# Bottlenose Dolphins' Clicks Comply with Three Laws of Efficient Communication
## 1 Introduction  
The communicative aspect of various animal vocalisations is commonly evaluated through the degree of conforming to the laws of efficient communication, a series of observed regularities in the communicative use of linguistic code that go back to [PERSON]'s influential work [1] and have been further elaborated in the context of information theory [2, 3]. In the following, we consider  
* [PERSON]'s rank-frequency law--also known as [PERSON]'s law [1], formulated as \[f(r,\alpha;\beta)=c\times(r+\beta)^{-\alpha}\] where \(r\) is the frequency rank of a word, \(f\) is its frequency in a natural corpus, \(\alpha\) is a positive parameter, and \(\beta\) is a _rank-shifting_ constant introduced by [PERSON] [3]. Under [PERSON]'s condition, [PERSON]'s distribution is necessary, but not sufficient, for optimalinformation transmission [4]. [PERSON]'s law formalises a negative correlation between \(f\) and \(r\) for \(\alpha>0\), which converges into a linear dependency with a negative slope on a logarithmic scale. [PERSON]'s law has been observed at various levels of structural organisation in natural language, and in the broader context of the biological organisation of living systems, as well as at macro levels, such as larger ecological systems [5]. * [PERSON]'s law of abbreviation--broadly states that the length of a linguistic unit reversely correlates with its frequency of use; viz., more frequent sequences tend to be shorter [1]. Language studies on the relation between word length in terms of letters in English and Swedish and their frequencies [6] and the length of sentences in terms of the number of words for American English [7] found that it follows the gamma distribution for probability density: \[f(L,\alpha;\beta)=K\times L^{\alpha-1}\times e^{\frac{-L}{\beta}}\] where \(L\) is the sequence length, \(\alpha\) is the shape parameter determining the function's peak existence and location, and \(\beta\) is the scale parameter determining the spread of the distribution. * [PERSON]'s law--indicates a reverse correlation between a sequence's size and its constituent parts. In other words, the longer a sequence, the shorter its constituents, and vice versa. In natural language, the law applies at different levels of linguistic description, e.g., sentences (size measured in terms of number of clauses), words (measured in terms of number of syllables), and other linguistic units. [PERSON] [8] proposed the following formulation of this law: \[f(L,a,b,c)=a\times L^{b}\times e^{-cx},\] (1) where \(f\) is the size of a constituent, \(L\) is the size of the entire sequence, and \(a\), \(b\), and \(c\) are parameters. From the information-theoretic perspective, [PERSON]'s and [PERSON]'s laws are commonly seen as realisations of the tendency to compress or minimise the linguistic code without losing information [9]. Compression was even suggested as a universal principle guiding animal behaviour in general [9], as well as communication, although, concerning the latter, the results were somewhat dependent on the species and selected unit of analysis [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. These and other studies have reported that the laws of communication efficiency and structure apply to certain primates [8; 12; 16], mammals [14], cetaceans [15], and birds [17; 18], but not to others [13; 15]. This can be attributed to evolutionary, social, cognitive, environmental, phylogenetic, and other factors, which collectively contribute to the diversity in animal sound communication systems and may condition the need to align with linguistic laws for some species, but not for others. These factors are briefly outlined below:  
1. Evolutionary Pressures: Species that rely heavily on vocal communication for survival, such as primates, cetaceans, and certain birds, have evolved complex vocal systems optimised for efficiency and structure to enhance communication effectiveness. 2.