This work was released as UNM Computer Science Department Tech Report TR-CS-2001-23. These reports are archived at:

The Distribution of Variable-length Phatic Interjectives on the World Wide Web

Dennis Chao
Computer Science Department
University of New Mexico
Albuquerque, NM 87131 USA
Patrik D'haeseleer
Department of Genetics and Lipper Center
for Computational Genetics
Harvard Medical School
Boston, MA 02115 USA


If one uses a commercial internet search engine to search for increasingly long versions of variable-length interjectives on the web (e.g. ``whee'', ``wheee'', ``wheeee'', etc.), the number of pages found containing these longer words falls off as a power law. The exponents for the length frequency distributions of different interjectives are not the same, although they may cluster around a few exponents. Surprisingly, the exponents are much larger than the -1 predicted by Zipf's Law. We believe that the restricted domain of variable-length phatic interjectives is an interesting subset of English that can provide an alternative simple model system of word length distributions.


The variability in the written length of phatic interjectives (e.g. ``whee'', ``wheeee'', ``aah'', ``aaah'') is of great interest to linguists and computer scientists. 1In this paper, we examine the distribution of variable-length phatic interjectives on the largest corpus of such words ever assembled: the World Wide Web. By using commercial internet search engines, we can quickly get a rough estimate of the distribution of these words on the internet.

Method and results

The web search engines Google [4] and AltaVista [1] were used to estimate the number of occurrences of length variations on the interjectives ``aah'', ``ahh'', ``cool'', ``ooh'', and ``whee'' on the World Wide Web. Ideally, the search engines return the number of web pages that contain a particular word, while in reality a single search engine probably covers less than 20% of the web [5]. Two different search engines were employed in order to verify the consistency of their results and to increase the search coverage. These words all feature a single letter that can be repeated to create longer versions of the words, for example, ``ahh'' can become ``ahhhhhhh''. These particular words were chosen because their extended versions can be found at relatively high frequencies on the Web. The authors recorded the number of hits returned by the search engines for these words and varied the number of repeated letters in them, from three to thirty-nine (e.g. from ``wheee'' to ``wheeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee''). Variations with only one or two repetitions, such as ``whe'' and ``whee,'' were not considered due to the appearance of ``whe'' in non-interjectives such as ``whet'' and ``wheel.'' ``Wheee'' is a reasonable starting point as it does not begin any words in the standard English lexicon. We also investigated the frequencies of interjectives with two-letter repeated elements: ``haha'' and ``hoho,'' repeating the two-letter units three to thirty-nine times. The data for ``hoho'' was truncated after 20 repeated units because the longer words had frequencies close to zero. The results are summarized in Table 1.

The frequency distributions of the lengths of these phatic interjectives exhibit clear power-law relationships (Figure 1). With the exception of the words based on ``ooh'', the interjectives with a single repeating letter (i.e. ``aah'', ``ahh'', ``cool'', and ``whee'') have slopes of approximately -3.0, which we call We. The slopes of interjectives that have two-letter repeating units (i.e. ``haha'' and ``hoho'') is higher, from -3.42 to -4.30.


We had expected the frequency distributions of the lengths of phatic interjectives to obey Zipf's law [8], which observes (but does not explain) that the ranked frequencies of words in a corpus follow a power law with an exponent of -1. However, we found much higher exponents, ranging from -2.86 to -4.30. Therefore, we do not believe that the published explanations of Zipf's law (such as [6]) explain the distribution of variable-length interjectives on the Web. We believe that this group of words comprises a simple model system for studying word lengths that provides an alternative to ``monkey languages'' [3], in which text is simply a random stream of letters and spaces like the proverbial monkey banging on a typewriter.

The frequency of the variations of words on the World Wide Web depend on at least two factors: the frequencies with which authors of web content use these variations and the frequencies at which this content is duplicated and reached by search engines.

We believe that the first distribution is either exponential or follows a power law. If each typist of interjectives follows a Poisson process in which after each character the typist has constant probability of quitting, the distribution of lengths would be exponential. If, instead of typing each character of the word individually, the typist doubles the number at each step by copying and pasting the block of all repeated letters at each step, then the word lengths would have a power law distribution.

