Last update: 16 July, 2022.
Launched: 27 December, 2011.
Note: the largest public bibliography of references on Zipf’s law for word frequencies is available here. Here we only offer a selection of references on Zipf’s law on animal behavior and organic chemistry.
Hint for browsing: Heaps’ law is another name for Herdan’s law.
1999
McCowan, Brenda; Hanser, Sean F; Doyle, Laurance R
Quantitative tools for comparing animal communication systems: information theory applied to bottlenose dolphin whistle repertoires Journal Article
In: Animal Behaviour, vol. 57, no. 2, pp. 409 - 419, 1999, ISSN: 0003-3472.
Abstract | Links | BibTeX | Tags: Zipf's law for word frequencies
@article{MCCOWAN1999409,
title = {Quantitative tools for comparing animal communication systems: information theory applied to bottlenose dolphin whistle repertoires},
author = {Brenda McCowan and Sean F Hanser and Laurance R Doyle},
url = {http://www.sciencedirect.com/science/article/pii/S0003347298910004},
doi = {https://doi.org/10.1006/anbe.1998.1000},
issn = {0003-3472},
year = {1999},
date = {1999-01-01},
journal = {Animal Behaviour},
volume = {57},
number = {2},
pages = {409 - 419},
abstract = {Comparative analysis of nonhuman animal communication systems and their complexity, particularly in comparison to human language, has been generally hampered by both a lack of sufficiently extensive data sets and appropriate analytic tools. Information theory measures provide an important quantitative tool for examining and comparing communication systems across species. In this paper we use the original application of information theory, that of statistical examination of a communication system’s structure and organization. As an example of the utility of information theory to the analysis of animal communication systems, we applied a series of information theory statistics to a statistically categorized set of bottlenose dolphinTursiops truncatus, whistle vocalizations. First, we use the first-order entropic relation in a Zipf-type diagram (Zipf 1949Human Behavior and the Principle of Least Effort) to illustrate the application of temporal statistics as comparative indicators of repertoire complexity, and as possible predictive indicators of acquisition/learning in animal vocal repertoires. Second, we illustrate the need for more extensive temporal data sets when examining the higher entropic orders, indicative of higher levels of internal informational structure, of such vocalizations, which could begin to allow the statistical reconstruction of repertoire organization. Third, we propose using ‘communication capacity’ as a measure of the degree of temporal structure and complexity of statistical correlation, represented by the values of entropic order, as an objective tool for interspecies comparison of communication complexity. In doing so, we introduce a new comparative measure, the slope of Shannon entropies, and illustrate how it potentially can be used to compare the organizational complexity of vocal repertoires across a diversity of species. Finally, we illustrate the nature and predictive application of these higher-order entropies using a preliminary sample of dolphin whistle vocalizations. The purpose of this preliminary report is to re-examine the original application of information theory to the field of animal communication, illustrate its potential utility as a comparative tool for examining the internal informational structure of animal vocal repertoires and their development, and discuss its relationship to behavioural ecology and evolutionary theory.},
keywords = {Zipf's law for word frequencies},
pubstate = {published},
tppubtype = {article}
}
Comparative analysis of nonhuman animal communication systems and their complexity, particularly in comparison to human language, has been generally hampered by both a lack of sufficiently extensive data sets and appropriate analytic tools. Information theory measures provide an important quantitative tool for examining and comparing communication systems across species. In this paper we use the original application of information theory, that of statistical examination of a communication system’s structure and organization. As an example of the utility of information theory to the analysis of animal communication systems, we applied a series of information theory statistics to a statistically categorized set of bottlenose dolphinTursiops truncatus, whistle vocalizations. First, we use the first-order entropic relation in a Zipf-type diagram (Zipf 1949Human Behavior and the Principle of Least Effort) to illustrate the application of temporal statistics as comparative indicators of repertoire complexity, and as possible predictive indicators of acquisition/learning in animal vocal repertoires. Second, we illustrate the need for more extensive temporal data sets when examining the higher entropic orders, indicative of higher levels of internal informational structure, of such vocalizations, which could begin to allow the statistical reconstruction of repertoire organization. Third, we propose using ‘communication capacity’ as a measure of the degree of temporal structure and complexity of statistical correlation, represented by the values of entropic order, as an objective tool for interspecies comparison of communication complexity. In doing so, we introduce a new comparative measure, the slope of Shannon entropies, and illustrate how it potentially can be used to compare the organizational complexity of vocal repertoires across a diversity of species. Finally, we illustrate the nature and predictive application of these higher-order entropies using a preliminary sample of dolphin whistle vocalizations. The purpose of this preliminary report is to re-examine the original application of information theory to the field of animal communication, illustrate its potential utility as a comparative tool for examining the internal informational structure of animal vocal repertoires and their development, and discuss its relationship to behavioural ecology and evolutionary theory.
