The origins of Zipf’s meaning-frequency law
In his pioneering research, G. K. Zipf observed that more frequent words tend to have more meanings, and showed that the number of meanings of a word grows as the square root of its frequency. He derived this relationship from two assumptions: that words follow Zipf’s law for word frequencies (a power law dependency between frequency and rank) and Zipf’s law of meaning distribution (a power law dependency between number of meanings and rank). Here we show that a single assumption on the joint probability of a word and a meaning suffices to infer Zipf’s meaning-frequency law or relaxed versions. Interestingly, this assumption can be justified as the outcome of a biased random walk in the process of mental exploration.
G. K. ? (?) investigated many statistical regularities of language. Some of them have been investigated intensively such as Zipf’s law for word frequencies (?, ?, ?, ?, ?) or Zipf’s law of abbreviation (?, ?, ?). Some others, such as Zipf’s law of meaning distribution have received less attention. In his pioneering research, ? (?) found that more frequent words tend to have more meanings. The functional dependency between , the number of meanings of a word, and , the frequency of a word, has been approximated with (?, ?, ?, ?)
where is a constant such that . Eq. 1 defines Zipf’s meaning-frequency law. Equivalently, the meaning-frequency law can be defined as
G. K. Zipf derived the meaning-frequency law assuming two laws, the popular Zipf’s law for word frequencies and the law of meaning distribution. On the one hand, Zipf’s law for word frequencies states the relationship between the frequency of a word and its rank (the most frequent word has rank , the 2nd most frequent word has rank and so on) as
where is a constant (?, ?, ?). On the other hand, the law of meaning distribution (?, ?, ?) states that
where . Notice that is still the rank of a word according to its frequency. The constants , and can be estimated applying some regression method as in Zipf’s pioneering research (?, ?, ?).
where and are constants such that . Eq. 5 can be seen as relaxation of Eq. 3. Similarly, Heaps’ law on the relationship between , the number of types, as function of , the number of tokens, is defined as with (?, ?), a relaxed version of (see ? (?) for a critical revision of the power law model for Heaps’ law).
The meaning-frequency law (Eq. 1) and the law of meaning distribution (Eq. 4) predict the number of meanings of a word using different variables as predictors. The meaning-frequency law has been confirmed empirically in various languages: directly through Eq. 1 in Dutch and English (?, ?) or indirectly through Eq. 4 and the assumption of Zipf’s law (?, ?, ?) in Turkish and English. Qualitatively, the meaning-frequency law defines a positive correlation between frequency and the number of meanings. Using a proxy of word meaning the qualitative version of the law has been found in dolphin whistles (?, ?) and in chimpanzee gestures (?, ?). Thus, the law is a candidate for a universal property of communication.
? (?) argued that Eq. 1 with follows from Eq. 3 with and Eq. 4 with . Indeed, it has been proven that Eq. 1 with follows from Eqs. 1 and 4 (?, ?). Here we consider alternative derivations of Eq. 1 or relaxed versions of Eq. 1 from the assumption of a biased random walk (?, ?, ?) over word-meaning associations. The remainder of the article is organized as follows.
First, we will present the mathematical framework.
Second, we will present a minimalist derivation of the meaning-frequency law (Eq. 1) with law that is based on just one assumption on the joint probability of a word and a meaning. Suppose a word that is connected to meanings and a meaning that is connected to words. Assuming that the joint probability of and is proportional to if and are connected and zero otherwise, suffices to obtain Eq. 1 with . A problem of the argument is that the definition is somewhat arbitrary and theoretically superficial.
Third, we will replace this simplistic assumption by a more fundamental assumption, namely that the joint probability of and is proportional to if and are connected and zero otherwise. This assumption is a more elegant solution for two reasons: it corrects the arbitrariness of the assumption of the minimalist derivation, fits into standard network theory and it can be embedded into a general theory of communication. From this deeper assumption we derive the meaning-frequency law following three major paths. The 1st path consists of assuming the principle of contrast (?, ?) or the principle of no synonymy (?, ?, p. 67), namely, for all words, which leads exactly to Eq. 1. The 2nd path consists of assuming that meaning degrees are mean independent of word degrees, which leads to a mirror of the meaning-frequency law (Eq. 2)
where is the expectation of , the probability of a word, knowing that its degree is . Notice that is linked with as , where is the length of a text in tokens, i.e. total number of tokens (the sum of all type frequencies) (?, ?). The 3rd path makes no assumption to obtain a relaxed version of the meaning-frequency law, namely the number of meanings is bounded above and below by two power-laws over , i.e.
where and are constants such that . The result can be summarized as
Put together, these three paths strongly suggest that languages are channeled to reproduce Zipf’s meaning-frequency law.
