Unified Linguistic Space Framework

• Unified linguistic space is a conceptual framework that integrates diverse linguistic subsystems into a single multilayer network structure.
• It employs metrics like link overlap, weighted overlap, and triad significance profiles to quantify both universal and language-specific structural properties.
• The approach facilitates comparative studies by revealing universal word-level patterns and distinct subword (syllable and grapheme) organization across languages.

A unified linguistic space is a conceptual and computational framework in which diverse linguistic subsystems and their interactions are simultaneously represented and analyzed within a single, multilayer network structure. This approach models word-level and subword-level phenomena as distinct but interlinked network layers, enabling direct quantification of both subsystem-specific properties and their structural interdependencies. The framework allows for modeling, comparison, and cross-linguistic generalization of core linguistic processes such as syntax, word co-occurrence, syllabic structure, and grapheme connectivity.

1. Formal Structure of the Multilayer Linguistic Network

The unified linguistic space is operationalized as a multilayer network $M = (V_M, E_M, V, L)$, extending the formalism described in Kivelä et al. (2014):

• $V_M$ is the set of node-layer tuples, with each node residing in a specific layer of the network.
• $E_M$ comprises directed, weighted edges between nodes within or across layers.
• $V$ is the disjoint union of all lexical units (words, syllables, graphemes) across levels.
• $L = \{L_a\}_{a=1}^d$ is the set of aspects (dimensions) along which layers are distinguished. Aspects include network construction principle (e.g., syntactic, co-occurrence), linguistic subsystem (word, syllable, grapheme), and language (e.g., Croatian, English). Layers are thus indexed by tuples $$(\text{construction}, \text{subsystem}, \text{language})$$.

Word-level layers (syntax, co-occurrence, shuffle) are multiplex: nodes (words) are shared 1:1 across word-level layers; subword-level layers (syllables, graphemes) are modeled as monoplex networks with separate nodes, as word-syllable mappings are generally $N:M$.

Aspect	Nodes	Edges (directed, weighted)	Layers
Syntax	Words	Syntactic dependencies (head → dependent)	syntax-word-<language>
Co-occurrence	Words	Adjacent words in text	co-occurrence-word-<language>
Shuffle	Words	Adjacent words, but from shuffled text	shuffle-word-<language>
Syllable	Syllables	Adjacent syllables within words	syllable-<language>
Grapheme	Graphemes	Adjacent graphemes within words	grapheme-<language>

2. Layer Construction Principles and Definitions

• Nodes:
  • Word-level: Each node is a word type in the language's vocabulary (size $N_w$).
  • Subword-level: Each node is a syllable or grapheme type (sizes $N_s, N_g$).
• Edges:
  • Syntax (SIN): Edges from the head word to dependent, weight is frequency of the dependency.
  • Co-occurrence (CO): Edges between adjacent words in sentences; direction is word order, weight is frequency.
  • Shuffle (SHU): As CO but constructed from randomly shuffled sentences (vocabulary and sentence boundaries preserved).
  • Syllable (SYL): Edges between sequential syllables within each word; direction is left-to-right, weight as frequency.
  • Grapheme (GR): Analogous to SYL but at the grapheme level.

This formalization enables rigorous comparison of structural properties both within and across subsystems.

3. Quantitative Measures for Inter-Layer Similarity and Structure

To evaluate how similar or distinct different subsystems (layers) are, several mathematically defined metrics are employed:

a) Link Overlap (Jaccard Index)

For layers $\alpha$ and $\alpha'$: $$J(E_\alpha, E_{\alpha'}) = \frac{|E_\alpha \cap E_{\alpha'}|}{|E_\alpha \cup E_{\alpha'}|}$$ This measures the proportion of shared edges.

b) Preserved Weighted Overlap (WO)

First, the preserved weighted ratio for intersected links: $$PW(E_\alpha, E_{\alpha'}) = \sum_{i,j} \frac{\min(w_{ij}^\alpha, w_{ij}^{\alpha'})}{\max(w_{ij}^\alpha, w_{ij}^{\alpha'})}$$ Then normalized by number of overlapping edges: $$WO(E_\alpha, E_{\alpha'}) = \frac{PW(E_\alpha, E_{\alpha'})}{|E_\alpha \cap E_{\alpha'}|}$$ High WO indicates not only structural similarity but also quantitative agreement in edge weights.

c) Motif Analysis (Triad Significance Profile)

Directed triads (three-node subgraphs) are enumerated for each layer. The Z-score for each motif $i$ is: $$Z_i = \frac{N_i^{\text{orig}} - \langle N_i^{\text{rand}} \rangle}{\sigma_i^{\text{rand}}}$$ Normalize to obtain the triad significance profile (TSP): $TSP_i = \frac{Z_i}{\sqrt{\sum_i Z_i^2}}$ Correlations between TSPs across layers assess local structural similarity.

4. Empirical Insights and Linguistic Patterns

a) Word-Level Universality vs. Subword Diversity

• Word-level layers (syntax, co-occurrence, shuffle) exhibit high preserved weighted overlap (WO ~90% in both Croatian and English), suggesting shared structural principles—robust degree distributions, similar motif spectra—across Indo-European languages.
• Syllabic and graphemic subword layers are highly language-dependent; e.g., Croatian (inflection-rich, syllabically regular) shows denser, more clustered syllable networks than English.

b) Subsystem Interaction

• High WO between syntax and co-occurrence layers indicates that syntactic dependency structure is strongly mirrored in word adjacency patterns.
• TSP correlations reveal that syntactic and syllabic layers, although modeling distinct aspects, share unexpectedly close local topological organization, suggesting universal processing constraints or cognitive pressures.

5. Theoretical Implications and Scope of Unified Linguistic Space

The multilayer framework substantiates a unified linguistic space at several levels:

• Subsystems are modeled jointly: The architecture captures both the autonomy of subsystems and the systematicity of their interaction.
• Structural differences and universals are quantifiable: Sensitive metrics distinguish universal structural parameters (degree, selectivity, motif profiles) from subsystem- and language-specific effects.
• Comparative linguistics: Cross-language applications reveal both shared core properties at the word level and distinctive subword organization, advancing empirical typology.

The approach transcends the limitations of isolated linguistic network analyses, enabling integrations relevant to language theory, typology, language evolution, and the interplay between linguistic structure and cognitive processes.

6. Applications and Future Research Directions

• Empirical validation: The framework supports systematic exploration of linguistic universals, subsystem divergence, and the influence of morphological, phonological, or syntactic properties in large text datasets.
• Comparative and evolutionary studies: The model is extensible to additional layers (morphology, semantics) and further languages, supporting hypotheses about universal cognitive pressures and the evolution of linguistic complexity.
• Computational and cognitive modeling: The unified linguistic space offers a foundation for theories of language acquisition, processing (e.g., co-activation of syntactic and phonological cues), and the development of realistic language technology architectures.

Summary Table: Key Metrics and Findings

Measure	Formula / Role	Insight
Link Overlap	$J(E_\alpha, E_{\alpha'})$	Edge-level structural similarity
Weighted Overlap	$WO(E_\alpha, E_{\alpha'})$	Quantifies weight/frequency alignment
Triad Profile	$TSP_i$ (normalized motif Z-scores)	Local (motif-scale) structural similarity
Multiplexity	1:1 node matching across word-level layers	Node (entity) preservation across subsystems
Language dependence	Word-level: low; subword-level: high	Subsystem specificity vs. cross-linguistic universals

A pivotal implication is that the unified linguistic space approach systematically quantifies both the universality and diversity of linguistic networks, providing a powerful instrument for advanced linguistic analysis, cognitive modeling, and comparative studies.