited: Edge cannot exist for scientific or definitional reasons.

Sparse omission of edges is not self-interpreting: an absent or unstored edge can mean measured absence, zero value, unavailable observation, threshold exclusion, or simply no materialized edge under the representation policy.

Directedness matters:

• Undirected edges represent relations where endpoint order is irrelevant under the adopted semantics.
• Directed edges preserve ordered source→target roles, which can differ in semantics or measured values.

A reciprocal pair of directed edges is not equivalent to one undirected edge because the two directions can carry distinct values, supports, types, or uncertainty.

A generic directed weighted adjacency convention is:

where:

• u is the declared source-node identity,
• v is the declared target-node identity,
• Auv is the adjacency or relation-matrix entry under the source-row/target-column convention,
• w(u→v) is the declared edge value if the edge exists, or a schema-defined default/masked state otherwise.

Note that some software frameworks use the opposite row/column convention. For an undirected relation, symmetry Auv = Avu holds only when the relation is symmetric. Adjacency matrix position never replaces stable node identity.

Edge values encode various semantic quantities, for example:

• Strength or similarity,
• Distance or cost,
• Count or duration,
• Probability-like quantities,
• Signed relations,
• Descriptor values in a declared domain.

Larger numeric values can mean stronger relations in one graph and greater separation or cost in another. It is essential to preserve units, domain, sign convention, normalization, reference states, and any transformation from the originating relational quantity.

Edge relation values are distinct from uncertainty or quality measures such as:

• Confidence,
• Statistical uncertainty,
• Edge-existence probability,
• Posterior probability,
• p-value or significance evidence,
• Quality score or validity.

Several of these attributes may coexist on one edge. Overloading these distinct concepts into a single generic weight field destroys interpretability.

Graph Forms and Relational Domain Structure

Graph structures vary by directionality and multiplicity of edges:

• Simple Undirected Graphs: Edges are undirected with no parallel edges allowed.
• Simple Directed Graphs: Edges are directed, no parallel edges allowed.
• Multigraphs: Allow parallel edges between the same unordered node pair; undirected.
• Directed Multigraphs: Allow parallel edges between the same ordered node pair.
• Bipartite/Multipartite Graphs: Nodes are partitioned into disjoint types or roles; edges only allowed between specific partitions.
• Heterogeneous Graphs: Contain multiple node and edge types with type-specific constraints.
• Hypergraphs/Higher-Order Structures: Edges (hyperedges) relate more than two nodes simultaneously.

When multiple scientifically distinct relations share endpoints, representation should use separate typed attributes, layers, or parallel edges as specified by the schema. Silent overwriting or aggregation risks misrepresenting distinct relations.

Self-loops are valid only when relation semantics admit self-relation, persistence, self-transition, or within-entity relation. Self-loops differ from diagonal entries in a relation matrix; diagonal entries may encode unity, zero, variance, self-similarity, or missingness without necessarily corresponding to a graph edge.

Bipartite and multipartite graphs restrict node membership to declared disjoint role/type sets, constraining admissible edges. Examples include participant↔event, sensor↔modality, person↔object, or descriptor↔context relations. Node-set membership is a graph schema property and must not be inferred solely from node naming, index ranges, or observed edges.

Homogeneous graphs share one node and relation schema; typed graphs retain node or edge classes within a domain; heterogeneous graphs contain multiple node and relation types with type-specific endpoint and attribute rules. Numeric edge values across different relation types are not automatically comparable even if stored under the same field name.

Higher-order relational structures involve hyperedges relating multiple entities simultaneously. Incidence representations retain explicit membership of nodes in such hyperedges. Group relations should not be decomposed into pairwise edges unless scientifically justified; decomposition can invent pairwise semantics not present in the original relation.

Attributes, Matrix Views, Incidence, and Serialization

Attributes exist at multiple semantic scopes:

• Node Attributes: Describe node-specific state, geometry, modality, quality, or context.
• Edge Attributes: Describe relation value, type, lag, frequency band, support, confidence, or uncertainty.
• Graph-Level Attributes: Describe common support, construction policy, population or group identity, coordinate frame, layer schema, or provenance.

Varying node or edge evidence must not be stored only as graph-global metadata, as this loses entity- or relation-specific interpretability.

