A quantification of small-worldedness in Afrikaans cultural networks in comparison with other complex networks

Abstract

Since the late nineties of the last century the theory of complex networks has become an indispensable approach to complex systems and has found applications in almost every field of science. One of the main findings of the study of complex networks that has emerged since the nineties is that the majority of networks are characterised by the so-called small world phenomenon. Watts and Strogatz’s seminal study (1998) was inspired by Stanley Milgram’s famous “six degrees of separation” study (1967), where Milgram suggested that anyone can reach almost anyone else in an average of just six steps. Watts and Strogatz established a model that postulates that nodes can reach other nodes in a network in a short path, on average, even if it is not a social network as in Milgram’s case, and that a high degree of clustering or transitivity occurs between nodes. In order to determine small-worldedness the average path length (L) and transitivity (C^∆) within the network has to be compared with a network of the same size (in terms of both the number of nodes and the number of edges), as constructed using the Erdös and Rényi (1960) network model, in other words where link formation occurs at random. According to Humphries and Gurney (2008), a small world network is then characterised by the fact that L ≥ L_rand and C^∆ >> C^∆_rand. Humphries and Gurney (2008) also suggest a more precise manner in which small-worldedness can be quantified, namely the small world index (S^∆), which quantifies the relationship between L and L_rand and between C^∆ and C^∆_rand. They suggest that networks where S^∆ > 1 can be called small world networks. However, cases where 1 ≤ S^∆ ≤ 3 are borderline cases, and therefore S^∆ ≥ 3 can be taken as a statistically significant result showing a clear small-worldedness.

Newman (2003; 2010) distinguishes between four types of networks: biological, technological, information and social networks. The small world phenomenon occurs in all four types, and Humphries and Gurney’s (2008) study also found that all 27 networks for which they calculated S^∆ (and covering all four types of networks) were small world networks.

The international film actor network is one of the most studied networks from a physics point of view, which is included, for example, in a variety of studies published in Physica A. Locally, some aspects of the Afrikaans film industry and film actor network have been studied as a complex network (Senekal 2015), but there has been little investigation into the incidence of small-worldedness in other cultural networks, such as the literary system. Literature operates within a system that includes not only literary works and writers but also publishers, critics, literary scholars, newspapers, journals and so on, and it is in this “web” (Senekal 1987:44) of interactions that a literary work arises, exists, and is preserved for posterity in literary histories. Much has been written about the literary system (e.g. Even-Zohar 1979; 1990), and especially about the Afrikaans literary system (Senekal 1987; Venter 2006; Kleyn 2013), and the Afrikaans literary system has been studied as a network (e.g. Senekal 2014b), but the small world phenomenon has not been studied in detail with reference to any literary system (whether Afrikaans or any other language), and nor has the small world phenomenon yet been quantified with respect to any Afrikaans cultural network.

This article investigates the small world phenomenon in Afrikaans cultural networks – in following Watts and Strogatz's (1998) seminal study – and the average path length (L) and average transitivity (C^∆) is calculated for a number of Afrikaans cultural networks, including the literary system and the Afrikaans film industry. Networks dating from different periods are investigated, as well as across genres where the literary system is involved, and where the Afrikaans film industry is examined, the investigation is not limited to the film actor network as in foreign studies, but the whole film industry is taken into account, which includes every person credited with a contribution to almost every film. Following Muller, Destexhe and Rudolph-Lilith (2014), Humphries and Gurney’s (2008) quantitative index of small world networks (S^∆) is then used to determine whether Afrikaans cultural networks really are small world networks, and if so, to what extent this is the case.

The following table represents the findings of the current study, and indicates the network that has been studied, the number of nodes (n), number of edges (m), average path length (L) and transitivity (C^∆), as well as average path length and transitivity of the equivalent network as constructed using the Erdös and Rényi random network model (L_rand and C^∆_rand respectively), as well as the small world index (S^∆) as calculated by following Humphries and Gurney (2008).

It was found without exception that S^∆ > 12, which means that the Afrikaans cultural networks that were investigated not only are all small world networks, but also that there are no borderline cases: all these networks are undoubtedly small world networks, and S^∆ values lie in the range 12 ≤ S^∆ ≤ 43. As such, it is the first quantification of small-worldedness in Afrikaans cultural networks. It was also found that Afrikaans literary networks differ considerably from the Afrikaans film networks, because C^∆ is consistently lower for the literary networks, and L < L_rand in literary networks, while film networks follow the more typical L ≥ L_rand pattern. Film networks generally exhibit a higher degree of transitivity, which identifies them as social networks, while the lower transitivity in literary networks suggests that these should rather be classified as information networks. The Afrikaans prose network from 1900 to 1978 is also singled out in the article as the literary network where the largest difference exists between C^∆ and C^∆_rand, which means that triangles occur more frequently in this network than in other literary networks. In practice, this means that if person A writes about the work of B and C, there is a significant probability that B and C will also discuss each other’s works. Prose writers are found to be more likely to participate in discussions on other literary works than playwrights or poets.

There are still many Afrikaans cultural networks that have not been analysed as networks, including the music industry and theatre (especially as found at Afrikaans cultural festivals). One could also study film, music, theatre and literature together to get an overall picture of the structure of the Afrikaans cultural network. This article represents the largest study of Afrikaans cultural networks to date, but suggestions are made for further research that will seek to map and study the whole Afrikaans cultural milieu.

Keywords: Afrikaans culture, Afrikaans film, Afrikaans literature, complex networks, literary system, small-worldedness, Watts and Strogatz

Lees die volledige artikel in Afrikaans: ’n Kwantifisering van kleinwêreldsheid in Afrikaanse kultuurnetwerke in vergelyking met ander komplekse netwerke.

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