The duplication of content may induce a power law, dominating the original underlying distribution produced by the web authors. We assume there is a distribution of pages that favors shorter phatic interjectives. An important source of web content duplication is the quoting of messages, especially e-mail, in replies. Replies to messages also get replies, and the original message is often still present in these chains of replies for many ``generations.'' This would contribute to bits of text that are replicated an exponential number of times. As in Barabási's model of the formation of scale-free networks [2], the positive feedback loop of often-quoted messages getting quoted even more often can cause a power law distribution. As more mail and other such messages are archived on the World Wide Web, these effects will gain importance.


There are a large number of similar experiments that can be performed to obtain a more accurate characterization of interjective length frequencies on the World Wide Web. These searches should be tried on different search engines. Google relies heavily on crawling the web from its current database, which may form a scale-free network a priori. Studying the number of letters that people actually use in long phatic interjectives would also contribute to our understanding of this phenomenon. This data may be difficult to acquire because standard corpuses consisting of published works typically do not have long interjectives. 2Variable-length interjectives are a small subset of the distribution of words than is usually studied, yet yield interesting behavior and should prove easier to investigate. Further research in unbounded phatic interjectives will give insight into the mechanisms behind Zipf's Law, the nature of the web, and the patience of authors.



A. Barabási and R. Albert.
Emergence of scaling in random networks.
Science, 286:509-512, 1999.

J. L. Casti.
Bell curves and monkey languages.
Complexity, 1(1):12-15, 1995.


S. Lawrence and C. L. Giles.
Accessibility of information on the web.
Nature, 400:107-109, 1999.

W. Li.
Random texts exhibit zipf's law-like word frequency distribution.
IEEE Transactions on Information Theory, 38(6):1842-1845, 1992.

R. Rothman.
In D. Eggers, editor, Timothy McSweeney's Quarterly Concern, volume 5, pages 262-263, 277, 288. McSweeney's, 2000.

G. K. Zipf.
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Addison-Wesley, 1949.

search engine word slope std err intercept std err
Google aah -2.88 0.0333 14.4 0.0978
Google ahh -2.89 0.0336 15.6 0.0988
Google cool -3.03 0.0560 14.2 0.165
Google ooh -3.98 0.0728 17.1 0.214
Google whee -2.86 0.0647 13.4 0.190
Google haha -3.42 0.0668 16.5 0.196
Google hoho -4.30 0.172 14.2 0.410
AltaVista aah -3.04 0.0522 13.2 0.154
AltaVista ahh -3.18 0.0343 14.9 0.101
AltaVista cool -3.03 0.0651 13.0 0.192
AltaVista ooh -4.04 0.0622 15.7 0.183
AltaVista whee -2.94 0.0691 11.8 0.203
AltaVista haha -3.48 0.593 15.2 0.174
AltaVista hoho -3.95 0.271 12.0 0.643
Table 1: The slopes and intercepts of the lines fitted to the points with the number of repeated letters (or two-letter units) on the x axis and number of pages found by search engines on the y axis. All data sets except for those involving ``hoho'' contain thirty-seven data points (three to thirty-nine repeated units). The word ``hoho'' and its variants are less common on the web, so the data was truncated after eighteen points (three to twenty repeated ``ho''s).

a) plot1 b) plot2
Figure 1: The number of hits for the words a) ``aah'', ``ahh'', ``cool'', and ``whee'' and b) ``haha'' and ``hoho'' on Google and AltaVista. The slopes of all lines in a) are approximately -3. The slopes for ``haha...'' and ``hoho...'' in b) are higher. Note that the plots are plotted on a log-log scale.

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The Distribution of Variable-length Phatic Interjectives on the World Wide Web

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... scientists. 1
Well, no, not really.
... interjectives.2
An exception to this may be found in an excerpt of Rodney Rothman's Rollercoaster [7], which has an exceptionally long instance of the word ``whee'' that contains over eight thousand e's.

Dennis Chao 2001-09-04