An edge-list representation consists of rows specifying stable endpoint IDs, optional edge keys or types, edge values, support/time stamps, attributes, and validity flags. Row order typically reflects serialization order rather than topology or temporal order unless explicitly declared.

An adjacency or relation-matrix view is a node-by-node indexed matrix whose rows and columns map to node identities, and entries encode edge presence or relation values under a declared convention. Adjacency views differ from pairwise relation matrices lacking explicit edge-existence or topology semantics. Node permutation changes numeric layout but preserves graph identity if mapping is consistent.

A generic node-edge incidence abstraction is:

where:

• v is a node identity,
• e is an edge or higher-order relation identity,
• Bve is the incidence matrix entry,
• ρ(v, e) is the declared incidence rule, e.g., membership or directed endpoint role.

Signed source/target conventions vary by domain and must be declared explicitly rather than imposed universally. Incidence indexes nodes against relation instances, not against nodes directly.

Alternative structural views or serializations such as edge lists, adjacency matrices, incidence matrices, sparse coordinate formats, node-plus-edge tables, or object-graph forms represent different perspectives on the same relational information. Conversion between forms can be lossy when the target cannot represent multiedges, typed relations, isolates, uncertainty, or higher-order memberships.

Construction, Thresholding, Sparsification, and Topology

Graph construction from relational evidence maps eligible entity pairs or higher-order entity sets and declared relation evidence into candidate relations, followed by an explicit edge-inclusion policy. This process preserves endpoint identity, relation definition and version, direction, support, validity, value, uncertainty, and any transformation applied before inclusion. The act of graph construction inherently changes organizational or topological semantics even when edge values directly reuse existing relational descriptors.

Edge-inclusion policies include:

• All valid relations: Include every scientifically valid relation.
• Domain-rule inclusion: Include edges based on scientific rules or constraints.
• Absolute or signed threshold: Include edges if relation value exceeds a threshold.
• Fixed-density rule: Include top edges to achieve a fixed density.
• Top-k/Nearest-neighbor-like rule: Include edges to the k strongest or nearest neighbors.
• Statistical-evidence rule: Include edges based on significance or confidence criteria.

Threshold value, comparator, sign handling, tie rules, units/domain, missingness treatment, directionality, and whether the threshold is fixed or adaptive constitute the Graph Definition. Changing these changes topology.

Representational sparsification intentionally retains a subset of otherwise eligible relations to achieve structural, interpretive, storage, or computational goals. Sparsification changes topology and can alter connectivity, degree distribution, path structure, components, or downstream graph descriptors. It is not a neutral storage optimization unless omitted edges remain recoverable and their semantics preserved outside the sparse graph.

Threshold and selection sensitivity refer to the fact that small perturbations near an inclusion boundary can switch edges on or off, causing abrupt topology changes even if underlying relation values change smoothly. Near-threshold relations, uncertainty, alternative defensible thresholds, and topology sensitivity must be preserved when scientific interpretation depends on stable relational structure.

Temporal, Layered, Hierarchical, and Dynamic Graph Structure

Temporal and dynamic graph representations encode graphs in which node presence, edge presence, relation values, attributes, or support vary over time or episodes. These include:

• Sequences of graph snapshots at discrete times,
• Interval-valid edges active only during specified periods,
• Timestamped edge events marking relational occurrences,
• Continuously varying attributes over time,
• Aggregated graphs over temporal support.

Such temporal graph representations record changing relational structure but do not by themselves constitute dynamical models or imply transition laws.

Multilayer and multiplex graph representations retain relation layers corresponding to modalities, frequency bands, contexts, time scales, or other declared domains while preserving layer identity. Interlayer relations and intralayer relations often have different semantics. Collapsing layers by sum, mean, maximum, or logical union produces new representations whose edge meaning must be explicitly defined.

Hierarchical relational representations arise through node groups, supernodes, nested structures, part–whole relations, or multiple resolutions when scientifically defined. Coarsening or aggregation must preserve membership and lineage because an edge between groups can represent aggregated member relations, a separately defined group relation, or another construction. Group-level topology should not be projected back automatically to every member pair.

The following inline SVG illustrates five identified entities represented in four relational forms: Pairwise Relation Matrix, Undirected Weighted Graph, Directed Typed Graph, and Two-Layer Temporal/Context Graph. Stable entity labels (A, B, C, D, E) are consistent across forms. The matrix contains relation values but graph topology appears only after an edge-inclusion rule; one isolated node E is included; one reciprocal directed pair with unequal edge values (A→B and B→A); and one relation present in only one layer (C→D in Layer 2). Labels highlight critical distinctions.

