Introduction

Quality assessment of literature is a highly contested matter. Positions in the debate range from constructivist context dependency ('the success of a work of literature depends entirely on its context') to work internalism ('success depends on work-internal features') [3,36]. While context dependency is evidenced by the variety and seemingly chaotic dynamics of, e.g., bestseller lists, the constructivist argument ignores the convergence of an empirical 'canon' for many readers over time and space [34,59]. Moreover, to attribute the longevity or popularity of certain books to purely contextual factors would seem to be at odds with large-scale consensus among readers, which appear far from volatile [1,60,39].

Several shifts have played a role in making terms like "literary quality" or "classics" unpopular in the discipline of literary studies, even said to belong to the "precritical era of criticism itself" [23]: Methodological shifts that move the scholarly focus from evaluation to interpretation [9], an expansion of the conceptual boundaries of "literature" to encompass texts that challenge traditional ideas of beauty or enjoyment [62], but also constructivist and postcolonial shifts that bring attention to the context of literary evaluation and tradition, canon representativity [24] and the inequality of cultural production [55,53].

Context-dependent positions on literary appreciation range from simply prioritizing context over work-internal factors [23], to beliefs on the contextual determinacy of aesthetic evaluation [19,15]. An extreme perspective holds that critical evaluation reflects culturally dominant voices, arguing that canon judgments "are only the instruments of entrenched interests" [24, p. iv]. In this sense, a disparity appears to have arisen between a scholarly "denial of quality" [62,46], and the multitude of quality judgements that are practiced within the literary culture (literary awards, classics book series, prescriptive creative writing courses, etc.).

Conversely, work-internalist positions are closer, at least in their purest form, to universalist claims, where aesthetic judgments are seen as arising solely from the interaction between the individual reader and writer [10]. On this position, aesthetic pleasure tends to be considered as an a-historical or culturally universal experience and sidetracks as unimportant any "historically contingent" aspect of literature [16].

We can simplify the concept of literary quality by modeling perceived literary quality, thereby differentiating between a contested intrinsic value and the perceived value of a work, which anchors the quality assessment in the reader's aesthetic experience (i.e., a situated experience).

Empirical aesthetics offers well-formed theoretical work supported by empirical findings on the nature of aesthetic experience [22] and preferences at the psychological level [47,11], as well as cultural and contextual influences on the aesthetic experience [12]. Yet, empirical aesthetics has predominantly focused on the visual modality, in particular of pictorial art, attempting to find patterns correlating with the appreciation of paintings [22]. Within the domain of paintings, prevalent research topics include aesthetic appreciation and judgment [37], their perceptual and cognitive underpinnings [38,61], and, finally, emotional response [54,58,47]. The appreciation of literary prose has received significantly less attention than pictorial art [22,11], and studies tend to focus on general aspects of aesthetic appreciation of language [33] and literary response [41,42,18].

Approaches in empirical aesthetics that are based on fractal theory are particularly promising since they cross aesthetic modalities, that is, they seem to be valid for pictorial [56,51] literary [43,44] and musical [40,25] arts. Collectively, these approaches state that the appreciation of art depends on the presence of fractal-like scale-invariant properties. Although the explanation for this can vary, it is typically grounded in neural mechanisms of sensory coding [51]: a prototypical example is Jackson Pollock's abstract expressionism, where his drip painting technique was used to compose self-similar patterns repeated at multiple scales, which enables "Pollock authentication" through fractal analysis [56]. The same properties can be found in literature, where linguistic properties tend to display fractal behavior [43], but where positively evaluated literary narratives tend to be characterized by a higher degree of self-similarity [8,4,5]. These findings of multiscale self-similar repetitions in a set of linguistic properties (e.g., sentence structure, literary entities, or lines) are generally translated into global coherence or consistency of the aesthetic object [5,7,50]. In literary prose, for example, it is argued that a high-quality story is characterized by a narrative coherence and multiscale predictability that distinguish it from bland and unpredictable stories [28]. Similarly, high-quality paintings have visual elements that remain consistent at different magnification levels. A recent addition to this global quality property is a local property in literary prose that can identify quality [43,44]. More specifically, canonical works show higher degrees of sequential unpredictability compared to noncanonical fiction [44]. Combining these two quality indicators, we expect that high-quality literature, that is, literature that is appreciated by a majority, should display a tension between global coherence and local unpredictability and, furthermore, that a model of perceived literary quality should account for both. This paper will presents theoretical framework on two models of literary quality that utilize simplified narrative representations in the form of story arcs [29,49,45]. The first model uses approximate entropy to capture local unpredictability of the story arc [44], and the second model is based on fractal analysis to characterize the global coherence of the story arc [28]. Finally, we present an application of the models to a large collection of literary texts that when applied to the texts' story arcs, covary in non-trivial ways.