1996
Burgos, Javier D.
Fractal representation of the immune B cell repertoire Journal Article
In: Biosystems, vol. 39, no. 1, pp. 19–24, 1996, ISSN: 0303-2647.
Abstract | Links | BibTeX | Tags: Zipf's law for word frequencies
@article{BURGOS199619,
title = {Fractal representation of the immune B cell repertoire},
author = {Javier D. Burgos},
url = {http://www.sciencedirect.com/science/article/pii/0303264795015744},
doi = {10.1016/0303-2647(95)01574-4},
issn = {0303-2647},
year = {1996},
date = {1996-01-01},
journal = {Biosystems},
volume = {39},
number = {1},
pages = {19--24},
abstract = {The immune repertoire is characterized by a complex and dynamic organization. Here I suggest, based on the presence of well-defined immune reactivity patterns (RP) analyzed using a linguistic analogy, that the B cell repertoires could possess a fractal structure. A simple estimation method of the fractal dimension (D) for a given immune B repertoire was developed, and D demonstrated to be an useful tool to define what complexity is regarding the immune repertoire, and to evaluate the complexity level of the immune B repertoire among age-structured groups of mice sensitized with the hemagglutinin (HA) molecule of influenza virus. Also, since a fractal-linguistic point of view, considerations are made about the ontogeny, connectivity, chaotic regimes, and size of the immune repertoire. Fractal analysis indicates that the immune repertoire shows a Zipf-like scaling behavior, a statistical property of natural languages. This result suggests that immune repertoire is structured like a powerful language.},
keywords = {Zipf's law for word frequencies},
pubstate = {published},
tppubtype = {article}
}
The immune repertoire is characterized by a complex and dynamic organization. Here I suggest, based on the presence of well-defined immune reactivity patterns (RP) analyzed using a linguistic analogy, that the B cell repertoires could possess a fractal structure. A simple estimation method of the fractal dimension (D) for a given immune B repertoire was developed, and D demonstrated to be an useful tool to define what complexity is regarding the immune repertoire, and to evaluate the complexity level of the immune B repertoire among age-structured groups of mice sensitized with the hemagglutinin (HA) molecule of influenza virus. Also, since a fractal-linguistic point of view, considerations are made about the ontogeny, connectivity, chaotic regimes, and size of the immune repertoire. Fractal analysis indicates that the immune repertoire shows a Zipf-like scaling behavior, a statistical property of natural languages. This result suggests that immune repertoire is structured like a powerful language.
Burgos, Javier D.; Moreno-Tovar, Pedro
Zipf-scaling behavior in the immune system Journal Article
In: Biosystems, vol. 39, no. 3, pp. 227 - 232, 1996, ISSN: 0303-2647.
