More frequent verbs are associated with more diverse valency frames: Efficient principles at the lexicon-grammar interface (2024)

Abstract: A substantial body of work has provided evidence that the lexicons of natural languages are organized to support efficient communication. However, existing work has largely focused on word-internal properties, such as Zipf’s observation that more frequent words are optimized in form to minimize communicative cost. Here, we investigate the hypothesis that efficient lexicon organization is also reflected in valency, or the combinations and orders of additional words and phrases a verb selects for in a sentence. We consider two measures of valency diversity for verbs: valency frame count (VFC), the number of distinct frames associated with a verb, and valency frame entropy (VFE), the average information content of frame selection associated with a verb. Using data from 79 languages, we provide evidence that more frequent verbs are associated with a greater diversity of valency frames, suggesting that the organization of valency is consistent with communicative efficiency principles. We discuss our findings in relation to classical findings such as Zipf’s meaning-frequency law and the principle of least effort, as well as implications for theories of valency and communicative efficiency principles. 1

heading: 1 Introduction
text: The idea that functional pressures shape the organization of natural language lexicons has a long tradition, a prominent early example being Zipf’s law of abbreviation: the length of the word is inversely related to the frequency of its use, motivated by the “principle of least effort” (Zipf, 1935, 1949). Further evidence has been provided by a functionalist line of research studying the emergence of the lexicon from the frequency of use (e.g. Bybee, 1998; Hopper and Bybee, 2001) as well as more
1Code for reproducing our results is available under GNU GPL Version 2 at https://github.com/siyutao/ verbal-valency-ud
recent work that formalizes such ideas using information theory (e.g. Regier et al., 2015; Wu et al., 2019; Mahowald et al., 2022; Pimentel et al., 2021). Common across this literature is the idea that the lexicon reflects a pressure for efficient language use under the constraints posed by human cognition. Evidence has been adduced from domains such as word lengths (Piantadosi et al., 2011; Pimentel et al., 2023), morphological complexity (Wu et al., 2019), orthographic forms (Mahowald et al., 2018), and the partitioning of semantic space into word meanings (e.g. Regier et al., 2015; Kemp et al., 2018; Zaslavsky et al., 2018).
However, a key aspect of the lexicon has gone unaddressed in this literature: words are not used in isolation but systematically integrated within the broader structure of a sentence. Verbs in particular play a key role in determining the syntactic and semantic structure of a sentence, and the study of valency features prominently in theoretical linguistics (e.g. Chomsky et al., 1970; Levin and Rappaport Hovav, 2005). In broad strokes, the valency of a verb indicates the type and number of dependents it takes. Certain features of valency are shared across many but not all languages, e.g., a majority of verbs in many languages minimally requires a subject (Chomsky, 1982; McCloskey, 1994); and a transitive category of verbs may take on one or more objects in addition (Bowers, 2002). Typological literature has documented the many ways in which valency differs across verbs within a language, and across languages (e.g. Hartmann et al., 2013).
Fig. 1 shows how the sentences equivalent to English The teacher helped the children with the homework are rendered differently in several languages, with dependency relations of verb annotated according to Universal Dependencies (UD, v.2 Nivre et al., 2020) guidelines. We observe both cross-lingual similarities and differences: for example, there is a subject (UD relation nsubj), an ob-
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ject (obj) and an oblique dependent (obl) in both English and Finnish, but all three dependents are marked for case in Finnish (nominative, partitive and inessive, respectively). Other languages use other combinations of relation types to express the same meaning; for instance, the verb has only two dependents in Japanese and Chinese: in Japanese the verb takes homework as obj and in Chinese the sentence must be formulated with another verb as a clausal complement (ccomp).
Within a language, different verbs allow different valency frames: specific configurations of morphosyntactic features they appear with. For example, Fig. 2 shows some other valency frames observed with the English verb help (also known as its diathesis alternations). There is general consensus that the syntactic frames can at least in part be predicted by semantics (e.g. Fillmore, 1968; Levin, 1993), but divergent explanations of the findings have been offered (Gropen et al., 1989; Goldberg, 1992). A substantial body of work has also classified verbs within a language based on their diathesis alternations (e.g. Levin, 1993).
Here, we investigate whether valency properties of verbs show signatures of efficient organization and test the hypothesis that verbs will systematically differ in the diversity of valency frames they are associated with, so that high-frequency verbs will be associated with a greater diversity of frames.
A range of theoretical considerations converge on this prediction. Our hypothesis is related to Zipf’s meaning-frequency law, which states that more frequent words tend to have more meanings and can be derived assuming two other Zipfian laws, the law for word frequencies and the law of meaning distribution (Zipf, 1945; Ferrer-iCancho and Vitevitch, 2018). It has been empirically verified in corpus-based studies for various languages, among others Dutch, English (Baayen and Prado Martin, 2005) and Turkish (Ilgen and Karaoglan, 2007). Psycholinguistic evidence has similarly suggested a correlation between semantic ambiguity and verb frequency – more frequent verbs tend to be more ambiguous semantically (Hoffman et al., 2013). As the syntactic frames of a verb correlate with its semantics, a corollary of the meaning-frequency law is that the more frequency verbs will have more diverse valency frames, i.e., our hypothesis.
Like other Zipfian observations of statistical regularities in languages, our hypothesis follows from
the principle of least effort. The greater semantic ambiguity associated with the verbs requires more freedom with respect to the syntactic frames to allow speakers to convey nuanced meanings or contextual information more efficiently. From a comprehension perspective, this kind of pattern also ensures frames are predictable when verbs are not, reducing spikes in information density. We argue that similar considerations apply from the perspective of language production and learning.
We operationalize valency at a lexeme-level, and use two information-theoretic metrics of the diversity of valency frames: (i) valency frame count (VFC) that measures the number of distinct frames associated with a verb; and (ii) valency frame entropy (VFE) that measures the average surprisal of valency frames as conditioned by the verb, determined by the diversity of frames associated with that verb. This allows us to quantitative test our hypothesis regarding the correlation between these metrics and the diversity of valency frames in experiments and show that effects predicted by learnability and efficiency considerations shape valency systems of languages cross-linguistically.