Proposal and Methodology

In this section, we present the theoretical foundation for models of literary quality, focusing on the integration of global coherence and local uncertainty within narrative structures. By employing metrics from empirical aesthetics, fractal theory and information theory, we propose two complementary approaches to quantify these aspects: approximate entropy and the Hurst exponent. Specifically, we suggest the for literature to be appreciated, it should manifest a tension, or positive correlation, between global coherence and local unpredictability (Fig. 1).

Approximate entropy -a measure of local predictability

Approximate entropy (𝐴𝑝𝐸𝑛) is a statistical metric that quantifies the complexity and regularity inherent in the time series data. Within the framework of narrative analysis, 𝐴𝑝𝐸𝑛 provides a means to assess the predictability and structural characteristics of story arcs [44]. A lower 𝐴𝑝𝐸𝑛 value signifies a more predictable and regular story arc, indicative of a repetitive and 'boring' narrative structure. In contrast, a higher 𝐴𝑝𝐸𝑛 value denotes a more complex and less predictable arc, suggesting a novel and intricate storyline, 1. This metric facilitates the quantitative analysis of local narrative predictability, enabling a formal evaluation of literary quality through empirical methodologies. For the specific question of evaluating literary quality, we expect that higher local 𝐴𝑝𝐸𝑛 will be associated with higher literary appreciation, c.f., [44].

In order to estimate 𝐴𝑝𝐸𝑛 for story arc 𝑋 = 𝑥(1), … , 𝑥(𝑛), subsequences of length 𝑚, 𝑦

(𝑟) = 1 𝑛 − 𝑚 + 1 𝑛−𝑚+1 ∑ 𝑗=1 𝐻 (𝑟 − 𝑑 𝑚 𝑖,𝑗 )

where 𝐻 (.) is the Heaviside function

𝐻 (𝑥) = { 1, 𝑥 > 0 0, 𝑥 ≤ 0 then define 𝜙 𝑚 (𝑟) = 1 𝑛 − 𝑚 + 1 𝑛−𝑚+1 ∑ 𝑖=1 log(𝐶 𝑚 𝑖 (𝑟))

where log(.) is the natural logarithm. Repeat the above for all sub-sequences of length 𝑚 + 1 to compute 𝜙 ( 𝑚 + 1)(𝑟), then 𝐴𝑝𝐸𝑛 is

𝐴𝑝𝐸𝑛(𝑚, 𝑟) = 𝜙 𝑚 (𝑟) − 𝜙 𝑚+1 (𝑟)

A note on parameter selections, the choice of 𝑚 and 𝑟 can influence the estimation of 𝐴𝑝𝐸𝑛. Typically, 𝑚 is set to 2 or 3 (in this study 𝑚 = 2), while 𝑟 is chosen as a small percentage of the standard deviation of the story arc (in this study 0.2 * 𝑆𝐷). It is important to note that the values of the optimal parameters may vary depending on the specific application and the characteristics of the data.