Abstract | Links | BibTeX | Tags: Zipf's law for word frequencies
@article{BURGOS1996227,
title = {Zipf-scaling behavior in the immune system},
author = {Javier D. Burgos and Pedro Moreno-Tovar},
url = {http://www.sciencedirect.com/science/article/pii/0303264796016188},
doi = {https://doi.org/10.1016/0303-2647(96)01618-8},
issn = {0303-2647},
year = {1996},
date = {1996-01-01},
journal = {Biosystems},
volume = {39},
number = {3},
pages = {227 - 232},
abstract = {The Zipf law, an approach to analyzing linguistic texts, was applied to study the Cytotoxic T Lymphocytes (CTL) repertoire, one of the major components of the immune system. A Zipf-scaling behavior was observed in those CTL repertoires obtained in normal or conventional mice. In contrast, CTL repertoires from irradiated chimeric animals, showed a non-scaling statistical behavior when tested by the Zipf analysis,suggesting that normal repertoires work like a structured biological language. The fractal dimension (D) was derived from the Zipf plots, and correlates well with the nature of the repertoires: high, D > 0.8, in conventional mice. Non-fractal, D > 1, in chimeric animals, and low, D < 0.3, in athymic mice. Then, D seems to be an useful tool to characterize the current status of the immune system.},
keywords = {Zipf's law for word frequencies},
pubstate = {published},
tppubtype = {article}
}
The Zipf law, an approach to analyzing linguistic texts, was applied to study the Cytotoxic T Lymphocytes (CTL) repertoire, one of the major components of the immune system. A Zipf-scaling behavior was observed in those CTL repertoires obtained in normal or conventional mice. In contrast, CTL repertoires from irradiated chimeric animals, showed a non-scaling statistical behavior when tested by the Zipf analysis,suggesting that normal repertoires work like a structured biological language. The fractal dimension (D) was derived from the Zipf plots, and correlates well with the nature of the repertoires: high, D > 0.8, in conventional mice. Non-fractal, D > 1, in chimeric animals, and low, D < 0.3, in athymic mice. Then, D seems to be an useful tool to characterize the current status of the immune system.
1994
Ficken, M S; Hailman, E D; Hailman, J P
The chick-a-dee call system of the Mexican chickadee Journal Article
In: Condor, vol. 96, pp. 70-82, 1994.
Abstract | Links | BibTeX | Tags: Zipf's law for word frequencies
@article{Ficken1994,
title = {The chick-a-dee call system of the Mexican chickadee},
author = {M S Ficken and E D Hailman and J P Hailman},
doi = {10.2307/1369065},
year = {1994},
date = {1994-01-01},
journal = {Condor},
volume = {96},
pages = {70-82},
abstract = {Chick-a-dee calls of the Mexican Chickadee (Parus sclateri) are composed of combinations of three common note types (A, C and D) and one very rare type (B). Calls have the invariant sequence of notes A-B-C-D, where any note type may be omitted, given once or repeated a variable number of times before transiting to the next type. The B and C notes are phonologically similar to the B and C notes of chick-a-dee calls of the Black-capped Chickadee (P. atricapillus), but the A note is markedly different and the D note somewhat different from equivalent notes of the congener. A total of 2,071 calls recorded yielded 60 different call types, and Zipf-Mandelbrot plots show that the call system is "open"; as the sample size is increased new call types will be found without demonstrable bound. In relatively undisturbed contexts (with mate on territory, in fall flocks, alone in fall) birds gave mainly [A][D] calls with lesser numbers of [A] and [C] calls, where brackets indicate variable repetition of note types. In disturbed contexts (mobbing plastic Great Horned Owl, mobbing speaker playing calls of the Northern Pygmy-Owl, observer sitting under the nest cavity) the birds gave more [C] calls with [A][C] as well. In the longest mobbing session to owl calls, birds gave mainly [A] calls when approaching, switched to [C] calls while flying about the speaker, and then resumed [A] calls and moved off when the playback was stopped. Outside of human language, this is the second truly combinatorial system of vocal communication found in animals, the first being chick-a-dee calls of the Black-capped Chickadee. This study provides the first data substantiating quantitative differences in calls from different contexts, an important step toward understanding what kinds of information combinatorial chick-a-dee calls encode.},
keywords = {Zipf's law for word frequencies},
pubstate = {published},
tppubtype = {article}
}
Chick-a-dee calls of the Mexican Chickadee (Parus sclateri) are composed of combinations of three common note types (A, C and D) and one very rare type (B). Calls have the invariant sequence of notes A-B-C-D, where any note type may be omitted, given once or repeated a variable number of times before transiting to the next type. The B and C notes are phonologically similar to the B and C notes of chick-a-dee calls of the Black-capped Chickadee (P. atricapillus), but the A note is markedly different and the D note somewhat different from equivalent notes of the congener. A total of 2,071 calls recorded yielded 60 different call types, and Zipf-Mandelbrot plots show that the call system is "open"; as the sample size is increased new call types will be found without demonstrable bound. In relatively undisturbed contexts (with mate on territory, in fall flocks, alone in fall) birds gave mainly [A][D] calls with lesser numbers of [A] and [C] calls, where brackets indicate variable repetition of note types. In disturbed contexts (mobbing plastic Great Horned Owl, mobbing speaker playing calls of the Northern Pygmy-Owl, observer sitting under the nest cavity) the birds gave more [C] calls with [A][C] as well. In the longest mobbing session to owl calls, birds gave mainly [A] calls when approaching, switched to [C] calls while flying about the speaker, and then resumed [A] calls and moved off when the playback was stopped. Outside of human language, this is the second truly combinatorial system of vocal communication found in animals, the first being chick-a-dee calls of the Black-capped Chickadee. This study provides the first data substantiating quantitative differences in calls from different contexts, an important step toward understanding what kinds of information combinatorial chick-a-dee calls encode.