heading: 2 Background and Motivation
text: The Notion of Valency The notion of valency, and closely related notions such as argument structure, subcategorization and diathesis alternation, have received a variety of formalizations across formal linguistic theories (e.g. Tesnière, 1959; Chomsky, 1965; Fillmore, 1982; Levin, 1993; Goldberg, 1995). While many theories make a formal distinction between arguments and adjuncts (Chomsky et al., 1970; Ackema, 2015), others treat them in a similar fashion (van Noord and Bouma, 1994). For this study, we abstract away from such theoretical questions, and adopt a broad definition of verb valency within UD, a dependency grammar framework. In doing so, we aim for a broad descriptive basis for cross-linguistic comparison, as UD classifies dependency relations into cross-linguistically meaningful types (cf. Figures 1–2); pragmatically, this also allows applicable operationalization on large-scale cross-linguistic usage data available as UD treebanks. We formalize as valency the set of such relation types of dependents a verb co-occurs with, and a valency frame as a set of UD relations corresponding to dependents of a verb (see § 4.1).
Computational Studies of Valency A key line of computational research on valency has cre-
ated datasets of the valency frames associated with verbs, such as FrameNet (Baker et al., 1998; Fillmore and Baker, 2015) and VerbNet (KipperSchuler, 2005; Kipper et al., 2006, 2008). Mu et al. (2017) automatically classifies verbs in terms of their valency as listed in such resources to recover classifications from the theoretical literature (Levin, 1993). Cross-linguistically, Baker and Lorenzi (2020); Ellsworth et al. (2021) explores the alignment of frames based on FrameNet. In contrast to their approach, our operationalization of valency frame is morphosyntactic in nature, directly in terms of UD relations. This allows us to match valency frames with large scale usage data across many languages and for the full set of verbs appearing in any given treebank – something that is not available when operationalizing frames using
curated lexical resources such as FrameNet.
Theories of Communicative and Processing Efficiency One justification for efficient organization of the lexicon comes from learnability considerations. For instance, it has been suggested that infrequent irregular verbs are poorly acquired, leading to morphological irregularity being focused on high-frequency words (e.g. Bybee, 1998; Wu et al., 2019). Similar considerations apply to valency: When learning from a finite sample, a learner should be able to acquire a larger variety of frames for frequent verbs, as more data is available for them. Thus, if learnability affects the organization of valency, one expects that infrequent verbs should be associated with smaller diversity of frames.
Another key justification for efficient organization of the lexicon (and language in general) comes from online processing, in both language production and comprehension. On the production side, accessibility, i.e. the ease of retrieval, has been shown to play a key role in shaping language form and structure (Kathryn Bock and Warren, 1985; MacDonald, 2013). To the extent that verbs may be planned in advance of their arguments in sentence production (e.g. Momma and Ferreira, 2019), reducing the diversity of frames associated with a verb should increase the accessibility of any of these frames. Such a pressure should be stronger for the long tail of low-frequency verbs, as producers will be less attuned to their production. Conversely, being compatible with a larger number of frames could make a verb more versatile and therefore encourage its use.
On the comprehension side, we expect consequences for sentence interpretation as well as the distribution of information density. Psycholinguistic evidence shows that argument structure information is used in online comprehension to eliminate implausible interpretations (Boland et al., 1995). The more frequent verbs, being more ambiguous or vague (Hoffman et al., 2013), would require additional information to disambiguate or narrow down the meaning. Additionally, the Uniform Information Density (UID) hypothesis (Fenk and FenkOczlon, 1980; Levy and Jaeger, 2006) posits that language speakers prefer a more even distribution of surprisal values across utterances in order to maximize but not overload the capacity of the communication channel. Less frequent verbs will, on average, have high surprisal; high entropy in the valency frames associated with it would lead to an undesirable spike in overall surprisal.