The Hurst exponent -a measure of global coherence

Story arcs can be modeled as fractal processes to understand their time-dependent selfsimilarity. By representing narratives as one-dimensional time series, we can apply fractal analysis to model the underlying pattern of coherence [28,8,6]. This approach allows us to quantify the degree of self-similarity in the narrative flow. Using the Hurst exponent, 𝐻 , which quantifies the persistence of a story arc, we can differentiate between persistent, anti-persistent, and short-memory processes, thus characterizing the temporal dynamics in three qualitative ranges [28], 1. We expect that literary appreciation will be associated with coherence, which translates to a persistent story arc such that narrative fluctuations at shorter time scales are approximate copies of narrative fluctuations at longer time scales. The Hurst 𝐻 exponent quantifies persistence or memory in time series, where 0 < 𝐻 < 0.5 is an anti-persistent process, 𝐻 = 0.5 is a short-memory process, and 0.5 < 𝐻 < 1 is a persistent process [20]. For the present proposal, a persistent process indicates continuity of the story arc (i.e., sentiment states will last for a long time). An anti-persistent indicates rigidity (i.e., sentiment states will rapidly decay to a mean state). Finally, short memory indicates a lack of continuity (sentiment states will only be correlated at short time scales).

Detrended fluctuation analysis (DFA) [48] is a widely used method for estimating the Hurst exponent for a time series. DFA consists of five steps, 1) initially a random walk process is constructed from the time series:

𝑢(𝑛) = 𝑛 ∑ 𝑘=1 (𝑥 𝑘 − 𝑥), 𝑛 = 1, 2, ⋯ , 𝑁 ,

where 𝑥 is the mean of the series 𝑥(𝑘), 𝑘 = 1, 2, ⋯ , 𝑁 ; 2) dividing the constructed random walk process into non-overlapping segments; 3) determining the local trends of each segment as the best polynomial fit; 4) getting the variance of the differences between the random walk process and the local trends; and 5) determining the average variance over all the segments. DFA may involve discontinuities at the boundaries of adjacent segments. Such discontinuities can be detrimental when the data contain trends [27], non-stationarity [32], or nonlinear oscillatory components [13,26]. Adaptive fractal analysis (AFA) provides a robust alternative to DFA that solves these issues [20]. The main advantage of AFA over DFA is that it identifies a global smooth trend, which is obtained by optimally combining local linear or polynomial fitting, and thus no longer suffers from DFA's problem of discontinuities of adjacent segments. As a result, AFA can automatically deal with arbitrary, strong nonlinear trends that are not unusual to encounter in story arcs [20,26,28].

AFA is based on a multiscale non-linear adaptive decomposition algorithm [20]. The first step involves partitioning the time series under study into overlapping segments of length 𝑤 = 2𝑛 + 1, where neighboring segments overlap by 𝑛 + 1 points. In each segment, the time series is fitted with the best polynomial of order 𝑀, obtained using the standard least squares regression; the fitted polynomials in overlapped regions are combined to yield a single global smooth trend. Denoting the fitted polynomials for the 𝑖 − 𝑡ℎ and (𝑖 + 1) − 𝑡ℎ segments by 𝑦 𝑖 (𝑙 1 ) and 𝑦 (𝑖+1) (𝑙 2 ), respectively, where 𝑙 1 , 𝑙 2 = 1, ⋯ , 2𝑛 + 1, we define the fitting for the overlapped region as