Fourth, we will review a family of optimization models of communication that was put forward to investigate the origins of Zipf’s law for word frequencies (?, ?) but that has recently been used to shed light on patterns of vocabulary learning and the mapping of words into meanings (?, ?). Interestingly, models from that family give Eq. 1 with (?, ?). Crucially, however, the true exponent is (?, ?, ?). The mismatch should not be surprising. Imagine that a speaker has to choose a word for a certain meaning. Those models assume that given a meaning, all the words with that meaning are equally likely (?, ?). However, this simple assumption is not supported by psycholinguistic research (?, ?). We will show how to modify their definition so that the words that are used for a certain meaning do not need to be equally likely and one can obtain Eq. 1 with or relaxed versions. Finally, we will discuss the results, highlighting the connection with biased random walks, and indicate directions for future research.
A mathematical framework
As in the family of models of communication above, we assume a repertoire of words, ,…,,… and a repertoire of meanings, ,…,,…,. Words and meanings are associated through an adjacency matrix : if and are associated ( otherwise). defines the edges of an undirected bipartite network of word-meaning associations. The degree of the -th word is
while the degree of the -th meaning is
In human language, the relationship between sound and meaning has been argued to be arbitrary to a large extent (?, ?, ?, ?). That is, there is no intrinsic relationship between the word form and its meaning. For example the word "car" is nothing like an actual automobile. An obvious exception are onomatopoeias, which are relatively rare in language. However, despite the immense flexibility of the world’s languages, some sound-meaning associations are preferred by culturally, historically, and geographically diverse human groups (?, ?). The framework above is agnostic concerning the type of association between sound and meaning. By doing that, we are borrowing the abstract perspective of network theory, that is a priori neutral concerning the nature or the origins of the edges (?, ?, ?). Our framework could be generalized to accommodate Peirce’s classic types of reference, i.e., iconic, indexical and symbolic (?, ?), or the state-of-the-art on the iconicity/systematicity distinction (?, ?). An crucial reason to remain neutral is that the distinctions above were not made when defining the laws of meaning that are the target of this article.
The framework allows one to model lexical ambiguity: a lexically ambiguous word is a word such that its degree is greater than one. Although the model starts from a flat hierarchy of concepts (by default all concepts have the same degree of generality), a word with an abstract meaning could be approximated either as a word linked to a single abstract concept or as a word linked to the multiple specific meanings it covers (?, ?). As for the latter approach, the word for vehicle would be linked to the meanings for car, bike, ship, airplane,…
Suppose that is the joint probability of the unordered pair formed by and . By definition,
The probability of is
and the probability of is
Our model shares the assumptions of distributional semantics that the meaning of a word is represented as a vector of the weights of different concepts for that word (?, ?). In our framework, the meaning of the word , is represented by the -dimensional vector
The joint probabilities for all words and meanings defines a weighted matrix of the same size of . In the coming sections, we will derive the meaning frequency-law defining as a function of . Put differently, we will derive the law from a weighted undirected bipartite graph that is build from the unweighted undirected graph defined by . This organization in two graphs (one unweighted and the other weighted) instead of a single weighted graph is borrowed from successful models of communication (?, ?) and allows one to apply the theory of random walks (?, ?, ?) as we will see later on.
A minimalist derivation of the law
The law of meaning distribution can be derived by making just one rather simple assumption, i.e.
Applying Eq. 10, one obtains
where is a normalization constant defined as
namely Eq. 1 with . Our derivation of the strong and relaxed version of the meaning-frequency law is simpler than that of Zipf’s in the sense that it requires assuming a smaller number of equations (we are assuming only Eq. 13 while Zipf assumed Eqs. 3 and 4). However, the challenge of our approach is the justification of Eq. 13.
A theoretical derivation of the law
The definition of in Eq. 13 suffices as a model but not for the construction of a real theory of language. Eq. 13 is simple but somewhat arbitrary: the degree of the word, , contributes raised to 1 but the degree of the meaning, has no direct contribution, or one may say that it contributes raised to 0. Therefore, a less arbitrary equation would be
Eq. 15 is theoretically appealing for various reasons. If is regarded as the weight of the association between and , it defines the general form of the relationship between the weight of and edge and the product of the degrees of vertices at both ends that is found in real networks (?, ?). For this reason, a unipartite version of Eq. 15 is assumed to study dynamics on networks (?, ?). When , it matches the definition of models about the origins of Zipf’s law for word frequencies (?, ?), the variation of the exponent of the law (?, ?, ?) and vocabulary learning (?, ?). When , it defines an approximation to the stationary probability of observing a transition involving and in a random walk on a network that is biased to maximize the entropy rate of the walks (Appendix A), thus suggesting that the meaning-frequency law could be a manifestation of a particular random walk process on semantic memory.