heading: 3 Data and Methodology

heading: 3.1 Data
text: We draw on Universal Dependencies (UD) as the source of data. It provids as a cross-linguistically consistent system for annotating morphosyntactic information within a dependency grammar framework (de Marneffe et al., 2021). We use the UD v2.11 release, and include all languages with at least 10,000 tokens across treebanks. We exclude L2 learner corpora and code-switched corpora. Korean corpora are excluded due to the lack of verb lemmatization, which is necessary for empirically estimating valency. In total, 79 languages out of the 138 languages remained.

heading: 3.2 Quantifying Diversity of Frames and Verbs
text: We hypothesize a positive relation between a verb’s frequency and the diversity of frames that it occurs with. In order to test this, we propose two measures of the diversity of frames associated with a verb.
The simplest metric of the diversity of frames associated with a verb is valency frame count (VFC): the number of distinct frames that a verb occurs with in the corpus.
By definition, VFC gives equal weight to more or less frequent frames. As discussed in §2, information-theoretic models of language comprehension such as UID predict an effect of the entropy of the frames, conditioned on the verb. Our second measure formalizes this. Let R be the set of
UD relations under consideration. Let F be a random variable ranging over subsets of R – that is, over frames. Let V be a random variable ranging over the verb lemmata v in a language. A dependency treebank defines, for each verb v, a probability distribution P (f |V = v) over different frames proportional to their frequencies of occurrence. We formalize the valency frame entropy (VFE) as H[F |V = v], the entropy of the frame F conditioned on a verb v:
− ∑
f⊆R P (f |V = v) log2 P (f |V = v) (1)
The measure we use follows the subcategorization entropy measure as used by Linzen et al. (2013) and Linzen and Jaeger (2015) as a predictor of online processing costs. These two measures are equivalent in information-theoretic terms, but we adapt our measure further to include so-called adjuncts (non-core dependencies in UD) that are not considered part of subcategorization frames in most generative grammars (Chomsky, 1965).
Valency frame entropy equals log(VFC) in the special case where all frames that appear with a verb do so at equal frequencies. However, keeping VFC constant, VFE is lowered when the frames appear at more skewed frequencies.

