 Regular article
 Open Access
Topology and evolution of the network of western classical music composers
 Doheum Park^{1},
 Arram Bae^{1},
 Maximilian Schich^{2} and
 Juyong Park^{1}Email author
https://doi.org/10.1140/epjds/s136880150039z
© Park et al.; licensee Springer. 2015
 Received: 8 November 2014
 Accepted: 9 April 2015
 Published: 22 April 2015
Abstract
The expanding availability of highquality, largescale data from the realm of culture and the arts promises novel opportunities for understanding and harnessing the dynamics of the creation, collaboration, and dissemination processes  fundamentally network phenomena  of artistic works and styles. To this end, in this paper we explore the complex network of western classical composers constructed from a comprehensive CD (Compact Disc) recordings data that represent the centuriesold musical tradition using modern data analysis and modeling techniques. We start with the fundamental properties of the network such as the degree distribution and various centralities, and find how they correlate with composer attributes such as artistic styles and active periods, indicating their significance in the formation and evolution of the network. We also investigate the growth dynamics of the network, identifying superlinear preferential attachment as a major growth mechanism that implies a future of the musical landscape where an increasing concentration of recordings onto highlyrecorded composers coexists with the diversity represented by the growth in the sheer number of recorded composers. Our work shows how the network framework married with data can be utilized to advance our understanding of the underlying principles of complexities in cultural systems.
Keywords
 complex network
 classical music
 network topology
 network evolution
1 Introduction
Networks have been used extensively in recent years for characterizing and modeling intricate patterns found in various social, technological, and biological complex systems originating from the functional and dynamical interdependence between their components. The methodology is expanding its horizon, being eagerly adopted in new fields such as culture for exploring novel answers to new and longstanding issues [1–4]. In broad terms, the potential of the network framework for understanding culture originates from the observation that the creation and transmission of cultural products are essentially network phenomena. Networks, therefore, may lead to a new fundamental understanding of the complex nature of culture.
There are several interesting questions one can explore regarding such ‘network in culture’  e.g., How does it evolve and change over time? Who are the most prominent or popular artists, and how do we measure their importance? How do the different styles combine to produce cultural products? How does information flow over network ties? [5]  the answers to which would contribute to deepening our understanding of culture and the arts.
There have been several notable works that highlight the importance of networks on understanding various issues in culture. Suárez, Sancho and de la Rosa [6] analyzed the linkage patterns from the data of 11,443 artworks from Spain and Latin America with respect to genre and theme, identifying religious theme to be a dominant factor connecting the paintings. Gleiser and Danon [7] studied the topology and the community structure of the collaboration network of Jazz musicians, uncovering the presence of communities based on the locations of the bands correlated with the racial segregation between the musicians. Park et al. [8] compared two networks of contemporary popular musicians  one representing the collaboration history of the musicians, and the other representing musical similarities as judged by human experts  finding that the significant topological differences are related closely to the nature of the connections. Salganik et al. [9] studied the role of social influence and success of cultural products. They found that social influence has a significant effect, elevating the inequality and the unpredictability of success. Uzzi and Spiro [10] investigated the smallworld property of the network of the creators of original Broadway musicals that reflect the level of their cohesion and its impact on the success of the musicals. They found that cohesion had a positive impact up to some point, but began to impede the creativity when it becomes too large. The characteristics of a system and growth or evolutionary dynamics are deeply intertwined [11]. This is likely to be the case for cultural system as well, and grasping the underlying growth principles may need a better understanding of their nature. This line of thinking is behind the work of Jeong et al. [12] who applied network evolution models to the movieactor network. Recently in the field of music, Serrà et al. [13] constructed a timevarying network of pitch transitions in contemporary western popular music (from 1955 to 2010) and analyzed the degree distributions of pitches, pointing out the limited use of various pitch transitions.