Hailman, J P
Constrained permutation in 'chick-a-dee' like calls of a black-lored tit em Parus em Xanthogenys Journal Article
In: Bioacoustics, vol. 6, pp. 33-50, 1994.
Abstract | Links | BibTeX | Tags: Zipf's law for word frequencies
@article{Hailman1994,
title = {Constrained permutation in 'chick-a-dee' like calls of a black-lored tit em Parus em Xanthogenys},
author = {J P Hailman},
doi = {10.1080/09524622.1994.9753270},
year = {1994},
date = {1994-01-01},
journal = {Bioacoustics},
volume = {6},
pages = {33-50},
abstract = {Examined the vocalizations of 2 wild-caught black-lored tits. It was found that the Ss gave chick-a-dee-like calls in response to a stuffed owl. The tit used combinations of 4 note types to create 65 different call types. The repertoire of call types had the language-like property of openness to formation of new call types without limit. Calling under constant stimulus conditions changed unidirectionally during the course of mobbing, with the 4 note types replacing each other in calls, as time progressed. Thus, the tit had a constrained type of permuted order of note types, which enlarged the potential for call variety and closely resembled properties of language.},
keywords = {Zipf's law for word frequencies},
pubstate = {published},
tppubtype = {article}
}
Examined the vocalizations of 2 wild-caught black-lored tits. It was found that the Ss gave chick-a-dee-like calls in response to a stuffed owl. The tit used combinations of 4 note types to create 65 different call types. The repertoire of call types had the language-like property of openness to formation of new call types without limit. Calling under constant stimulus conditions changed unidirectionally during the course of mobbing, with the 4 note types replacing each other in calls, as time progressed. Thus, the tit had a constrained type of permuted order of note types, which enlarged the potential for call variety and closely resembled properties of language.
1990
Markov, Vladimir I; Ostrovskaya, Vera M
Organization of Communication System in Tursiops Truncatus Montagu Incollection
In: Thomas, Jeanette A; Kastelein, Ronald A (Ed.): Sensory Abilities of Cetaceans: Laboratory and Field Evidence, pp. 599–622, Springer US, Boston, MA, 1990, ISBN: 978-1-4899-0858-2.
