Emotional Trajectory Graphs

• Emotional trajectory graphs are structured models that capture the progression of affective states over time within narratives and interactive systems.
• They utilize defined nodes and edges, annotated with metrics like VAD vectors and categorical labels, to quantify key emotional shifts.
• Applications span literary analysis, film studies, dialogue systems, and customer-service evaluation to enhance understanding and predictive modeling.

Emotional trajectory graphs are structured, quantitative representations of how emotional content evolves over time within narrative, interactional, or multimodal data. They provide a formal framework for analyzing, visualizing, and comparing the progression, fluctuation, and interaction of affective states in domains as diverse as literary fiction, film, dialogue, games, and customer interaction logs. Central to these representations is the alignment of discrete or continuous emotion measures against a temporal or logical axis, enabling both micro-level and macro-level inferences concerning affective structure, variability, and impact (Vishnubhotla et al., 4 Mar 2024, Vecchio et al., 2018, Labat et al., 2023, Qiu, 11 Dec 2025, Reagan et al., 2016, Hipson et al., 2021, Wen et al., 4 Aug 2025, Christ et al., 4 Jun 2024, Tan et al., 12 Nov 2025, Qian et al., 2023).

1. Formal Structures: Definitions, Nodes, Edges, and Annotations

Across modalities, an emotional trajectory graph consists of a sequence of nodes and edges, with nodes corresponding to temporally or semantically ordered units (e.g., narrative time-points, utterances, dialogue turns, film segments, or game-story nodes). Each node is annotated with emotion attributes, which may take the form of scalar sentiment scores, valence-arousal-dominance (VAD) vectors, categorical emotion labels, or learned embeddings. Edges often reflect temporal progression, but in dialogic or interactive contexts, they may encode additional attributes such as the magnitude of emotional change (e.g., Euclidean VAD shift), operator response strategy, or cause attribution (Labat et al., 2023, Qian et al., 2023).

A canonical example is provided in customer-service dialogue modeling:

Attribute	Node	Edge
Node label	emotion label	emotional shift (VAD norm)
Numeric feature	VAD vector	operator response type
Annotation extras	cause category

$$w_{i,i+1} =\|\,e_{i+1}-e_i\|_2 =\sqrt{(v_{i+1}-v_i)^2 + (a_{i+1}-a_i)^2 + (d_{i+1}-d_i)^2}$$ (Labat et al., 2023)

In narrative domains, nodes map to windows in narrative time, with emotion computed via a lexicon (e.g., NRC-VAD, labMT) or via neural regression, and edges are implied by the order of text or structured choices (Vishnubhotla et al., 4 Mar 2024, Qiu, 11 Dec 2025, Reagan et al., 2016). In graph-based modeling of dynamic emotion in multimodal sequences, nodes are frames/segments and edges are learned adjacency weights parameterized by a neural network (Shirian et al., 2020).

2. Computational Frameworks for Graph Construction

Emotional trajectory graphs are instantiated through a sequence of algorithmic steps tailored to domain and data representation:

Text and Narrative (Books, Film, Games):

• Tokenization and windowing: Sliding word windows or sentence bins construct nodes with uniform or adaptive coverage of narrative progression (Vishnubhotla et al., 4 Mar 2024, Reagan et al., 2016, Qiu, 11 Dec 2025, Vecchio et al., 2018).
• Lexicon scoring: Each token or bin is mapped to emotion dimensions via a lexicon (e.g., NRC-VAD, labMT, syuzhet) (Vishnubhotla et al., 4 Mar 2024, Qiu, 11 Dec 2025, Reagan et al., 2016, Vecchio et al., 2018).
• Smoothing: LOESS, DCT, or moving-average filters are applied to reduce local noise and expose global structure (Vecchio et al., 2018, Reagan et al., 2016, Qiu, 11 Dec 2025, Hipson et al., 2021).
• Time normalization: Narrative progress is rescaled to [0,1] or segmented into percentiles for comparability across texts of varying length (Vishnubhotla et al., 4 Mar 2024, Vecchio et al., 2018, Reagan et al., 2016).
• Aggregation: In multiparty settings (dialogue, novels), arcs are computed per speaker or character (Vishnubhotla et al., 4 Mar 2024, Hipson et al., 2021).

Dialogue and Interactive Systems:

• Turn segmentation yields utterance-level nodes (Labat et al., 2023, Hipson et al., 2021, Qian et al., 2023, Tan et al., 12 Nov 2025).
• Annotations add categorical emotion labels, VAD triplets, cause labels, and response strategies (Labat et al., 2023, Qian et al., 2023).
• Edge weights capture affective shifts; transition matrices model Markov emotion dynamics (Labat et al., 2023, Tan et al., 12 Nov 2025).

Multimodal and Data-Driven Graph Networks:

• Frame-to-node mapping preserves temporal structure (Shirian et al., 2020).
• Node features aggregate high-dimensional descriptors (e.g., facial landmarks, MFCCs, pose angles) (Shirian et al., 2020).
• Graph parameters (adjacency, pooling) are optimized jointly with trajectory prediction or classification objectives (Shirian et al., 2020).

3. Quantitative Analysis and Metrics

A range of summary and comparative metrics have been established for trajectory graphs:

Per-curve statistics:

• Mean (μ), standard deviation (σ), home-base region definition, displacement counts and lengths, peak distances, rise and recovery rates (Vishnubhotla et al., 4 Mar 2024, Hipson et al., 2021).
• UED (Utterance Emotion Dynamics) metrics summarize mean level, variability, deviation from baseline, and return rates (Vishnubhotla et al., 4 Mar 2024, Hipson et al., 2021).