𝑦 (𝑐) (𝑙) = 𝑤 1 𝑦 (𝑖) (𝑙 + 𝑛) + 𝑤 2 𝑦 (𝑖+1) (𝑙), 𝑙 = 1, 2, ⋯ , 𝑛 + 1,

where 𝑤 1 = (1 − 𝑙−1 𝑛 ) and 𝑤 2 = 𝑙−1 𝑛 can be written as (1 − 𝑑 𝑗 /𝑛) for 𝑗 = 1, 2, and where 𝑑 𝑗 denotes the distances between the point and the centers of 𝑦 (𝑖) and 𝑦 (𝑖+1) , respectively. Note that the weights decrease linearly with the distance between the point and the center of the segment. Such weighting ensures symmetry and effectively eliminates any jumps or discontinuities around the boundaries of neighboring segments. As a result, the global trend is smooth at the non-boundary points and has the right and left derivatives at the boundary [52]. The parameters of each local fit are determined by maximizing the goodness of fit in each segment. The different polynomials in the overlapping part of each segment are combined so that the global fit will be the best (smoothest) fit of the overall time series. Note that, even if 𝑀 = 1 is selected, i.e., the local fits are linear, the global trend signal will still be nonlinear. For an arbitrary window size 𝑤, we determine, for the random walk process 𝑢(𝑖), a global trend 𝑣(𝑖), 𝑖 = 1, 2, ⋯ , 𝑁 , where 𝑁 is the length of the walk. The residual of the fit, 𝑢(𝑖) − 𝑣(𝑖), characterizes fluctuations around the global trend, and its variance yields the Hurst parameter 𝐻 according to the following scaling equation:

𝐹 (𝑤) = [ 1 𝑁 𝑁 ∑ 𝑖=1 (𝑢(𝑖) − 𝑣(𝑖)) 2 ] 1/2 ∼ 𝑤 𝐻 .

By computing the global fits, the residual, and the variance between original random walk process and the fitted trend for each window size 𝑤, we can plot log 2 𝐹 (𝑤) as a function of log 2 𝑤. The presence of fractal scaling amounts to a linear relation in the plot, with the slope of the relation providing an estimate of 𝐻 .

Application

To illustrate the practical application of our proposed models, we apply the models to a substantial corpus of literary texts. By analyzing the sentiment arcs of the narratives, we quantify the local unpredictability using 𝐴𝑝𝐸𝑛 and the global coherence using the Hurst exponent. This application demonstrates how our models align with quality indices, providing empirical support for the notion that a balance of global coherence and local uncertainty enhances perceived literary quality.

Data

To extract the sentiment arcs, we use a corpus of 9, 089 novels published in the US from 1880 to 2000, making it one of the largest corpora of manually cleaned contemporary literature [57,14,2]. It was assembled by Hoyt Long and Richard Jean So and was based on the number of libraries worldwide that held a copy of each title, favoring titles with a higher number of holdings. It comprises high-brow fiction (e.g., Joyce Carol Oates, Philip Roth) as well as highly known "genre fiction" (e.g., J.R.R Tolkien, Philip K. Dick), bestsellers (e.g., Agatha Christie, George R.R. Martin) and prestigious experimental literature (e.g., James Joyce). The works span from 341 − 714, 444 words, but more than 97% of the novels are longer than 35,000 words. Overall the corpus totals over one billion words. The corpus has an Anglophone bias: most are US or UK authors writing in English. While this allowed us some form of control on cultural variability within the corpus, it also presents a narrower perspective on literary dynamics.

From each novel, we extracted the average valence sentence by sentence using the Syuzhet library [30]. 1 Syuzhet was chosen based on recent research showing that it performs particularly well on detrended narrative arcs, returning values close to human annotations [63].

For this corpus various quality proxies were collected: GoodReads average rating and rating count,2 library holding numbers, 3 and translation numbers.4

Findings

Figure 2: Spearman correlation between the Hurst exponent and approximate entropy (ApEn) of novels in the corpus and four quality proxies: GoodReads average rating and rating count, library holding numbers and translation counts. For all correlations, 𝑝 < 0.01

We estimated 𝐴𝑝𝐸𝑛 and Hurst exponent for each story arc. We correlated these measures with available metadata quality proxies: GoodReads average rating and rating count, library holdings, and translation counts (see Figure 2). As predicted, we find empirical support that global coherence and local predictability are moderately correlated (𝜌 = .33, 𝑝 < .0001). Furthermore, average ratings are positively associated with both measures (𝜌 𝐴𝑝𝐸𝑛 = .15 and 𝜌 𝐻 = .19, 𝑝 < .0001). Interestingly, we observe that 𝐴𝑝𝐸𝑛 is more strongly associated with the additional quality proxies.