Two equivalent linguistic principles, the principle of contrast (?, ?) and the principle of no synonymy (?, ?, p. 67) can be implemented in our model as . From an algebraic standpoint, the condition is equivalent to orthogonality of the word vectors of the matrix . If indicates the row vector of for the -th word, and are orthogonal if and only if , where the dot indicates the scalar product of two vectors. To simplify matters, we assume that there is no row vector of that equals , a vector that has in all components. So far we have used and to refer, respectively, to the degree of the -th word and the -th meaning. We define and as the degree of the word and the degree of the meaning of the -th edge. and are components of the vectors and , respectively. We have because by definition. A deeper insight can be obtained with the concept of remaining degree, the degree at one end of the edge after subtracting the unit contribution of the edge (?, ?). The vectors of remaining degrees are then
The condition is equivalent to . leads to but trivially for being null.
and finally Eq. 1 with
When , we get again. Interestingly, the principle of contrast follows from the principle of mutual information maximization, a more fundamental principle that allows one to predict vocabulary learning in children and that can be combined with the principle of entropy minimization to predict Zipf’s law for word frequencies (?, ?). With Eq. 15, we follow Bunge (?, ?, pp. 32-33) preventing scientific knowledge from becoming “an aggregation of disconnected information” and aspiring to build a “system of ideas that are logically connected among themselves”.
It is possible to obtain a relaxed meaning frequency-law under more general conditions. In particular, we would like to get rid of the heavy constraint that meaning degrees cannot exceed one. Suppose that is a constant such that . Some obvious but not very general conditions are for all or for all . It is easy to see that they lead again to Eq. 13 when . A more general condition can be defined as follows. First, we define as the conditional expectation of given for an edge. Here and are the degrees at both ends of an edge. Then suppose that is given and that the is mean independent of , namely for each value of (?, ?, ?), then the expectation of (as defined in Eq. 18) knowing is
which can be seen as a regression model (?, ?) for the meaning-frequency law (Eq. 1) with word degree as predictor. Notice that mean independence is a more general condition than mutual or statistical independence but a particular case of uncorrelation (?, ?).
So far, we have seen ways of obtaining the meaning-frequency law from Eq. 15 making further assumptions. It is possible to obtain a relaxed version of the meaning-frequency law making no additional assumption (except Eq. 15 or the biased random walk that justifies it). Eq. 18 can be expressed as
Eq. 19 leads to
Recalling , these results can be summarized using asymptotic notation as or . The power of the bounds above depends on the gap between and . The gap can be measured with the ratio
where and are the minimum and the maximum meaning degree, respectively. The principle of mutual information maximization between words and meanings, a general principle of communication (?, ?), puts pressure for concordance with the meaning-frequency law. To see it, we consider two cases: and . When , its maximization predicts (Appendix B). As unlinked meanings are irrelevant (they do not belong to the support set), we have . As pressure for mutual information maximization increases, tends to 1 and thus tends to . Put differently, the gap between the upper and the lower bound in Eq. 20 reduces as pressure for mutual information maximization increases. When , mutual information maximization predicts that , where is an integer such that (Appendix B). We have seen above that one obtains the meaning-frequency law (Eq. 1) immediately from Eq. 15 when is constant. We conclude that the chance of observing the meaning-frequency law increases as pressure for mutual information maximization increases.
A family of optimization models of communication
Here we revisit a family of optimization models of communication (?, ?) in light of the results of the previous sections. These models share the assumption that the probability that a word is employed to refer to meaning is proportional to , i.e.
to Eq. 21, we obtain
We adopt the convention when .
Eq. 22 defines the probability of transition of a standard (unbiased) random walk to a word (?, ?), i.e. given a meaning, all related words are equally likely. This is unrealistic in light of picture naming norms (?, ?, ?). Consider the picture-naming norms compiled by ? (?), who simply asked participants to name 260 black-and-white line drawings of common objects. For some objects (e.g., balloon, banana, sock, star) there was 100% agreement among the participants for the word used to name the pictured object. However, for other objects there was considerable variability in the word used to name the pictured object. Important for the present argument, the other words that were used in such cases were not selected with equal likelihood. For example, the picture of a wineglass had 50% agreement, with the word glass (36% of the responses) and the word goblet (14% of the responses) also being used to name the object, showing that all the words that could be used for a given meaning are not equally likely. Although subjects tend to provide more specific responses when the concept is presented in textual form with respect to a visual form presentation (?, ?), we used the visual case simply to challenge the assumption of an unbiased random walk in general and justify a more realistic approach.