In this paper we analyze the cooccurrence network of western classical composers constructed from the comprehensive data set of CD recordings. Using the network framework, we try to shed light on the following specific questions in order: Who are the prominent composers? What are the driving force behind composers being cofeatured on a common CD? Can we characterize the temporal growth of the network and the composers? We start by measuring the fundamental network properties that give us a bird’seye view of the general features of the network, including the size, clustering coefficient, assortativity, degree distribution, centralities, and community structure. The centralities identify influential composers with varying artistic styles, while the positive assortativity shows that period designations and artistic styles are the main driving force behind cofeatured composers. This is explored further using the community structure of the composers. We characterize the growth of the composerCD network by way of the temporal evolution of the bipartite degree distribution, and how it can be mapped to a superlinear preferential attachment model. It also allows us to forecast the future of the landscape of the composer network, where the growths of the degrees of prominent composers are accelerating and accordingly the recordings are becoming increasingly concentrated on those already well established, an effect similar to the socalled urbanscaling laws, where existing cultural centers win out over new centers [4, 14].
2 Data and network construction
3 Topology of the composer network
First we examine the fundamental properties of the composer network to understand its general features. We then analyze the relationship between the characteristics of connected composers to study how they affect the chances of connection, i.e. becoming featured on a common recording. We investigate the centralities, the assortativity, and the community structure of the composercomposer network to answer the question of who are the prominent composers, and what drives the formation of connection patterns.
3.1 Fundamental network properties reveal the smallworld property, a high clustering, and wide differences in composer prominence
Many networks exhibit the socalled ‘smallworld’ property [16], often expressed using ‘six degree of separation’ in common parlance. Mathematically defined as the mean geodesic distance (the length of the shortest path) between nodes increasing sublinearly (typically logarithmically) as a function of the size of the network, it is frequently represented by a small mean geodesic distance (a geodesic is the shortest path between two nodes. If they are directly connected, the geodesic distance is 1). Technically speaking the two definitions are unidentical [17], although for the purpose of our paper the distinction is not critical. The mean geodesic distance and the diameter (the longest geodesic) in the largest component (set of connected nodes) are 13.1 and 20 for the CDcomposer network, and 3.5 and 20 for the composercomposer network, respectively.
The clustering coefficient C is the probability that two nodes connected to a common neighbor are themselves neighbors. It is a measure of the abundance of triangles in the network, and a high value is a hallmark of social networks [18]. In our composercomposer network we have \(C=0.648\) compared with the random expectation \(\overline{C}_{\mathrm{random}}=0.001\), meaning that our network is indeed highly clustered. Such a clustered behavior is deeply related to patterns of mixing and community structure, which we later discuss in detail.
Top 20 composers for degree, eigenvector, and betweenness centralities
Rank  Bipartite degree ( q )  Projected degree ( k )  Eigenvector centrality  Betweenness centrality  

Name  Period  Name  Period  Name  Period  Name  Period  
1  WA Mozart  C  JS Bach  B  JS Bach  B  JS Bach  B 
2  JS Bach  B  WA Mozart  C  WA Mozart  C  WA Mozart  C 
3  Beethoven  R  Handel  B  Handel  B  Handel  B 
4  Brahms  R  Brahms  R  Brahms  R  Piazzolla  M 
5  Schubert  R  Mendelssohn  R  Mendelssohn  R  Brahms  R 
6  Verdi  R  Debussy  M  Schubert  R  Gershwin  M 
7  Tchaikovsky  R  Schubert  R  Debussy  M  Debussy  M 
8  R Schumann  R  Beethoven  R  Beethoven  R  Mendelssohn  R 
9  Handel  B  SaintSaëns  R  SaintSaëns  R  Schubert  R 
10  Wagner  R  Tchaikovsky  R  Tchaikovsky  R  Beethoven  R 
11  Chopin  R  Ravel  M  Ravel  M  VillaLobos  M 
12  Haydn  C  Gershwin  M  Fauré  R  Ravel  M 
13  Liszt  R  R Schumann  R  R Schumann  R  Tchaikovsky  R 
14  Mendelssohn  R  Fauré  R  Liszt  R  Copland  M 
15  Debussy  M  Liszt  R  Chopin  R  SaintSaëns  R 
16  Puccini  R  Vivaldi  B  Vivaldi  B  Vivaldi  B 
17  Vivaldi  B  Piazzolla  M  Rossini  R  Stravinsky  M 
18  Dvor̆ák  R  Rossini  R  Rachmaninoff  M  Britten  M 
19  Ravel  M  Chopin  R  Haydn  C  Hindemith  M 
20  R Strauss  R  Verdi  R  Gershwin  M  Bernstein  M 
3.2 Composer centralities reveal the relationship between the network and composer characteristics
Next, we study further types of composer centralities in the composercomposer network that capture various ‘flavors’ of a node’s importance in a network.