One of the most interesting findings is, in our opinion, the positive correlation of Hurst exponent and 𝐴𝑝𝐸𝑛 in our corpus in general, and with at least one of the proxies of reception. As we show in 1, the Hurst exponent and 𝐴𝑝𝐸𝑛 tend to be divergent measures, and in fact they have negative correlations for artificial or simpler time series [21]. In other words, a completely random series will have both a higher 𝐴𝑝𝐸𝑛 and a lower Hurst exponent than a completely trending series. However, Hurst exponent and 𝐴𝑝𝐸𝑛 do not measure the same thing, and in series with complex behaviors it is not the case that the highest Hurst corresponds to the lowest entropy (see, e.g., Kristoufek and Vosvrda [35]).

Overall, 𝐴𝑝𝐸𝑛 (as we have computed it) here compares each embedding to all the other embeddings in the series. We can imagine a case in which a time series that has some degree of persistence in its overall trend, but which consists, at the sentence-level, of relatively unpredictable shorter sequences. Such a case would not result in a negative correlation between the measures. If the level of "noise" (unpredictability) is both relatively low and constant throughout a trending series, it is likely to result in a high 𝐴𝑝𝐸𝑛 and a high Hurst exponent.

This effect may be due to the ability of 𝐴𝑝𝐸𝑛 to capture shorter-range variations in the sentimental fluctuations of a novel, closer to style than to overall narrative structure. For example, a relatively low 𝐴𝑝𝐸𝑛 could represent a text where the same small-scale sentiment patterns are repeated in a structured and predictable fashion. Reversely, a high Hurst with high 𝐴𝑝𝐸𝑛 might represent a text where the overall narrative arc is highly coherent (and predictable), but where the small-scale succession of positive and negative sentences is more chaotic or unpredictable. In this case, readers may find short-term uncertainty but an overall smoother and foreseeable experience: a trend is builds up through a very diverse set of sentimental "zig-zags", rather than through a simple linear succession going from the saddest to the happiest point. Narratives that manage to strike a balance between short-term surprising patterns and long-term coherent structures might be one of the reasons for the observed correlations. Finally, while 𝐴𝑝𝐸𝑛 slightly correlates with all our metrics of reception, a linear positive relation with the Hurst exponent of sentiment arcs is present only in the case of average GoodReads ratings. Interestingly, this is the only metric that is not related to spread or popularity, while the other three more or less indirectly measure how popular, known or disseminated a text is [17].

Concluding Remarks

Our theoretical proposal aims to avoid the contentious issue of universal features by focusing on the reader's perception of literary quality. By redefining literary quality as perceived literary quality, we emphasize the reader's experience and the subjective nature of aesthetic appreciation. Despite this focus on perception, our approach still leans towards intrinsic properties, as our models of approximate entropy and the Hurst exponent are grounded in the structural characteristics of the narratives themselves.

Our findings provide initial empirical support for the hypothesis that tension between global coherence and local unpredictability contributes to perceived literary quality, as reflected in readers' average ratings. However, there is ample room for further research to explain more of the variance in literary quality assessments. Context-sensitive variables such as genre and reader demographics could significantly shape literary appreciation. Less indirect quality proxies with a higher temporal resolution, e.g., reading time, can also provide important correctives to the current proposal.

By combining the strengths of empirical aesthetics with fractal theory and information theory, our proposal offers a robust framework for evaluating literary quality. For future research, our aim is to empirically investigate how the appreciation of literary language is influenced by or drawn to specific forms of temporal organization of language.