Recalling the definition of conditional probability
Recalling the definition of in Eq. 9, it is easy to realize that Eq. 22 is a particular case of Eq. 23 with . While the family of models above stems from a concrete definition of a conditional probability, i.e. in Eq. 22, the general model that we have presented in this article is specified by a definition of the joint probability, i.e. in Eq. 15.
Models within that family are generated through
assuming an unbiased random walk from a meaning to a word (Eq. 22) and making different assumptions on .
If one assumes that all meanings are equally likely ( with ) one obtains the 1st model (?, ?). If one assumes that the probability of a meaning is proportional to its degree () one obtains the 2nd model (?, ?). While in the 2nd model defines an unbiased random walk from to (all ’s connected to are equally likely), this is not necessarily the case for the 1st model (?, ?). Therefore, the 2nd model defines a pure unbiased random walk while the 1st model is unbiased from meaning to words but biased from words to meanings.
Now we will introduce a generalized version of the family of models above consisting of replacing Eq. 22 by Eq. 23 and generating the corresponding variants of the 1st and the 2nd model applying the same procedure as in the original family, namely via Eq. 25. Notice that Eq. 23 defines the probability of reaching from in a biased random walk when .
Concerning the 1st model, suppose that the probabilities of the meanings are given a priori (they are independent from the matrix), e.g., all meanings are equally likely. Then it is easy to show that the model yields a relaxed version of the meaning frequency law, namely , the number of meanings is bounded above and below by two power-laws, i.e. (Appendix C)
We have seen that it is possible to obtain the meaning-frequency law (Eqs. 1 and 2) from Eq. 15 making certain assumptions. We have also seen that a relaxed version of the law (Eq. 7 can be obtained from Eq. 15 without making any further assumption. Our findings suggest that word probabilities are channeled somehow to manifest the meaning-frequency law. We have seen that the principle of mutual information maximization contributes to the emergence of the law. Our derivation is theoretically appealing for various reasons. First, it is more parsimonious than G. K. Zipf’s concerning the number of equations that are assumed (we only need Eq. 15 while Zipf involved Eqs. 3 and 4). Second, it can help a family of optimization models of language to reproduce the meaning-frequency law.
Therefore, a crucial assumption is Eq. 16, that we have justified as the outcome of a random walk that is biased to maximize the entropy rate of the paths (Appendix A). A random walk is the correlate in network theory of the concept of mental exploration (navigation without a target or nonstrategic search) in cognitive science and related fields (?, ?). Semantic memory processes can be usefully theorized as searches over a network (?, ?, ?) or some semantic space (?, ?). These approaches support the hypothesis of a Markov chain process for memory search (?, ?), provide a deeper understanding of creativity (?, ?) and help to develop efficient navigation strategies (?, ?).
A random walk in a unipartite word network of word-word associations has been argued to underlie Zipf’s law for word frequencies (?, ?). Here we contribute with a new hypothesis linking random walks with a linguistic law: that the meaning-frequency law would be an epiphenomenon of a biased random walk over a bipartite network of word-meaning associations in the process of mental exploration. The bias consists of exploiting local information, namely the degrees of first neighbours (?, ?). Transitions to nodes with higher degree are preferred. Our model shows that it is possible to approximate the optimal solution to a problem (maximizing the entropy rate of the paths) following an apparently nonstrategic search (?, ?, ?).
The probability of a word in Eq. 18 defines the probability that a random walker visits the word in the long run. This probability is what the PageRank algorithm estimates in the context of a standard (non-biased) random walk (?, ?). The assumption of a random walk with the particular bias above could help to improve random walk/PageRank methods to predict the prominence in memory of a word (?, ?) or the importance of a tag (?, ?). A virtue of our biased random walk is that it predicts an uneven conditional probability of a word given a meaning (Eq. 23) as it happens in real language (?, ?). A standard (uniform) random walk cannot explain this fact and for that reason the family of optimization models of language revisited above fails to reproduce the meaning-frequency law with .
Although biased random walks have already been used to solve information retrieval problems (see ? (?) and references therein), a bias based on the degree of neighbours has not been considered as far as we know. We hope that our results stimulate further research on linguistics laws and biased random walks in the information sciences. Specifically, we hope that our article becomes the fuel of future empirical research.