heading: 4 Experiment

heading: 4.1 Dependency Relations
text: As we take a broad view on the notion of valency, we correspondingly include a broad range of UD relations (Figure 6). Definitions of valency vary across the literature, but conveniently UD classifies the relevant set of relations. We included the UD v2 relations defined by UD as core arguments and non-core dependents. Core arguments, including – among others – subjects and objects are most clearly associated with argument structure in typical theories of valency and syntax. Noncore dependents are other possible dependents of a clausal predicate, including – among others – oblique nominals and adverbial modifiers. UD further includes nominal dependents – dependents of nouns – which are by definition of no relevance to our study. Finally, UD also has the categories coordination, headless, loose, special, and other; we do not consider any of these to be relevant: for instance, coordination (conj and cc) defines a symmetric relationship rather than a hierarchical dependency structure; and special lacks cross-lingual
applicability. We examine correlations estimated from various subsets in Figure 5.

heading: 4.2 Controlling for Confounds via Cross-Entropy and Subsampling
text: Estimating count and entropy measures on finite samples is a difficult problem, and popular estimators for such quantities are generally biased (e.g. Paninski, 2003). As our goal here is to test a hypothesized relationship, we aim for estimators that avoid any bias towards false positives in the directions of our hypothesis – that is, estimators that lead to conservative estimates for the testing of our hypothesis. Indeed, testing our hypothesized relationship using naive estimators of VFC and VFE faces a circularity confound: if there are few observations for a verb in a corpus, the number of distinct frames observed with it will necessarily be small, leading to an underestimation bias for entropy and frame count. As precisely these verbs are expected to have low frame entropy and low frame count under our hypotheses, it may lead to spurious correlations. Stated differently, low-frequency verbs might have a low measured diversity of frames simply due to lack of observations. We use two strategies to eliminate this confound.
Estimation via Cross-Entropy Our first strategy is to empirically estimate the valency frame entropy using the cross-entropy. That is, for each verb, we randomly split its observations in the corpora into two halves, resulting in two empirical distributions P (F |V = v) and P (F ′|V = v). The cross-entropy between them, conditioned on a verb v, is:
− ∑
f⊆R P (xf |V = v) log2 P ′(f |V = v) (2)
We use Laplace smoothing with α = 1 over the full set of frames to ensure well-definedness.
How Cross-Entropy Counteracts the Confound Whereas naive estimation creates a bias in the direction of our hypothesis (more frequent verbs exhibit higher valency frame entropy), cross-entropy-based estimation introduces a conservative bias against our hypothesis: when a verb has very few observations, the observations in the two halves both are very noisy estimates of the actual distribution, increasing cross-entropy. In contrast, for verbs with many observations, both halves have sufficient data to closely estimate the distribution, eliminating the overestimation bias affecting low-frequency verbs.
In order to understand this formally, we consider two verbs, each with k frames and identical distributions P (F |V = v), but where v1 has N observations, and v2 has 2N observations. In the limit of an infinite corpus, both verbs would have the same values for both VFC and VFE. However, when N is not much larger than k, there is a substantial amount of probability that not all k frames were observed; that is, the diversity of the frames will be underestimated. Such underestimation is more likely to affect the less frequent verb v1; hence, the entropy of the observed distribution will tend be smaller for v1 than for v2. Underestimating both VFE and VFC more strongly for low-frequency verbs, this phenomenon introduces a circularity confound in the direction of our proposed relationship.
The bias reverses under cross-entropy estimation: cross-entropy (2) reflects the true entropy (1) plus DKL(P ||P ′). This KL Divergence converges to zero as the number of observations increases, and will tend to be larger for v1 than for v2. The bias is thus again stronger for the infrequent verb v1, but now is an overestimation bias, pointing into the direction opposite of our proposed relationship.
Cross-entropy estimation controls for the circularity confound in the sense that a systematic pattern in the direction of our hypothesized relationship must reflect a pattern in the underlying true (population-level) VFE values, as the estimation bias points in the opposite direction. Cross-entropy estimation has two limitations: it replaces one bias by another, and it applies only to VFE, not to VFC.
Subsampling As a second way of mitigating the circularity confound, we performed subsampling such that (i) for each verb with frequency above the subsampling threshold ∆, a sample of ∆ observations is taken, and (ii) verbs with frequency below ∆ are excluded from the analysis.
Considering again verbs v1, v2 with the same distribution over frames but different counts N , 2N , subsampling caps observations of both verbs at ∆. If both verbs are included (i.e., N ≥ ∆), they are on equal footing in estimation of VFE and VFC – the estimation bias is the same for v1 and v2.
Subsampling is applicable to both VFE and VFC, and eliminates effects of verb frequency on the estimation bias. However, verbs with insufficient observations need to be excluded, reducing the number of verbs for which correlations of these measures with verb frequency can be tested.
We determine the subsampling threshold ∆ for
each language by applying a ratio of 0.1 to the maximum frequency of any verb in that language, but limit the absolute value to between a ceiling of 25 and a floor of 10 to avoid wide fluctuations between languages, which would have ranged from 2 in Mbya Guarani to 663 in German.
On average for 79 languages, subsampling filters out 91.6% of all verbs (SD = 4.7%). The number of verbs filtered out ranges from Russian (9083 out of 10410 verbs, 87.3%) to Mbya Guarani (108 out of 113 verbs, 95.6%). We include only languages where more than 10 verbs remain after subsampling, excluding a further four languages2.