Missingness, Comparability, Quality, and Provenance

Missingness, uncertainty, and comparability are essential considerations in relational and graph representations.

Different missingness states include:

• Absent edge: Edge is known to be absent by definition or observation.
• Unobserved relation: Edge presence is unknown due to missing data.
• Unavailable endpoint: Node or entity unavailable for measurement or inclusion.
• Invalid relation estimate: Edge value known to be unreliable or erroneous.
• Threshold-excluded relation: Edge removed due to edge-inclusion policy.
• Censored value: Edge value partially observed or adjusted.
• Missing attribute: Node or edge attribute unmeasured or omitted.
• Structurally prohibited relation: Edge forbidden by schema or domain.

Direct comparison across Graph Instances requires:

• Compatible entity/node schema,
• Relation definitions and versions,
• Node-set membership policies,
• Edge-inclusion policies,
• Directionality conventions,
• Weight/value semantics,
• Support for time/layers,
• Missingness and uncertainty handling,
• Representation versioning.

Equal node count or density does not establish comparability.

Integrated Worked Comparison: Behavioral Episode Example

Consider a behavioral episode involving five entities:

• Participants: P1, P2
• Wearable Sensors: S1, S2
• Task Object: O1

Declared relational descriptors include:

• Pairwise correlations between physiological signals,
• Binary co-presence events,
• Directed predictive relations,
• Sensor-modality mappings,
• Temporal context separations.

1. Complete Pairwise Relation Matrix
   A full 5×5 matrix with entries representing correlation coefficients or relation values. This remains a matrix representation without imposed edge presence semantics.

2. Undirected Weighted Graph
   Using symmetric correlation values above a threshold to include edges. Node O1 (object) is present but isolated due to no significant correlations, illustrating that isolated nodes remain scientifically meaningful.

3. Directed Graph
   Using ordered predictive relations (e.g., Granger causality) between participants and sensors, edges are directed and may show reciprocal edges with different strengths, e.g., P1 → S1 stronger than S1 → P1.

4. Bipartite Participant↔Event or Sensor↔Modality Graph
   Nodes partitioned into sensor and modality types, edges represent sensor activation of modalities. This graph schema restricts edges to cross partitions only.

5. Two-Layer Graph Separating Relation Types or Contexts
   Layer 1 encodes communication relations; Layer 2 encodes physiological coupling. Some relations appear only in one layer, preserving layer identity.

This example features:

• Stable entity IDs distinct from local node indices,
• Presence of an isolated but present node (O1),
• Edges absent by thresholding versus unobserved edges due to missing data,
• Zero relation values distinguished from no edge presence,
• Reciprocal directed edges with differing values,
• Parallel edges representing distinct relation types (e.g., communication and influence),
• A threshold-induced topology change when edges near the cutoff appear or disappear,
• A lossy matrix-to-edge-list conversion if metadata such as attributes or support are omitted,
• Temporal or layer changes that should not be interpreted as dynamical laws without further modeling,
• A boundary case: graph centrality, community structure, path statistics, or embeddings are analyses derived from the graph, not the graph representation itself.

A provenance audit preserves:

• Representation definition and version,
• Stable entity IDs,
• Node keys and types,
• Node membership policy,
• Edge endpoint identities and roles,
• Relation types and definitions/versions,
• Directionality and edge key or parallel-edge policy,
• Self-loop policy,
• Edge presence and default semantics,
• Weight/value units, sign, and meaning,
• Confidence, uncertainty, and quality attributes,
• Node, edge, and graph attributes with their supports,
• Graph support and construction/inclusion policy,
• Sparse representation semantics,
• Matrix, incidence, edge-list serialization and node order,
• Temporal, layered, and hierarchical structure,
• Higher-order relation semantics,
• Missing, invalid, or censored states,
• Implementation and version information,
• Comparability constraints and sensitivity findings.

A defensible relational or graph behavioral representation states explicitly which entities exist, which relations are represented, what makes a relation an edge, what every edge value means, and which aspects of topology are observed, constructed, uncertain, or intentionally omitted. This transparency is essential for scientific validity and interpretability.