We are specially grateful to R. Pastor-Satorras and Massimo Stella for helpful comments and insights. We also thank S. Semple, M. Gentili and E. Bastrakova for helpful discussions. This research was supported by the grant APCOM (TIN2014-57226-P) from MINECO (Ministerio de Economía y Competitividad) and the grant 2014SGR 890 (MACDA) from AGAUR (Generalitat de Catalunya).
Appendix A Random walks
We will show that Eq. 15 defines the probability of observing a transition between and in any direction in a biased random walk. We will proceed in two steps. First, we will summarize some general results on biased random walks on unipartite networks and then we will adapt them to bipartite networks.
Suppose a unipartite network of nodes that is defined by an adjacency matrix such that if the -th and the -th node are connected and otherwise. Let be the degree of the -th node, namely,
Suppose a random walk over the vertices of a network where is the probability of jumping from to . A first order approximation to the that maximizes the entropy rate is (?, ?)
We choose a generalization (?, ?)
Now we adapt the results above to a bipartite graph of word-meaning associations. As the graph is bipartite, the random walker will be alternating between words and meanings. The probability that the vertex visited is a word is (the same probability for a meaning). Suppose that there are words and meanings. Recall that the bipartite network of word-meaning associations is defined by an adjacency matrix such that if the -th word and the -th meaning are connected and otherwise. is the degree of the -th word is (Eq. 8) whereas is the degree of the -th meaning (Eq. 9). The probability of jumping from to becomes (recall Eq. 28)
The probability of jumping from to is
where . Similarly, the probability of observing the transition from to is
Therefore the stationary probability of observing a transition between and in any direction (from to or from to ) is
with , as we wanted to show.
Finally, we will link , the probability of a word that is used in the main text to derive the meaning-frequency law, with . Notice that , the latter being the probability of visiting vertex knowing that it belongs to the partition , the partition of words. Since the graph is bipartite, , probability that the random walk is visiting a vertex of partition , is . The joint probability of visiting vertex and that the vertex belongs to is
Then is the stationary probability of visiting in a biased random walk knowing that the vertex is in .
Appendix B Mutual information maximization
Suppose that is the mutual information between words () and meanings (), that can be defined as
where is the entropy of words and is the conditional entropy of words given meanings. For the case , the configurations that maximize when are defined by two conditions (?, ?)
with for .
When , those configurations are the symmetric, i.e. (?, ?)
with for .
Here we will show that the configurations that maximize are the same as in the case when is a positive and finite real number (). By symmetry, it suffices to show it for the case . We will proceed in three steps. First, deriving the configurations minimizing . Second, showing that the configurations above yield maximum . Third, showing that they are the only configurations.
Step 1: Recall that
where is the conditional entropy of words given the meaning . Eq. 24 implies that is equivalent to and then
it is easy to see that when for : by continuity since as (?, ?, p. 14) and obviously . Eq. 23 implies that is equivalent to being the only neighbour of , i.e. . Therefore, implies for .
Step 2: notice that the 2nd condition of the case above implies (recall Step 1). The 2nd condition transforms Eq. 17 into
and Eq. 18 into
Adding the 1st condition, one obtains
and then (as all words are equally likely). Thus, is taking its maximum possible value whereas is taking its minimum value. As , it follows that is maximum.
Step 3: notice that
If the 2nd condition fails, then and thus even if is maximum because of Eq. 34. Thus, the 2nd condition is required to maximize .
If the 1st condition fails (while the 2nd condition holds), then words are not equally likely as the probability of a word is proportional to a power of its degree (Eq. B). Then one has that and it follows that is not maximum because .
Appendix C New models
when . Eq. 35 leads to
recalling the definition of in Eq. 8. Equivalently,
namely a relaxed version of the meaning-frequency law when .
- Abbott, Austerweil, GriffithsAbbott et al. Abbott, J. T., Austerweil, J. L., Griffiths, T. (2015). Random walks on semantic networks can resemble optimal foraging. Psychological Science, 122, 558–569.
- Allegrini, Gricolini, PalatellaAllegrini et al. Allegrini, P., Gricolini, P., Palatella, L. (2004). Intermittency and scale-free networks: a dynamical model for human language complexity. Chaos, solitons and fractals, 20, 95-105.
- Baayen Moscoso del Prado MartínBaayen Moscoso del Prado Martín Baayen, H., Moscoso del Prado Martín, F. (2005). Semantic density and past-tense formation in three Germanic languages. Language, 81, 666-698.
- Baeza-Yates NavarroBaeza-Yates Navarro Baeza-Yates, R., Navarro, G. (2000). Block addresing indices for approximate text retrieval. Journal of the American Society for Information Science, 51(1), 69-82.