heading: 5 Results
text: We report primarily results with both subsampling and cross-entropy estimation, as these choices lead to the most conservative estimates for testing our hypothesis. The effect of subsampling is examined below.
We quantify the hypothesized relationships using Spearman’s rank correlation coefficient (Spearman, 1904). The results show robust correlation between frequency and our valency metrics: when using VFE, we observe at least moderate correlations (ρ ≥ .40, following Schober et al., 2018) in 66 out of 75 languages and strong to very strong correlations (defined as ρ ≥ .70) in 18. The remaining 9 languages show only weak correlations3. The mean Spearman’s ρ is .59, with a standard
2Manx (10), Mbya Guarani (5), Upper Sorbian (10), Welsh (9)
3They are Akkadian (.17), Ancient Hebrew (.27), Bambara (.39), Chinese (.39), Classical Chinese (.39), Japanese (.20), Sanskrit (.38), Turkish (.36), and Urdu (.22).
deviation of .15. Using VFC measure results in similar correlations, marginally stronger than VFE on average across the 75 languages (mean ρ = .67, SD = .14), of which 71 show at least moderate correlations, and 31 strong to very strong.
We note that our correlation hypothesis takes the form of an overall trend rather than a strict rule. A more frequent verb is not guaranteed nor expected to always have higher diversity of frames. Outliers to this trend among high-frequency verbs illustrate this: in English, verbs like arrive (frequency = 237, VFC = 13, VFE = 3.94) have significantly lower VFC and VFE values than others with similar frequency, e.g. call (frequency = 238, VFC = 18, VFE = 5.22) and work (frequency = 221, VFC = 18, VFE = 5.23).
Fig. 3a shows how the correlation coefficients vary across the languages and in relation to the sample sizes, using VFE. Larger sample sizes, either measured by the number of verb lexemes or by the overall UD corpora size, reduce cross-lingual variance by allowing for better estimations of the entropy and count4 measures.
Effects of subsampling Our subsampling procedure as described in §3.2 eliminates the circularity confound by fixing the sample size, albeit at the expense of statistical power. Results without subsampling validate our cautious approach, especially for VFC: the mean ρ between VFC and frequency rank is a staggering 0.996 (SD=.004), i.e., almost perfect correlations for all languages. Performing subsampling but then including verbs below the cutoff results in similar numbers (mean ρ =.996,
4A version of the plot using the VFC measure, which shows a similar pattern, is shown in Appendix, Fig. 7a.
SD = .073 for VFC). These results are arguably a reflection of the circularity confound, eliminated using subsampling.
A visual confirmation of this effect is possible when we plot individual verb lexemes by their frequency ranks on the x-axis and VFE and VFC on y-axes. As seen in Fig. 4, subsampling introduces a flattening effect to the VFE and VFC metrics of verbs with frequency above the subsampling threshold ∆ (i.e., verbs to the left of the dotted lines), such that the plot density makes the correlations less visually clear (which can nevertheless be statistically confirmed); in contrast, the clean correlations for the excluded verbs with frequency below ∆ (i.e., verbs to the right) reflect how circularity confound would have clouded our analysis without subsampling.
For the VFE measure, cross-entropy estimation alone reverses the circularity confound, even without subsampling (§ 3.2). The stronger correlation (mean ρ =.849, SD = .073) we observe without subsampling helps to confirm that our hypothesis apply to less frequent verbs that were elsewhere excluded. Fig 3b shows cross-linguistic variation in the correlation coefficient (VFE) without subsampling. Compared to the subsampled version
(Fig 3a), we also observe a greater correlation between the sample size and correlation strength, as the larger number of less frequent verbs for languages with a larger sample size would mean a stronger confound.
Impact of UD Relations We now investigate whether the selection of the UD relations impact the results, i.e., whether certain UD relations drive
the observed correlations more than others. We randomly sampled subsets of sizes 5, 10, 15 from the full set of 16 UD relations, and compared their correlation results with those of a subset including only the 6 core UD relations and of another including only the 10 non-core UD relation. Across the random subsets, larger sets lead to stronger correlations (Figure 5). These results suggest that the observed correlation reflects an across-the-board statistical pattern across a larger set of relation types, and not an artifact of any particular relation type.
Next, in order to compare the effects of the specific relations, we fitted a separate linear mixedeffects model for each relation, predicting the regression coefficient from the presence or absence of that relation in a random subset, with random adjustments for each language. We fitted the model on the set of 1000 randomly sampled subsets of size 5. The models are defined formally in Appendix A.
Fig. 6 shows each relation as a linear function with the fixed effect coefficient as the slope and the fixed effect intercept. The value at UD relation = 1 is therefore the predicted correlation coefficient when the relation is present. Note that a negative slope in the figure only indicates that including the relation instead of other relations on average has a negative effect on correlation strength, i.e., it has a less positive effect than average on correlation strength.
Relations differ in their impact. The presence of nsubj and obl have the strongest positive effect on the correlation strength. The latter is expected, as obl encompasses a variety of features often part of frame encoding, including case markings on ad-
juncts, passive verb use, and dative alternation. The strong positive effect of nsubj is more surprising, but the variety of non-predicative uses of verbs, which often involve the absence of or different case markings on nsubj, may provide an explanation. There is no evidence that core and non-core relations have systematically different effects based on this metric.