Besides the degree that we have already seen, the Eigenvector Centrality and the Betweenness Centrality (also called the Freeman Centrality after its inventor) [19, 20] are widely used. The eigenvector centrality is a generalization of the degree that considers the ‘quality’ of a connection: Being a neighbor to a central node in turn raises one’s own eigenvector centrality. The name comes from its mathematical definition as the components of the leading eigenvector of the adjacency matrix. The betweenness centrality measures how often a node sits on the geodesic between two nodes, acting as an intermediary (e.g., in communication). One benefit of investigating different centralities is that while the centralities are often correlated, significant disagreements can point to unusual aspects of the network that in turn can lead to a deeper understanding of it. The highestranked composers in each centrality are given in Table 1. The lists do appear correlated, with Spearman Rank Correlations (SPR) equal to \(0.753\pm0.002\) between degree k and eigenvector centrality, and \(0.784\pm0.003\) between degree k and betweenness. But composers labeled ‘M’ for Modern (the composer’s period, to be discussed later) are ranked significantly higher in betweenness than in other centralities. It turns out that Modern composers form a tightknit group with many connections between them, elevating the betweenness of prominent Modern composers such as Aaron Copland (19001990) and Leonard Bernstein (19181990) although their degree is significantly lower than those from other periods. In order to understand the implications of this type of relationship between a node attribute and network topology, next we review the common period designation in western classical music and analyze them further.
3.3 Common artistic style and period designations in western classical music
 1.
Medieval (5001400). The period when primeval shape of musical notation appeared, along with advances in tonal material, texture, and rhythm. Polyphony took shape in terms of tonal material [21]. Notable composers include Guillaume de Machaut (13001377) and Francesco Landini (13251397).
 2.
Renaissance (14011600). The period of modes and rich textures in four or more parts blending strands in the musical texture, harmony, and progression of chords [27]. Notable composers include Thomas Tallis (15051585), William Byrd (15401623), and John Dowland (15631626).
 3.
Baroque (16011750). The period distinguished by the creation of tonality. During this period, composers used elaborate musical ornamentation and made changes in musical notation. Baroque music became more complex and expanded the range of instrumental performance [23]. Notable composers include Henry Purcell (16591695), Antonio Vivaldi (16781741), Johann Sebastian Bach (16851750), and George Frideric Handel (16851759).
 4.
Classical (17301820). The period characterized by a lighter, clearer texture than Baroque. Variety and contrast within a piece became more pronounced than before, and melodies tended to be shorter, with clearcut phrases and clearly marked cadences [28]. Notable composers include Wolfgang Amadeus Mozart (17561791) and Franz Joseph Haydn (17321809).
 5.
Romantic (18151910). The period when music was closely related with romanticism, the artistic and literary movement in Europe [29]. Romantic music is characterized by freedom of form, emotions, individuality, dynamic changes and nationalism. Notable composers include Ludwig van Beethoven (17701827), Franz Schubert (17971828), Frédéric Chopin (18101849), Robert Schumann (18101856), Franz Liszt (18111886), and Pyotr Ilyich Tchaikovsky (18401893).
 6.
Modern (1900current). The period characterized by musical innovations in organizing and approaching harmonic, melodic, sonic, and rhythmic aspects leading to many novel styles including expressionism, abstractionism, neoclassicism, futurism, etc. [30]. The rise of American classical music was also significant. Notable composers from this period include Claude Debussy (18621918), Maurice Ravel (18751937), Sergei Rachmaninoff (18731943), Igor Stravinsky (18821971), George Gershwin (18981937) and Leonard Bernstein (19181990).