- Baronchelli, Castellano, Pastor-SatorrasBaronchelli et al. Baronchelli, A., Castellano, C., Pastor-Satorras, R. (2011). Voter models on weighted networks. Physical Review E, 83, 066117.
- Baronchelli, Ferrer-i-Cancho, Pastor-Satorras, Chatter, ChristiansenBaronchelli et al. Baronchelli, A., Ferrer-i-Cancho, R., Pastor-Satorras, R., Chatter, N., Christiansen, M. (2013). Networks in cognitive science. Trends in Cognitive Sciences, 17, 348-360.
- Barrat, Barthélemy, Pastor-Satorras, VespignaniBarrat et al. Barrat, A., Barthélemy, M., Pastor-Satorras, R., Vespignani, A. (2004). The architecture of complex weighted networks. Proc. Nat. Acad. Sci. USA, 101(11), 3747-3752.
- BarthélemyBarthélemy Barthélemy, M. (2011). Spatial networks. Physics Reports, 499(1), 1 - 101. doi: http://dx.doi.org/10.1016/j.physrep.2010.11.002
- Blasi, Wichmann, Hammarström, Stadler, ChristiansenBlasi et al. Blasi, D. E., Wichmann, S., Hammarström, H., Stadler, P., Christiansen, M. (2016). Sound-meaning association biases evidenced across thousands of languages. Proceedings of the National Academy of Sciences, 113(39), 10818-10823.
- Bourgin, Abbott, Griffiths, K.A., VulBourgin et al. Bourgin, D. D., Abbott, J., Griffiths, T., K.A., K. S., Vul, E. (2014). Empirical evidence for Markov Chain Monte Carlo in memory search. In Proceedings of the 36th annual meeting of the cognitive science society (p. 224-229).
- BungeBunge Bunge, M. (2013). La ciencia. su método y su filosofía. Pamplona: Laetoli.
- Capitán et al.Capitán et al. Capitán, J. A., Borge-Holthoefer, J., Gómez, S., Martínez-Romo, J., Araujo, L., Cuesta, J. A., Arenas, A. (2012, 08). Local-based semantic navigation on a networked representation of information. PLOS ONE, 7(8), 1-10.
- ClarkClark Clark, E. (1987). The principle of contrast: a constraint on language acquisition. In B. MacWhinney (Ed.), Mechanisms of language acquisition. Hillsdale, NJ: Lawrence Erlbaum Associates.
- Conrad MitzenmacherConrad Mitzenmacher Conrad, B., Mitzenmacher, M. (2004). Power laws for monkeys typing randomly: the case of unequal probabilities. IEEE Transactions on Information Theory, 50(7), 1403-1414.
- Cover ThomasCover Thomas Cover, T. M., Thomas, J. A. (2006). Elements of information theory. New York: Wiley. (2nd edition)
- DeaconDeacon Deacon, T. W. (1997). The symbolic species: the co-evolution of language and the brain. New York: W. W. Norton & Company.
- Dingemanse, Blasi, Lupyan, Christiansen, MonaghanDingemanse et al. Dingemanse, M., Blasi, D. E., Lupyan, G., Christiansen, M. H., Monaghan, P. (2015). Arbitrariness, iconicity, and systematicity in language. Trends in Cognitive Sciences, 19(10), 603 - 615. doi: https://doi.org/10.1016/j.tics.2015.07.013
- Duarte Torres, Hiemstra, Weber, PavelDuarte Torres et al. Duarte Torres, S., Hiemstra, D., Weber, I., Pavel, S. (2014). Query recommendation in the information domain of children. Journal of the Association for Information Science and Technology, 65(7), 1368–1384.
- Duñabeitia et al.Duñabeitia et al. Duñabeitia, J. A., Meyer, D. C. A. S., Boris, B., Pliatsikas, C., Smolka, E., Brysbaert, M. (2017). MultiPic: A standardized set of 750 drawings with norms for six European languages. The Quarterly Journal of Experimental Psychology, in press.
- FedorowiczFedorowicz Fedorowicz, J. (1982). The theoretical foundation of Zipf’s law and its application to the Bibliographic Database Environment. J. Am. Soc. Inf. Sci., 33, 285-293.
- Ferrer-i-CanchoFerrer-i-Cancho Ferrer-i-Cancho, R. (2005a). The variation of Zipf’s law in human language. European Physical Journal B, 44, 249-257.
- Ferrer-i-CanchoFerrer-i-Cancho Ferrer-i-Cancho, R. (2005b). Zipf’s law from a communicative phase transition. European Physical Journal B, 47, 449-457.