heading: 6 Discussion
text: Results across 79 languages provide strong support to our hypothesis of a positive correlation between a verb’s frequency and the diversity of its valency frame. By controlling for the circularity confound, we are able to determine that this correlation stands even after we exclude the effect of the more frequent verbs having more opportunities to appear. The cross-linguistic coverage further confirms that the results indicate a general trend and are not specific to individual languages.
While the correlational nature of our findings precludes us from drawing definitive conclusions as to which causal pathway is behind the observed regularities, the results match predictions made by efficiency-based theories and thus corroborate them. As we have detailed in §2, considerations from learnability in language acquisition, online production and comprehension converge to predict the correlation between verb frequency and valency frame diversity and provide possible casual explanations for them. In particular, our results complement a number of existing studies focusing a correlation between word frequency and number of meanings (as a measure of semantic ambiguity) (Hoffman et al., 2013) but test our hypothesis independently on the morphosyntax; seen from a processing perspective, the valency frames may serve as additional morphosyntactic disambiguation for verbs with multiple or broad meanings, suggesting a close interaction between lexical semantics and morphosyntax. In this way, our study extends previous work on natural language lexicons and shows that they are optimized for efficiency not just in word-internal proprieties but also in their interface with grammar.
Additionally, our findings show that different syntactic relations differ in their impact on the correlation, but do not support a clear-cut distinction between core and non-core dependents, otherwise understood as arguments and adjuncts. The quantitative trend was supported both by core and
non-core dependents, and larger sets of relations strengthened the observed correlation. This is evidence in support of a more graded distinction between different verb dependencies rather than a binary one.
While the predicted pattern holds across languages, we observe variation in correlation strength between languages. Understanding whether these differences reflect typologically meaningful variation is an interesting task for future work. One possibility is that the UD annotation scheme does not sufficiently account for cross-linguistic differences in argument encoding: e.g. in Chinese, many outlier verbs that have high frequency but low VFE are coverbs, which serve semantic functions similar to prepositions in English while are syntactically verbs (e.g., 自‘from’, 隨‘follow / with’). Their have narrow semantic functions and are associated with a relatively small number of valency frames compared to similarly frequent verbs. This brings into question the strictness of word category boundaries, and fuller picture may need to better situate the verb category within the overall lexicon.