Abstract | Links | BibTeX | Tags: Zipf's law for word frequencies
@incollection{Markov1990,
title = {Organization of Communication System in Tursiops Truncatus Montagu},
author = {Vladimir I Markov and Vera M Ostrovskaya},
editor = {Jeanette A Thomas and Ronald A Kastelein},
url = {https://doi.org/10.1007/978-1-4899-0858-2_42},
doi = {10.1007/978-1-4899-0858-2_42},
isbn = {978-1-4899-0858-2},
year = {1990},
date = {1990-01-01},
booktitle = {Sensory Abilities of Cetaceans: Laboratory and Field Evidence},
pages = {599--622},
publisher = {Springer US},
address = {Boston, MA},
abstract = {The problem of the degree of complexity and semantic capabilities of the acoustic communicative system in bottlenose dolphins has been under discussion for over a quarter of a century, ever since John Lilly published his book ``Man and Dolphin'' (Lilly, 1962). Nowadays, there is an abundance of literature, with different viewpoints, but no consensus has been reached so far among researchers. The problem proved to be very complicated, both methodologically and experimentally, while the methods used turned-out to be labor-consuming and, on the whole, inefficient; all kinds of straightforward attacks failed. Meanwhile, one can try and assess potential capabilities of communicative system by analysing dolphins' mechanisms ensuring its productivity, i.e. the creation of signals and messages in amounts nessesary for communication. These mechanisms ensure the encoding of information and, in accordance with the theory, their functioning, in this way or other, affects the structure of signals and their sequences, in other words, it affects the organization of communicative system.},
keywords = {Zipf's law for word frequencies},
pubstate = {published},
tppubtype = {incollection}
}
The problem of the degree of complexity and semantic capabilities of the acoustic communicative system in bottlenose dolphins has been under discussion for over a quarter of a century, ever since John Lilly published his book ``Man and Dolphin'' (Lilly, 1962). Nowadays, there is an abundance of literature, with different viewpoints, but no consensus has been reached so far among researchers. The problem proved to be very complicated, both methodologically and experimentally, while the methods used turned-out to be labor-consuming and, on the whole, inefficient; all kinds of straightforward attacks failed. Meanwhile, one can try and assess potential capabilities of communicative system by analysing dolphins' mechanisms ensuring its productivity, i.e. the creation of signals and messages in amounts nessesary for communication. These mechanisms ensure the encoding of information and, in accordance with the theory, their functioning, in this way or other, affects the structure of signals and their sequences, in other words, it affects the organization of communicative system.
1989
Wilde, J; Schwibbe, H
Organizationsformen von Erbinformation Im Hinblick auf die Menzerathsche Regel Incollection
In: Altmann, G; Schwibbe, M H (Ed.): Das Menzerathsche Gesetz in informationsverarbeitenden Systemen, pp. 92-107, Olms, Hildesheim, 1989.
BibTeX | Tags: Menzerath-Altmann law
@incollection{Wilde1989a,
title = {Organizationsformen von Erbinformation Im Hinblick auf die Menzerathsche Regel},
author = {J Wilde and H Schwibbe},
editor = {G Altmann and M H Schwibbe},
year = {1989},
date = {1989-01-01},
booktitle = {Das Menzerathsche Gesetz in informationsverarbeitenden Systemen},
pages = {92-107},
publisher = {Olms},
address = {Hildesheim},
keywords = {Menzerath-Altmann law},
pubstate = {published},
tppubtype = {incollection}
}
1988
Howes-Jones, Daryl; Barlow, Jon C; Museum., Royal Ontario
The structure of the call note system of the warbling vireo Book
Royal Ontario Museum, Toronto, 1988, (The authors actually reports evidence of an exponential-like distribution, rather than the power-law like distribution of Zipf's law for word frequencies.).
Links | BibTeX | Tags: Zipf's law for word frequencies
@book{bhl113530,
title = {The structure of the call note system of the warbling vireo},
author = {Daryl Howes-Jones and Jon C Barlow and Royal Ontario Museum.},
url = {https://www.biodiversitylibrary.org/item/113530},
doi = {10.5962/bhl.title.53490},
year = {1988},
date = {1988-01-01},
pages = {52},
publisher = {Royal Ontario Museum},
address = {Toronto},
note = {The authors actually reports evidence of an exponential-like distribution, rather than the power-law like distribution of Zipf's law for word frequencies.},
keywords = {Zipf's law for word frequencies},
pubstate = {published},
tppubtype = {book}
}
1987
Hailman, J P; Ficken, M S; Ficken, R
Constraints on the structure of combinatorial "chick-a-dee" calls Journal Article
In: Ethology, vol. 75, pp. 62-80, 1987.