- Ferrer-i-CanchoFerrer-i-Cancho Ferrer-i-Cancho, R. (2006). When language breaks into pieces. A conflict between communication through isolated signals and language. Biosystems, 84, 242-253.
- Ferrer-i-CanchoFerrer-i-Cancho Ferrer-i-Cancho, R. (2016a). Compression and the origins of Zipf’s law for word frequencies. Complexity, 21, 409-411.
- Ferrer-i-CanchoFerrer-i-Cancho Ferrer-i-Cancho, R. (2016b). The meaning-frequency law in Zipfian optimization models of communication. Glottometrics, 35, 28-37.
- Ferrer-i-CanchoFerrer-i-Cancho Ferrer-i-Cancho, R. (2016c). The optimality of attaching unlinked labels to unlinked meanings. Glottometrics, 36, 1-16.
- Ferrer-i-Cancho Díaz-GuileraFerrer-i-Cancho Díaz-Guilera Ferrer-i-Cancho, R., Díaz-Guilera, A. (2007). The global minima of the communicative energy of natural communication systems. Journal of Statistical Mechanics, P06009.
- Ferrer-i-Cancho GavaldàFerrer-i-Cancho Gavaldà Ferrer-i-Cancho, R., Gavaldà, R. (2009). The frequency spectrum of finite samples from the intermittent silence process. Journal of the American Association for Information Science and Technology, 60(4), 837-843.
- Ferrer-i-Cancho, Hernández-Fernández, Baixeries, Dębowski, MačutekFerrer-i-Cancho et al. Ferrer-i-Cancho, R., Hernández-Fernández, A., Baixeries, J., Dębowski, Ł., Mačutek, J. (2014). When is Menzerath-Altmann law mathematically trivial? A new approach. Statistical Applications in Genetics and Molecular Biology, 13, 633-644.
- Ferrer-i-Cancho et al.Ferrer-i-Cancho et al. Ferrer-i-Cancho, R., Hernández-Fernández, A., Lusseau, D., Agoramoorthy, G., Hsu, M. J., Semple, S. (2013). Compression as a universal principle of animal behavior. Cognitive Science, 37(8), 1565-1578.
- Ferrer-i-Cancho McCowanFerrer-i-Cancho McCowan Ferrer-i-Cancho, R., McCowan, B. (2009). A law of word meaning in dolphin whistle types. Entropy, 11(4), 688-701. doi: 10.3390/e11040688
- Ferrer-i-Cancho SoléFerrer-i-Cancho Solé Ferrer-i-Cancho, R., Solé, R. V. (2003). Least effort and the origins of scaling in human language. Proceedings of the National Academy of Sciences USA, 100, 788-791.
- Font-Clos, Boleda, CorralFont-Clos et al. Font-Clos, F., Boleda, G., Corral, A. (2013). A scaling law beyond Zipf’s law and its relation to Heaps’ law. New Journal of Physics, 15, 093033.
- Font-Clos CorralFont-Clos Corral Font-Clos, F., Corral, A. (2015). Log-log convexity of type-token growth in Zipf’s systems. Phys. Rev. Lett., 114, 238701. doi: 10.1103/PhysRevLett.114.238701
- GoldbergGoldberg Goldberg, A. (1995). Constructions: a construction grammar approach to argument structure. Chicago: Chicago University Press.
- Gómez-Gardeñes LatoraGómez-Gardeñes Latora Gómez-Gardeñes, J., Latora, V. (2008). Entropy rate of difussion process on complex networks. Physical Review E, 78, 065102(R).
- Griffiths, Steyvers, FirlGriffiths et al. Griffiths, T., Steyvers, M., Firl, A. (2007). Google and the mind. Predicting fluency with PageRank. Psychological Science, 18, 1069-1076.
- Hills, Jones, ToddHills et al. Hills, T., Jones, M., Todd, P. (2012). Optimal foraging in semantic memory. Psychological Science, 119, 431–440.
- Hobaiter ByrneHobaiter Byrne Hobaiter, C., Byrne, R. W. (2014). The meanings of chimpanzee gestures. Current Biology, 24, 1596-1600.
- HockettHockett Hockett, C. F. (1966). The problem of universals in language. In Universals of language (p. 1-29). Cambridge, MA: The MIT Press.
- Ilgen KaraoglanIlgen Karaoglan Ilgen, B., Karaoglan, B. (2007). Investigation of Zipf’s “law-of-meaning” on Turkish corpora. In 22nd international symposium on computer and information sciences (iscis 2007) (p. 1-6).