heading: 7 Conclusion
text: Across 79 languages, we have provided evidence for a cross-linguistic quantitative trend in the organization of valency: More frequent verbs are associated with a larger diversity of valency frames, both when estimated by count-based or entropy-based metrics. Crucially, we showed that this pattern is not driven by differences in the number of observations between high- and low-frequency verbs, and persists even when equalizing the number of observations available for each verb. Extending a recent line of research studying the lexicon’s efficiency for human communication and language use, these results suggest that such considerations apply not just to word-internal structure, but also to the way words are integrated within the sentence.
Limitations
In §2, we derived our prediction from efficiency considerations in both language production and language comprehension. However, our methods did not allow us to determine which of these pressures are ultimately responsible for the observed pattern. Indeed, it is an ongoing debate to what extent apparently efficient properties of language respect optimization for production or comprehension. It is possible that, in the context of valency
organization, these perspectives will make distinct predictions. For instance, theories based on comprehension might induce asymmetries between dependents before and after the head. Teasing these apart in order to determine the contributions of production and comprehension is an interesting problem, beyond our scope here.
A second limitation is that, even though we considered data from 79 languages, we are not able to make truly universal claims. Some language families, such as the Indo-European languages, are substantially over-represented in available corpus data, whereas languages from some parts of the world, in particular Africa and indigenous languages of the Americas, are under-represented, both in the number of languages and language families, as well as in the corpora size.
A third limitation is that corpus data is finite and that in particular the circularity confound forced us to introduce biases to the estimation in order to eliminate that confound. While our results unambiguously show that the predicted correlation exists, a more precise estimation will be affected by these factors.
Fourth, we study a very simple operationalization of valency in terms of UD relations, far simpler than the sophisticated representations typically used in formal linguistic theories. This is due to the fact that such sophisticated representations are not currently available at scale and across languages. While we did verify that the pattern held across languages, we opted for cross-linguistic applicability over language-specific analysis.

heading: Acknowledgments
text: ST would like to thank Annemarie Verkerk for her help and patient guidance in getting this project off the ground. We also thank the anonymous reviewers and action editor for their helpful feedback and suggestions for our paper.

heading: A Linear Mixed-Effect Model
text: Formally, for the i-th relation across K languages, we fit a linear mixed-effect model with with N different randomly subsampled subsets of relations,
y = α(i) + β(i)X + Zu+ ϵ
where y ∈ [−1, 1]N is the outcome variable, i.e. the strength of the rank correlation between verb frequency and valency frame entropy; X ∈ {0, 1}N indicates, for each of the N subsets, whether the i-th relation is in that subset; α(1) is the fixed effect intercept; β(1) is a fixed effect coefficient; Z is a N ×K design matrix of the random effect; u is a K × 1 vector of the random effect of language; and ϵ is a N × 1 column of residuals.
B Additional Figures

heading: Future Work
text: Understanding whether these differences reflect typologically meaningful variation is an interesting task for future work. One possibility is that the UD annotation scheme does not sufficiently account for cross-linguistic differences in argument encoding: e. g. in Chinese, many outlier verbs that have high frequency but low VFE are coverbs, which serve semantic functions similar to prepositions in English while are syntactically verbs (e. g. , 自‘from’, 隨‘follow / with’). Their have narrow semantic functions and are associated with a relatively small number of valency frames compared to similarly frequent verbs. This brings into question the strictness of word category boundaries, and fuller picture may need to better situate the verb category within the overall lexicon