The composer metadata (period and active years) were available for 878 composers, leaving us with 13,667 edges between those in the composercomposer network. While accounting for 6.3% of the entire composer group, these are still the most prominent and significant ones who would be of primary interest; the average bipartite degree for this group is \(\overline{q}=64.8\), nearly twenty times larger than the remainder for which \(\overline{q}=3.5\).
3.4 Assortativity and community structures reveal artistic styles and periods as the main factor behind connections between composers
The relationship between node characteristics and network topology can be quantified by the assortativity coefficient for discrete node characteristics [31] given by \(r\equiv\cfrac{\sum_{i} e_{ii}\sum_{i} a_{i} b_{i}}{1\sum_{i} a_{i} b_{i}}\), where \(\mathbf{e}=\{e_{ij}\}\) is a matrix whose element \(e_{ij}\) is the fraction of edges in a network that connect a node of type i to one of type j, and \(a_{i}\) and \(b_{i}\) are the fraction of each type of end of an edge that is attached to nodes of type i. For the composers’ periods we have \(r=0.257 \pm0.005\), meaning that composers belonging to a common period tend to be connected preferentially to one another. The Pearson Correlation Coefficient (PCC) between connected composers’ active years (the middle point between their birth and death years) is even higher, with \(0.451 \pm0.009\).

Community 1A: William Byrd (15401623, Renaissance) and Henry Purcell (16591695, Baroque)

Community 1B: Antonio Vivaldi (16781741, Baroque), Johann Sebastian Bach (16851750, Baroque), George Frideric Handel (16851759, Baroque) from the Baroque period, and Wolfgang Amadeus Mozart (17561791, Classical), and Franz Joseph Haydn (17321809, Classical) from the Classical period.

Community 2: Ludwig van Beethoven (17701827) and Franz Schubert (17971828) who are considered transitional between Classical and Romantic; Robert Schumann (18101856, Romantic), Frédéric Chopin (18101849, Romantic), Franz Liszt (18111886, Romantic), Johannes Brahms (18331897, Romantic), and Pyotr Ilyich Tchaikovsky (18401893, Romantic) from Romantic.

Community 3: A UScentric Modern community, with two highestdegree Modern composers being George Gershwin (18981937, Modern) of Rhapsody in Blue and Leonard Bernstein (19181990, Modern) of West Side Story. Scott Joplin (18671917, Modern) and Billy Strayhorn (19151967, Modern), both prominent Jazz composers, and Richard Rodgers (19021979, Modern) and Irving Berlin (18881989, Modern), both Broadway composers, are also included.

Community 4: Another UScentric community. Including the likes of Charles Ives (18741954, Modern) of The Unanswered Question, Aaron Copland (19001990, Modern) of Appalachian Spring, Samuel Barber (19101981, Modern) of Adagio for Strings, and John Cage (19121992) of 4′33″, this can be said to represent the 20thcentury American vernacular style of classical music [38]. More contemporary US composers, Terry Riley (1935current, Modern), Steve Reich (1936current, Modern) and Philip Glass (1937current, Modern), are also in this module.

Community 5: Composed of mainly Modern (89.3%) and Romantic (10.2%) composers, it includes transitional figures such as Gabriel Fauré (18451924, Romantic), Claude Debussy (18621918, Modern), and Maurice Ravel (18751937, Modern). In contrast with Community 4, this community represents the nonUS branch of modern music, including Arnold Schoenberg (18741951, Modern, Austria), Manuel de Falla (18761946, Modern, Spain), Béla Bartók (18811945, Modern, Hungary), Igor Stravinsky (18821971, Modern, Russia), Heitor VillaLobos (18871959, Modern, Brazil), Paul Hindemith (18951963, Modern, Germany), Francis Poulenc (18991963, Modern, France), Ástor Piazzolla (19211992, Modern, Argentina), and Luciano Berio (19252003, Modern, Italy).
