Network-Based Music Representations

• Network-based representations of music are formal models that map musical elements into graphs, capturing sequential and co-occurrence relationships.
• Graph metrics such as degree, clustering, and centrality quantitatively reveal compositional styles, genre evolution, and human perceptual patterns.
• These models enable practical applications in music classification, generative composition, and recommendation systems, driving both analytical and creative insights.

Network-based representations of music are formal models in which musical data—ranging from notes and chords to higher-order rhythmic, metric, or timbral events—are mapped onto graphs or networks. In these models, nodes represent musical objects (such as pitches, intervals, chords, durations, or more complex combinations), and edges typically capture sequential, co-occurrence, or similarity relationships. Over the past decade, this approach has become central for quantitative and comparative analysis in music theory, music information retrieval, cognitive science, and music technology. By extracting graph-theoretic or topological metrics from these networks, researchers can rigorously paper structure and complexity, compare genres or epochs, illuminate aspects of perception, and even develop generative and recommendation systems.

1. Fundamental Principles and Modeling Paradigms

The core construct in network-based representations is the mapping of symbolic (or audio-derived) music data into one (or more) graphs. The simplest paradigm, widely used in monophonic or polyphonic sequence analysis, assigns nodes to distinct notes or musical events, and draws a directed edge from node $x$ to node $y$ whenever $y$ immediately follows $x$ in a performance or score—a principle formalized in (Ferretti, 2016, Ferretti, 2016), and (Ferretti, 2017). If sequences are repeated, edges receive integer weights indicating transition frequency.

Nodes can represent various musical units:

• Single notes (possibly parameterized by pitch class, octave, duration)
• Rests or chords (as tuples)
• Higher-order objects, such as rhythm patterns, intervals, or multi-note combinations (e.g. n-simplices in a simplicial complex, see (Mrad et al., 10 Jun 2025))

Edges are directed in most frameworks, encoding temporal order. Some models assign weights (for frequency or transition strength), while chordal co-occurrence graphs or social/influence networks may use undirected or multiplex edges (Zhang et al., 2022).

A significant modeling choice is the feature set or “viewpoint” used to define vertices. Networks may be constructed on:

• Pitch only (resulting in a compact state space)
• Integrated features (pitch+duration, or pitch+octave+duration, etc.)
• Interval-based nodes
• Multidimensional “split” models that treat each note/chord as unique (Rosselló et al., 17 Sep 2025)

This network abstraction generalizes to higher-order frameworks using simplicial complexes. Here, $k$-simplices encode simultaneous participation of $k+1$ musical elements in a chord, with resulting topological invariants (Betti numbers, Euler characteristics) capturing polyphony and harmonic organization (Mrad et al., 10 Jun 2025).

2. Network Metrics and Quantitative Analysis

A rich toolbox of graph-theoretical and topological metrics is used to describe, compare, and interpret network-based music representations. Key measures include:

• Degree (in/out/total): Quantifies the number of transitions involving a node. In melodic sequence networks this reflects, for each note, how often it is preceded or succeeded by others (Ferretti, 2016, Ferretti, 2017).
• Degree Distribution: $P(k)$, the fraction of nodes with degree $k$, can follow a power law ($P(k) \sim k^{-λ}$), indicating scale-free structure where some notes act as “hubs.”
• Weighted Degree: Incorporates transition frequency, capturing recurrence/repetitiveness.
• Distances and Diameter: Average shortest path lengths (sometimes on the undirected version) signal how “connected” or “linear” the music is.
• Clustering Coefficient: $$C = \frac{3 \times \text{number of triangles}}{\text{number of connected triplets}}$$, interpretable musically as the tendency to group notes or chords in tightly-knit motifs.
• Centrality Measures: Betweenness and eigenvector centralities identify influential notes or chords (structurally or stylistically crucial transitions).
• Entropy: Nodes’ local or global Shannon entropy, $S_i = - \sum_j P_{ij} \log P_{ij}$, quantifies uncertainty in transitions (Rosselló et al., 17 Sep 2025, Marco et al., 13 Jan 2025, Alcalá-Alvarez et al., 1 Apr 2024).
• Modularity and Community Detection: Partitioning the network detects subgraphs (motivic, harmonic, or timbral modules).
• Small-world Coefficient: $\sigma = \frac{cc / cc_{RG}}{L / L_{RG}}$ compares network clustering and path length to random graphs, diagnosing small-world structure (Ferretti, 2017).
• Curvature and Topological Invariants: In higher-order networks (simplicial complexes), quantities such as Forman–Ricci curvature and Euler characteristic $\chi = \sum_m (-1)^m \beta_m$ provide a geometric/topological reading (Mrad et al., 10 Jun 2025).

These metrics capture stylistic and structural traits, discriminate genres, and expose both macrolevel features (e.g., modularity, scale-freeness) and microlevel (e.g., frequent “licks,” local entropy gradients).

3. Cognition, Perception, and Feature Efficiency

A critical question is how network representations relate to cognitive processing of music. Comparative studies of different modeling choices show that the feature set used to construct the network directly impacts both its complexity and alignment with human music perception (Rosselló et al., 17 Sep 2025).