Trajectory similarity and structure:

• Pairwise Spearman correlation ρ between curves; arc similarity within and across stories (Vishnubhotla et al., 4 Mar 2024).
• Cluster assignment (e.g., via k-means) to six archetypal trajectories (Rags to Riches, Riches to Rags, Man in a Hole, Icarus, Cinderella, Oedipus) (Reagan et al., 2016, Vecchio et al., 2018).
• SVD or NMF decompositions allow for basis projection and reconstruction error analysis (Reagan et al., 2016).

Interaction and longitudinal evaluation:

• Markov transition matrices for emotional state evolution over turns (Tan et al., 12 Nov 2025).
• BEL (Baseline Emotional Level), ETV (Emotional Trajectory Volatility), ECP (Emotional Centroid Position) for evaluation in support and adaptation tasks (Tan et al., 12 Nov 2025).

4. Visualization Techniques and Interpretive Practices

Emotional trajectory graphs are conventionally visualized as one or several continuous line plots:

• X-axis: normalized narrative time, sentence or chapter index, game node position, or dialogue turn (Vishnubhotla et al., 4 Mar 2024, Qiu, 11 Dec 2025, Reagan et al., 2016, Wen et al., 4 Aug 2025, Labat et al., 2023).
• Y-axis: emotion value (scalar or vector component, e.g., valence, arousal, dominance) (Vishnubhotla et al., 4 Mar 2024, Qiu, 11 Dec 2025, Reagan et al., 2016, Labat et al., 2023).
• Multiple dimensions (V, A, D) in subplots or overlaid with color-coding (Vishnubhotla et al., 4 Mar 2024, Qiu, 11 Dec 2025).
• Color/line-style conventions: narration vs. character (distinct colors, solid/dashed), positive/negative (green/red), major/minor actors (bright/pale) (Vishnubhotla et al., 4 Mar 2024, Hipson et al., 2021).
• Confidence intervals or smoothing bands for aggregated plots (Hipson et al., 2021).
• Inflection points, peaks/troughs, annotated with vertical lines or text for interpretive emphasis (Qiu, 11 Dec 2025, Hipson et al., 2021).

Custom visualizations for more complex structures (e.g., branching DAGs in games) present sequence nodes as circles colored by emotional phase, with directed edges and triggers (Wen et al., 4 Aug 2025).

5. Empirical Insights and Story Structure

Analysis based on emotional trajectory graphs has revealed substantive empirical and structural insights:

• Emotional arcs of narration and dialogue within novels are largely uncorrelated, with arc similarities among character-pairs normally distributed around zero, indicating a plurality of overlapping trajectories, not a single canonical type (Vishnubhotla et al., 4 Mar 2024).
• The six archetypal emotional arcs (as determined by SVD, hierarchical clustering, or k-means) dominate both literary and cinematic narratives, with “Man in a Hole” (fall-then-rise) associated with heightened discussion and box office revenue, though not with maximal likability (Vecchio et al., 2018, Reagan et al., 2016).
• Movie dialogues show a “negativity bias” towards story climaxes (≈90-91% story progress), character discordance increases before narrative crises, and variability in home-base adherence separates protagonist types (Hipson et al., 2021).
• Interactive or game story generation pipelines use trajectory graphs to ensure the correspondence of gameplay difficulty and narrative emotion, with trajectory compliance verified via model and human perception (Wen et al., 4 Aug 2025).
• In customer-service, node-level emotion shifts correlate with response strategies, and trajectory graphs support predictive tasks such as next-emotion, dialogue success, or adaptive response-recommendation (Labat et al., 2023).

6. Applications, Evaluation, and Open Questions

Emotional trajectory graphs serve as analytic substrates for:

• Character arc analysis and gender/authorial difference studies (e.g., higher valence/lower arousal in female-authored/female-character dialogue) (Vishnubhotla et al., 4 Mar 2024).
• Modeling and improving empathetic dialogue via concept transition graphs, optimizing response generation for narrative-conformant and emotionally-appropriate turns (Qian et al., 2023).
• Algorithmic evaluation of AI systems’ emotional support via trajectory-based metrics capturing the stabilization and enhancement of user mood, moving beyond static, turn-level evaluation (Tan et al., 12 Nov 2025).
• Weakly-supervised modeling of continuous valence/arousal trajectories using neural regressors, informed by crowd-labeled benchmarks and evaluated by concordance correlation coefficient (CCC) (Christ et al., 4 Jun 2024).
• Applications to procedural generation, anomaly detection, behavioral analysis, and conversational agent tuning (Shirian et al., 2020, Labat et al., 2023, Tan et al., 12 Nov 2025).

Open questions concern the modeling of long-term continuity across modalities, the integration of external knowledge for trajectory prediction, the generalization across interaction types (Twitter, live chat, in-person), and the joint learning of coupled emotional and causal structures.

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Emotional trajectory graphs, through their formalization, computational instantiation, and diverse application, have emerged as foundational tools for the diachronic analysis of emotion in stories, interactive systems, and multimodal data, enabling both theoretical insight and practical advances in narrative and affective computing (Vishnubhotla et al., 4 Mar 2024, Reagan et al., 2016, Vecchio et al., 2018, Qiu, 11 Dec 2025, Labat et al., 2023, Shirian et al., 2020, Hipson et al., 2021, Wen et al., 4 Aug 2025, Christ et al., 4 Jun 2024, Tan et al., 12 Nov 2025, Qian et al., 2023).