That the communities appear to correspond well to periods indicate the existence of correlation between the two partitions. The overlap can be quantified, for instance, via the normalized mutual information measure proposed by Danon et al. [36, 39]. It is given as \(I_{\mathrm{norm}}(\mathcal{X},\mathcal{Y})=\frac{2I(X,Y)}{H(X)+H(Y)}\), where \(\mathcal{X}\) and \(\mathcal{Y}\) are the partitions of the nodes by community detection and period designations, \(I(X,Y)\) is the mutual information, and \(H(X)\) and \(H(Y)\) are the Shannon entropies of X and Y. Widely used in tests of community detection algorithms, the normalized mutual information equals 1 if the partitions are identical and 0 if the partitions are independent. In other words, in our network of composers, the normalized mutual information is 1 if the members in each community are completely identical in periods, and 0 if they are uncorrelated. The normalized mutual information of our result using Louvain method is 0.325, indicating a significant level of overlap, as was hypothesized.
Our investigation of the centralities, assortativity, and community structure shows how largescale data built primarily for commercial purposes can yield a coherent and useful picture of the landscape of western classical music. This demonstrates that the quantitative analysis of a large collection of cultural artifacts such as CDs can indeed yield meaningful results, validated by agreements with qualitative musicology. A deeper understanding of the network of artists based on commercial cultural products may help in devising new ways of approaching the market which can in turn result in larger, more elaborate commercial data that can further help advance our understanding of the subject.
4 Growth and evolution of the composer network
The classical musical sphere is constantly evolving, with new composers entering the scene, and old composers gaining further prominence or fading out in popularity. As a consequence it is interesting to understand the dynamics of the evolution of popularity or success [40].
4.1 Network growth process hints at the uneven growth of oldboys and newbies
4.2 Evolution of bipartite degree distribution indicates predictability for topranked composers and explains the richgetricher phenomenon in classical music industry
Extending the skewed bipartite degree distribution in the CDcomposer network shown in Figure 1(B), we now focus on a more detailed figure of the temporal evolution of \(P(q)\). In Figure 3(B) we show the degree distribution of five snapshots of the network taken every five years. The degree distribution approximates a truncated power law that approaches a true power law (with power exponent \(\gamma =1.89\pm0.01\)) as the network grows. For comparison with another example in culture, in the movie actor network the power exponent is \(\gamma_{\mathrm{actor}}=2.3\). The highestdegree (most recorded) composers are WA Mozart, JS Bach, L van Beethoven, F Schubert, and J Brahms throughout the observational period (with final degrees 4,851, 4,292, 3,778, 2,328, and 2,429, respectively). The mean degree of the rest of the composers stays nearly constant at 3.2, again confirming the significant discrepancy between the ‘major minority’ and the ‘minor majority’ in the network landscape of classical music.
In Figure 3(C) we show the growths of the number of edges in the network and the degrees of six highestdegree composers. The curves appear to be quadratic (i.e. \({\propto} t^{2}\), dotted curves), suggesting a constant acceleration. Although it remains to be seen if the trend continues, if it does then in the year 2019 WA Mozart would have a bipartite degree of \(q\simeq10\mbox{,}264\), JS Bach would have \(q\simeq 9\mbox{,}211\), and so forth (see Figure 3(C) for all six top composers). Regarding the number of edges, we find that they are becoming increasingly concentrated between the topdegree nodes: The top 1% of the nodes in 1990 (8 composers out of 882) account for 20.3% of all the degrees, while in 2009 the top 1% (139 composers of 13,981) account for 57.1%. These observations tell us that a reasonable growth dynamics of the network must incorporate at minimum two properties, namely, powerlaw degree distribution and the increasing concentration of degrees on the topdegree nodes.
A popular model of a growing network that exhibits a skewed degree distribution (such as the power law) is the Cumulative Advantage (CA) or Preferential Attachment (PA) [41, 42]. Our CDcomposer network, too, boasts typical features that render PA a reasonable mechanism of its growth: It has a skewed distribution and a fixed fraction of the highdegree nodes take up a larger portion of the total degrees as it grows, evidenced by Figures 3(B) and 3(C).