• Networks constructed from single musical features (e.g., pitch-only, duration-only) more closely match human inference models: cognitive efficiency is measured by lower KL divergence between the network’s transition matrix and a “fuzzy” human-inference matrix $\hat{P} = (1–η)P(I–ηP)^{-1}$.
• Multidimensional feature combinations yield structurally complex networks but introduce “cognitive inefficiencies”: the human brain is likely to process music via modular, parallel networks (“channels”) for different features, consistent with predictive processing and free energy minimization theories.
• Local entropy gradients, $\nabla S_i = S_i - S_{funnel, i}$ (difference between a node’s local entropy and the mean entropy of its predecessors), reveal how music organizes points of structural uncertainty and directs attention toward expressive transitions—governing the alternation of tension and release.

Thus, the network’s topological and entropic structure is both a mirror of compositional style and a mechanistic model of attention and prediction in listening.

4. Application Domains: Analysis, Generation, and Retrieval

Network-based representations have been widely applied across several domains:

• Music Classification and Style Recognition: By extracting network statistics (degree distributions, centralities, clustering), machine learning models achieve high performance in classifying genres, composers, and even recognizing performers from short melodic fragments (Pimenta-Zanon et al., 2021, Ferretti, 2017, Ferretti, 2016).
• Interpretive Musicology and Comparative Studies: Quantitative network metrics distinguish between styles (Classical, Jazz, Pop, etc.), trace genre evolution, and reveal how musical complexity, diversity, and structural connectivity have shifted over centuries and under technological/market pressures (Marco et al., 13 Jan 2025).
• Generative Modeling and Algorithmic Composition: Network parameters constrain and inspire generative systems, which can create new music matching the structural “signature” of a given piece, genre, or performer (Ferretti, 2017, Nardelli, 2020, Nardelli, 2019). Higher-order networks enable more structurally faithful output by capturing polyphony and intricate harmonic interrelations (Mrad et al., 10 Jun 2025).
• Music Recommendation and Retrieval: Instrument- and tag-specific network representations permit disentangled similarity measures tailored to user controls and preferences; prototypical network models align recommendations with interpretable musical features (Hashizume et al., 21 Mar 2025, Öncel et al., 31 Jul 2025).
• Visualization and Didactics: Visualization of network graphs (nodes sized by degree, communities colored, transitions weighted) provides an effective pedagogic and analytical tool, revealing relationships often missed by traditional score-based or waveform-based representations (Alcalá-Alvarez et al., 1 Apr 2024, Kim et al., 2021).

5. Historical Evolution, Cultural Trends, and Genre Analysis

Long-term, large-scale network analyses trace the historical dynamics of musical structure:

• Analysis of $\sim$20,000 MIDI files (Marco et al., 13 Jan 2025) demonstrates that earlier Classical and Jazz works exhibit higher node counts, entropy, and efficiency, indicative of greater melodic diversity, while recent decades (especially in Pop, Rock, Hip Hop, Electronic genres) show marked simplification and homogenization. The trends are quantified using metrics such as density, weighted reciprocity, and global efficiency.
• Digital production tools and streaming platforms have contributed both to democratization (emergence of new forms) and to structural convergence within and between genres—a measurable densification and reduction in complexity.
• Influence networks (Zhang et al., 2022) provide a sociomusical perspective, mapping artist influence with temporal and centrality-adjusted weights and demonstrating correlation between network centrality and “revolutionary” shifts in genre formation.

Network representations thus serve as quantitative frameworks for studying both the microevolution of compositional practices and macroevolutionary trends in cultural history.

6. Extensions: Higher-Order and Topological Models

Recent works extend network models to capture higher-order interactions via simplicial complexes, especially in the analysis of polyphonic music (Mrad et al., 10 Jun 2025):

• Vertices (0-simplices) encode single notes, edges (1-simplices) two-note chords, triangles (2-simplices) three-note chords, and so on, adhering to closure properties.
• Algebraic topology invariants, such as Betti numbers and Euler characteristic, and geometric quantities (Forman–Ricci curvature) are computed dynamically as the piece unfolds.
• Distinct signatures in the topological evolution (e.g., exponential decay of Euler characteristic in fugues, stepwise plateaus in binary-form dances) identify genre-specific organizational schemes and compositional strategies.

This expansion from dyadic to higher-dimensional frameworks enables structural analyses that more faithfully mirror the multi-layered character of tonal and post-tonal works.

7. Future Directions and Open Problems

Primary open challenges and research directions include:

• Bridging topological and cognitive perspectives: optimizing feature selection for network construction to balance analytical richness with cognitive relevance.
• Developing real-time, interpretable network-based music generation and retrieval systems that scale to large catalogs and interact with users in meaningful ways.
• Extending network-based frameworks to non-Western, non-tonal, or multimodal musics—adapting node/edge definitions to capture distinctive structural features.
• Integrating higher-order geometric/topological methods (e.g., persistent homology) to compare works at deeper hierarchical levels.
• Analyzing the relationship between network evolution and cultural, technological, or social changes, opening up musicological insights at scale.

The theoretical and applied richness of network-based music representation continues to offer a powerful lens for studying the structure, perception, and evolution of music across disciplinary and methodological boundaries.