Abstract | Links | BibTeX | Tags: Zipf's law of abbreviation
@article{Hailman1987a,
title = {Constraints on the structure of combinatorial "chick-a-dee" calls},
author = {J P Hailman and M S Ficken and R Ficken},
doi = {10.1111/j.1439-0310.1987.tb00642.x},
year = {1987},
date = {1987-01-01},
journal = {Ethology},
volume = {75},
pages = {62-80},
abstract = {Chickadee calls include 4 note types (A, B, C, and D) that occur in fixed sequence. Quantitative analyses determining the nature of departures from 1st-order transitional probabilities between successive notes in calls of the blackcapped chickadee found that (1) some constraint limited the length of calls (i.e., shortening of repetition strings, omitting of note types) and (2) there was an opposing tendency to include at least 1 D-note at the end of a call. The 2nd feature suggests semantic properties of D-notes. A chicakadee call is thought to resemble a natural sentence of spoken human language.},
keywords = {Zipf's law of abbreviation},
pubstate = {published},
tppubtype = {article}
}
Chickadee calls include 4 note types (A, B, C, and D) that occur in fixed sequence. Quantitative analyses determining the nature of departures from 1st-order transitional probabilities between successive notes in calls of the blackcapped chickadee found that (1) some constraint limited the length of calls (i.e., shortening of repetition strings, omitting of note types) and (2) there was an opposing tendency to include at least 1 D-note at the end of a call. The 2nd feature suggests semantic properties of D-notes. A chicakadee call is thought to resemble a natural sentence of spoken human language.
1986
Bender, M. L.; Gill, Pritmohinder
The genetic code and Zipf's Law Journal Article
In: Current Anthropology, vol. 27, no. 3, pp. 280–283, 1986, ISSN: 00113204, 15375382.
Links | BibTeX | Tags: Zipf's law for word frequencies
@article{10.2307/2742889,
title = {The genetic code and Zipf's Law},
author = {M. L. Bender and Pritmohinder Gill},
url = {http://www.jstor.org/stable/2742889},
issn = {00113204, 15375382},
year = {1986},
date = {1986-01-01},
journal = {Current Anthropology},
volume = {27},
number = {3},
pages = {280--283},
publisher = {University of Chicago Press, Wenner-Gren Foundation for Anthropological Research},
keywords = {Zipf's law for word frequencies},
pubstate = {published},
tppubtype = {article}
}
1985
Hailman, J P; Ficken, M S; Ficken, R W
The 'chick-a-dee' calls of em Parus em atricapillus: a recombinant system of animal communication compared with written English Journal Article
In: Semiotica, vol. 56, pp. 121-224, 1985.
Links | BibTeX | Tags: Zipf's law of abbreviation
@article{Hailman1985,
title = {The 'chick-a-dee' calls of em Parus em atricapillus: a recombinant system of animal communication compared with written English},
author = {J P Hailman and M S Ficken and R W Ficken},
doi = {10.1515/semi.1985.56.3-4.191},
year = {1985},
date = {1985-01-01},
journal = {Semiotica},
volume = {56},
pages = {121-224},
keywords = {Zipf's law of abbreviation},
pubstate = {published},
tppubtype = {article}
}
1978
Ficken, Millicent S; Hailman, Jack P; Ficken, Robert W
A model of repetitive behaviour illustrated by chickadee calling Journal Article
In: Animal Behaviour, vol. 26, pp. 630 - 631, 1978, ISSN: 0003-3472.
Abstract | Links | BibTeX | Tags: Zipf's law of abbreviation
@article{FICKEN1978630,
title = {A model of repetitive behaviour illustrated by chickadee calling},
author = {Millicent S Ficken and Jack P Hailman and Robert W Ficken},
url = {http://www.sciencedirect.com/science/article/pii/0003347278900751},
doi = {10.1016/0003-3472(78)90075-1},
issn = {0003-3472},
year = {1978},
date = {1978-01-01},
journal = {Animal Behaviour},
volume = {26},
pages = {630 - 631},
abstract = {If the probability of an animal's continuing (or repeating) a given behavioral pattern is independent of the elapsed time (or number of previous acts), the data conform to the exponential distribution generated by a Poisson process. The number of chickadee calls/bout was recorded for 1,524 bouts. Results show a negative linear relation between the number of calls/bout and the cumulative frequency of bouts when plotted on logarithmic axes.},
keywords = {Zipf's law of abbreviation},
pubstate = {published},
tppubtype = {article}
}
If the probability of an animal's continuing (or repeating) a given behavioral pattern is independent of the elapsed time (or number of previous acts), the data conform to the exponential distribution generated by a Poisson process. The number of chickadee calls/bout was recorded for 1,524 bouts. Results show a negative linear relation between the number of calls/bout and the cumulative frequency of bouts when plotted on logarithmic axes.