Trope-based Graphs in Interactive Narratives

• Trope-based graphs are formal structures that encode narrative tropes as labeled nodes and relationships as labeled edges for clear story representation.
• They utilize context-free graph grammars and evolutionary algorithms, such as Constrained MAP-Elites, to systematically generate and diversify narrative architectures.
• Applications include automated narrative generation and rigorous structural analysis in games and interactive media to enhance storytelling quality.

Trope-based graphs are formal structures designed to abstract, encode, and generate narrative architectures in interactive media, particularly games, by representing recurring narrative elements—tropes—as labeled nodes and their interrelations as labeled edges. Systems such as TropeTwist mechanize this approach, utilizing context-free graph grammars and evolutionary algorithms to systematically generate, evaluate, and diversify narrative blueprints built from interconnected tropes, thereby facilitating both analytic and generative treatments of story structures (Alvarez et al., 2022).

1. Formal Semantics of Trope-Based Narrative Graphs

A trope-based narrative graph is defined as a directed, labeled graph with specialized edge semantics for representing narrative relations. The formal tuple $(V,E,\ell_V,\ell_E)$ comprises:

• $V$: a finite set of nodes, each labeled by the mapping $\ell_V: V \to \mathcal{T}$, where $\mathcal{T}$ is a finite set of tropes derived from resources such as TVTropes (e.g., HERO, MCG, NEO).
• $E\subseteq V\times V$: a set of labeled edges, with the edge labeling $$\ell_E: E \to \{\to, \leftrightarrow, \diamondsuit\text{---}\}$$. Here, $\to$ models unidirectional narrative causality, $\leftrightarrow$ models bidirectional or reflexive narrative links, and $\diamondsuit\text{---}$ encodes entailment (causal/hierarchical dependencies).

For interpretive clarity, a motif such as: $$\text{HERO} \;\to\; \text{MCG} \;\diamondsuit\text{---}\; \text{NEO}$$ denotes "the hero pursues a MacGuffin, which in turn entails the rise of the Chosen One" (Alvarez et al., 2022).

2. Graph-Grammar Foundation and Encoding

Trope-based narrative graphs are generated indirectly by means of a context-free graph grammar, formally expressed as a quadruple $V$0:

• $V$1: nonterminal symbols representing abstract subgraph configurations—effectively, narrative "slots" or unconcretized patterns.
• $V$2: terminal symbols, exactly comprising the trope set $V$3.
• $V$4: a finite set of (hyper)graph production rules $V$5, where both $V$6 and $V$7 are labeled subgraphs over $V$8. Production rules specify local subgraph rewritings according to the standard hypergraph rewriting paradigm.
• $V$9: the designated start symbol, corresponding to a proof-of-concept or root narrative graph.

Each grammar instance encodes possible expansions of abstract narrative structures into concrete graphs by sequentially applying production rules to matching subgraphs, gluing in new tropes or relations at the "open slots" defined by nonterminals (Alvarez et al., 2022).

3. Evolutionary Generation with MAP-Elites

TropeTwist employs the Constrained MAP-Elites framework to evolve populations of graph grammars, sampling and archiving narrative graphs with specified behavioral diversity and quality characteristics. Central operational details include:

• Chromosome Representation: Each evolutionary individual is a variable-length list of production rules $\ell_V: V \to \mathcal{T}$0.
• Genotype-to-Phenotype Mapping: For each individual, a random "recipe"—a list of rule indices—is sampled and applied sequentially, constructing a unique narrative graph.
• Variation Operators: Crossover exchanges either the $\ell_V: V \to \mathcal{T}$1 or $\ell_V: V \to \mathcal{T}$2 subgraph in a selected production rule; mutation can (with 10% probability) add or remove an entire rule, or (90%) locally rewire a subgraph's structure via node/edge insertion or deletion.
• Feasibility Constraints:

1. Narrative graphs must be weakly connected.
2. No Conflict Pattern may exhibit more than a single self-conflict. Infeasible individuals are tracked in a specialized MAP-Elites population, with fitness scores based on the degree of infeasibility (e.g., the number of disconnected components).

Two principal behavioral axes underpin MAP-Elites:

1. Step Distance from a root graph $\ell_V: V \to \mathcal{T}$3, measured by a Levenshtein-type edit distance on the node–edge sequences, with a cap $\ell_V: V \to \mathcal{T}$4 to limit divergence.
2. Interestingness, a weighted composite of meso-pattern metrics (Active Plot Devices, Plot Points, Plot Twists) described below (Alvarez et al., 2022).

4. Narrative Graph Evaluation Metrics

Trope-based graphs are evaluated for coherence and interestingness using a suite of graph-structural and alignment metrics:

• Coherence $\ell_V: V \to \mathcal{T}$5 incorporates:
  • Consistency $\ell_V: V \to \mathcal{T}$6: Quantifies alignment of micro-patterns (e.g., trope frequencies, conflict motifs) with a designated root graph, with penalty terms for "fake" conflict motifs.
  • Cohesion $\ell_V: V \to \mathcal{T}$7: Penalizes auxiliary or dangling narrative structures by measuring the proportion of incomplete or broken connections.

Formally: $\ell_V: V \to \mathcal{T}$8

$\ell_V: V \to \mathcal{T}$9

• Interestingness $\mathcal{T}$0 aggregates three meso-pattern attributes—Active Plot Devices ($\mathcal{T}$1), Plot Points ($\mathcal{T}$2), and Plot Twists ($\mathcal{T}$3)—with a weighted sum: $\mathcal{T}$4 where weights are $\mathcal{T}$5.

Micro-metrics such as repetition quality ($\mathcal{T}$6, penalizing trope overuse) and involvement quality ($\mathcal{T}$7, gauging participation in higher-order structures) contribute to these meso-scores, supporting fine-grained narrative pattern matching and diversity evaluation (Alvarez et al., 2022).

5. Exemplification: Root and Elite Narrative Graphs

Concrete instantiation within TropeTwist employs hand-crafted "root" narrative graphs based on existing works, against which evolved graphs are evaluated for divergence and coherence.

• Example Root Graph ($\mathcal{T}$8): For "The Legend of Zelda: Ocarina of Time"

$\mathcal{T}$9

$E\subseteq V\times V$0

• Example Elite Graph ($E\subseteq V\times V$1): An evolved variant may use

$E\subseteq V\times V$2

$E\subseteq V\times V$3

where bidirectional and entailment edges introduce new narrative reveals.

Adjacency matrices for these graphs express the edge structure explicitly, supporting algorithmic comparison and edit-distance computation.

6. Applications and Implications

Trope-based narrative graphs support systematic investigation and generative experimentation with narrative architectures in games and interactive media. By formulating narrative objects as labeled graphs governed by hypergraph grammars, and leveraging Constrained MAP-Elites for evolutionary search, designers and researchers gain access to a structured archive of quality-diverse, high-coherence narrative blueprints. A plausible implication is that this framework could facilitate both automated content generation and rigorous structural analysis in computational narratology, with extensibility to other media where narrative tropes are salient (Alvarez et al., 2022).