- Jäschke, Marinho, Hotho, Schmidt-Thieme, StummeJäschke et al. Jäschke, R., Marinho, L., Hotho, A., Schmidt-Thieme, L., Stumme, G. (2007). Tag recommendations in folksonomies. In J. N. Kok, J. Koronacki, R. L. de Mantaras, S. Matwin, D. Mladenič, A. Skowron (Eds.), Knowledge discovery in databases: Pkdd 2007: 11th european conference on principles and practice of knowledge discovery in databases, warsaw, poland, september 17-21, 2007. proceedings (pp. 506–514). Berlin, Heidelberg: Springer Berlin Heidelberg.
- Kenett AusterweilKenett Austerweil Kenett, Y., Austerweil, J. (2016a). Examining search processes in low and high creative individuals with random walks. In Proceedings of the 38th annual meeting of the cognitive science society (p. 313-318).
- KolmogorovKolmogorov Kolmogorov, A. N. (1956). Foundations of the theory of probability (2nd ed.). New York: Chelsea Publishing Company.
- Lund BurgessLund Burgess Lund, K., Burgess, C. (1996). Producing high-dimensional semantic spaces from lexical co-occurrence. Behavior Research Methods, Instruments, and Computers, 28(2), 203â208.
- Moreno-Sànchez, Font-Clos, CorralMoreno-Sànchez et al. Moreno-Sànchez, I., Font-Clos, F., Corral, A. (2016, 01). Large-scale analysis of Zipf’s law in English texts. PLOS ONE, 11(1), 1-19.
- NewmanNewman Newman, M. E. J. (2002). Assortative mixing in networks. Phys. Rev. Lett., 89, 208701.
- NewmanNewman Newman, M. E. J. (2010). Networks. an introduction. Oxford: Oxford University Press.
- Noh RiegerNoh Rieger Noh, J. D., Rieger, H. (2004). Random walks on complex networks. Physical Review Letters, 92, 118701.
- Page, Brin, Motwani, WinogradPage et al. Page, L., Brin, S., Motwani, R., Winograd, T. (1998). The PageRank citation ranking: bringing order to the web (Tech. Rep.). Stanford, CA: Stanford Digital Library Technologies Project.
- PinkerPinker Pinker, S. (1999). Words and rules: The ingredients of language. New York: Perseous Books.
- PoirierPoirier Poirier, D. J. (1995). Intermediate statistics and econometrics: A comparative approach. Cambridge: MIT Press.
- Ritz StreibigRitz Streibig Ritz, C., Streibig, J. C. (2008). Nonlinear regression with r. New York: Springer.
- SaussureSaussure Saussure, F. (1916). Cours de linguistique générale (C. Bally, A. Sechehaye, A. Riedlinger, Eds.). Lausanne and Paris: Payot.
- Sinatra, Gómez-Gardeñes, Lambiotte, Nocosia, LatoraSinatra et al. Sinatra, R., Gómez-Gardeñes, J., Lambiotte, R., Nocosia, V., Latora, V. (2011). Maximal-entropy random walks in complex networks with limited information. Physical Review E, 83, 030103(R).
- Smith, Huber, VulSmith et al. Smith, K. A., Huber, D. E., Vul, E. (2013). Multiply-constrained semantic search in the Remote Associates Test. Cognition, 128(1), 64 - 75.
- Snodgrass VanderwartSnodgrass Vanderwart Snodgrass, J. G., Vanderwart, M. (1980). A standardized set of 260 pictures: norms for name agreement, image agreement, familiarity, and visual complexity. Journal of Experimental Psychology: Human Learning and Memory, 6, 174-215.
- Strauss, Grzybek, AltmannStrauss et al. Strauss, U., Grzybek, P., Altmann, G. (2006). Word length and word frequency. In P. Grzybek (Ed.), Contributions to the science of text and language: Text, speech and language technology (Vol. 31, p. 277-294). Berlin: Springer.
- Thompson KelloThompson Kello Thompson, G., Kello, C. (2014). Walking across Wikipedia: a scale-free network model of semantic memory retrieval. Frontiers in Psychology, 5, 86.
- Tversky HemenwayTversky Hemenway Tversky, B., Hemenway, K. (1983). Categories of environmental scenes. Cognitive Psychology, 15, 121-149.
- ZipfZipf Zipf, G. K. (1945). The meaning-frequency relationship of words. Journal of General Psychology, 33, 251-266.
- ZipfZipf Zipf, G. K. (1949). Human behaviour and the principle of least effort. Cambridge (MA), USA: Addison-Wesley.