Top 10 pairs of composers for edge weights
Rank  Weight  Pair  

Name  Period  Active year  Name  Period  Active year  
1  680  Giacomo Puccini  Romantic  1891  Giuseppe Verdi  Romantic  1857 
2  613  Beethoven  Romantic  1798  WA Mozart  Classical  1773 
3  417  Giuseppe Verdi  Romantic  1857  Gaetano Donizetti  Romantic  1822 
4  389  Beethoven  Romantic  1798  Johannes Brahms  Romantic  1865 
5  384  JS Bach  Baroque  1717  WA Mozart  Classical  1773 
6  381  Gioachino Rossini  Romantic  1830  Giuseppe Verdi  Romantic  1857 
7  368  JS Bach  Baroque  1717  GF Handel  Baroque  1722 
8  355  Giuseppe Verdi  Romantic  1857  WA Mozart  Classical  1773 
9  352  Maurice Ravel  Modern  1906  Claude Debussy  Modern  1890 
10  350  Franz Schubert  Romantic  1812  WA Mozart  Classical  1773 
Here we identified two main components of the evolution of the classical composer networks: growth and attachment. We observe that the powerlaw degree distribution becoming clearer as the network grows, which allows us to forecast the future degrees of the most recorded composers. We also find an effective superlinear preferential attachment behavior which can partially explain the observations.
5 Conclusion
In this paper, we studied the network of classical music composers constructed from the comprehensive recording data from ArkivMusic. We presented the basic properties of the CDcomposer and composercomposer networks, finding that they exhibit characteristics common to many realworld networks, including the smallworld property, the existence of a giant component, high clustering, and heavytailed degree distributions. We also explored the global association patterns of composers via centrality, assortative mixing, community structure analyses, which suggest an intriguing interplay between the networks of musicians and our musicological understanding of the western musical tradition in which both are undoubtedly continuously influencing each other. We then examined the growth of the CDcomposer bipartite network over time. We identified superlinear preferential attachment as a strong candidate for explaining the increasing concentration of edges around topdegree nodes and the powerlaw degree distributions. The growth of edges and composer degrees exhibits a quadratic growth, allowing us to forecast the future of several prominent composers. If this behavior persists further into the future, it would suggest an interesting future research direction regarding the growth dynamics of the network.
An analytical investigation as ours on new, largescale data can provide either new lessons or more rigorous answers to questions that are only partially understood on a subject matter (in our case classical music), although the boundary between the two can often be fuzzy: New findings challenge us to find rigorous answers to already known issues, which in turn can bring about novel discoveries about a system. In our paper we have quantified the overlap (correlation) between manually designated periods and the computationally identified communities; found the superlinear preferential attachment behavior and a quadratic growth of network edges; and showed that the growth of the composer pool is sublinear against the network size, leading to a concentration of edge weights onto specific composer pairs. These findings pose interesting possibilities and opportunities for the type of commercial databases such as the one on classical music that we have used here which, we believe, will play an increasingly important role as a source of many more academic findings than those presented here. We are living in an era where technology is greatly facilitating the consumption of culture by the public, evidenced by the increasing adoption of technology by artistic institutions worldwide for public outreach. This will lead to larger and higherquality data that can allow us to learn more about culture and art. Classical music as presented here is merely one of many cultural subjects for this kind of investigation, and we believe that it would be interesting to apply our quantitative methodology to other subjects, e.g. visual arts and literature, and explore their nature in novel ways. Going further, it would also be interesting to compare different systems and find universal characteristics of cultural systems as well as those unique to each. We believe that our work highlights the potential of network science coupled with wellcurated largescale data in answering many pertinent questions.
Declarations
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF20100004910 and NRF2010330B00028), IT R&D program of MSIP/KEIT (10045459), and BK21 Plus Postgraduate Organization for Content Science.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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