info 2950, lecture 19power laws in log-log space y = cx k (k=1/2,1,2)log 10 y = k ∗ log 10 x +log...
TRANSCRIPT
Prob Set 6: tomorrow sometime (likely just first part)
tentative due dates for the rest:ps6 due Mon 23 Apr [longest]ps7 due Wed 2 Mayps8 due Wed 9 May (last day classes, auto-extension to end week)
Info 2950, Lecture 19 12 Apr 2018
Z = X . Y10c = 10a . 10b = 10a+b
c = a + blog10Z = log10X + log10Y
Y = cXk
log10Y = klog10X + log10c(of the form y = ax + b)
Power Laws in log-log space
y = cxk (k=1/2,1,2) log10 y = k ∗ log10 x + log10 c
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Power Laws in log-log space
y = cxk (k=1/2,1,2) log10 y = k ∗ log10 x + log10 c
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Power Laws in log-log space
y = cx−k (k=1/2,1,2) log10 y = −k ∗ log10 x + log10 c
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100/sqrt(x)100/x
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Power Laws in log-log space
y = cx−k (k=1/2,1,2) log10 y = −k ∗ log10 x + log10 c
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Zipf’s law
Now we have characterized the growth of the vocabulary incollections.
We also want to know how many frequent vs. infrequentterms we should expect in a collection.
In natural language, there are a few very frequent terms andvery many very rare terms.
Zipf’s law (linguist/philologist George Zipf, 1935):The i th most frequent term has frequency proportional to 1/i .
cf i ∝1i
cf i is collection frequency: the number of occurrences of theterm ti in the collection.
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Zipf’s law
Now we have characterized the growth of the vocabulary incollections.
We also want to know how many frequent vs. infrequentterms we should expect in a collection.
In natural language, there are a few very frequent terms andvery many very rare terms.
Zipf’s law (linguist/philologist George Zipf, 1935):The i th most frequent term has frequency proportional to 1/i .
cf i ∝1i
cf i is collection frequency: the number of occurrences of theterm ti in the collection.
3 / 25
http://en.wikipedia.org/wiki/Zipf’s law
Zipf’s law: the frequency of any word is inversely proportional toits rank in the frequency table. Thus the most frequent word willoccur approximately twice as often as the second most frequentword, which occurs twice as often as the fourth most frequentword, etc. Brown Corpus:
“the”: 7% of all word occurrences (69,971 of!>1M).
“of”: ∼3.5% of words (36,411)
“and”: 2.9% (28,852)
Only 135 vocabulary items account for half the Brown Corpus.
The Brown University Standard Corpus of Present-Day American English
is a carefully compiled selection of current American English, totaling
about a million words drawn from a wide variety of sources . . . for many
years among the most-cited resources in the field.
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http://en.wikipedia.org/wiki/Zipf’s law
Zipf’s law: the frequency of any word is inversely proportional toits rank in the frequency table. Thus the most frequent word willoccur approximately twice as often as the second most frequentword, which occurs twice as often as the fourth most frequentword, etc. Brown Corpus:
“the”: 7% of all word occurrences (69,971 of!>1M).
“of”: ∼3.5% of words (36,411)
“and”: 2.9% (28,852)
Only 135 vocabulary items account for half the Brown Corpus.
The Brown University Standard Corpus of Present-Day American English
is a carefully compiled selection of current American English, totaling
about a million words drawn from a wide variety of sources . . . for many
years among the most-cited resources in the field.
4 / 25
http://en.wikipedia.org/wiki/Zipf’s law
Zipf’s law: the frequency of any word is inversely proportional toits rank in the frequency table. Thus the most frequent word willoccur approximately twice as often as the second most frequentword, which occurs twice as often as the fourth most frequentword, etc. Brown Corpus:
“the”: 7% of all word occurrences (69,971 of!>1M).
“of”: ∼3.5% of words (36,411)
“and”: 2.9% (28,852)
Only 135 vocabulary items account for half the Brown Corpus.
The Brown University Standard Corpus of Present-Day American English
is a carefully compiled selection of current American English, totaling
about a million words drawn from a wide variety of sources . . . for many
years among the most-cited resources in the field.
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(Recall ps3#3 for 120 article dataset)
Zipf’s law
Zipf’s law: The i th most frequent term has frequencyproportional to 1/i .
cf i ∝1i
cf is collection frequency: the number of occurrences of theterm in the collection.
So if the most frequent term (the) occurs cf1 times, then thesecond most frequent term (of) has half as many occurrencescf2 =
12cf1 . . .
. . . and the third most frequent term (and) has a third asmany occurrences cf3 =
13cf1 etc.
Equivalent: cf i = cik and log cf i = log c + k log i (for k = −1)
Example of a power law
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Zipf’s law for Reuters
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01
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67
log10 rank
log1
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Fit far from perfect, but nonetheless key insight:Few frequent terms, many rare terms.
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more from http://en.wikipedia.org/wiki/Zipf’s law
“A plot of word frequency in Wikipedia (27 Nov 2006). The plot is in log-log coordinates. x is rank of a word in the
frequency table; y is the total number of the words occurrences. Most popular words are “the”, “of” and “and”, as
expected. Zipf’s law corresponds to the upper linear portion of the curve, roughly following the green (1/x) line.”
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Another Wikipedia count (15 May 2010)
http://imonad.com/seo/wikipedia-word-frequency-list/
All articles in the English version of Wikipedia, 21GB in XMLformat (five hours to parse entire file, extract data from markuplanguage, filter numbers, special characters, extract statistics):
Total tokens (words, no numbers): T = 1,570,455,731
Unique tokens (words, no numbers): M = 5,800,280
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“Word frequency distribution follows Zipf’s law”
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rank 1–50 (86M-3M), stop words (the, of, and, in, to, a, is,. . .)
rank 51–3K (2.4M-56K), frequent words (university, January,tea, sharp, . . .)
rank 3K–200K (56K-118), words from large comprehensivedictionaries (officiates, polytonality, neologism, . . .)above rank 50K mostly Long Tail words
rank 200K–5.8M (117-1), terms from obscure niches,misspelled words, transliterated words from other languages,new words and non-words (euprosthenops, eurotrochilus,lokottaravada, . . .)
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Some selected words and associated counts
Google 197920
Twitter 894
domain 111850
domainer 22
Wikipedia 3226237
Wiki 176827
Obama 22941
Oprah 3885
Moniker 4974
GoDaddy 228
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Project Gutenberg (per billion)
http://en.wiktionary.org/wiki/Wiktionary:Frequency_lists#Project_Gutenberg
Over 36,000 items (Jun 2011), average of > 50 new e-books / weekhttp://en.wiktionary.org/wiki/Wiktionary:Frequency_lists/PG/2006/04/1-10000
the 56271872
of 33950064
and 29944184
to 25956096
in 17420636
I 11764797
that 11073318
was 10078245
his 8799755
he 8397205
it 8058110
with 7725512
is 7557477
for 7097981
as 7037543
had 6139336
you 6048903
not 5741803
be 5662527
her 5202501
. . . 100, 000th
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If a city is 10 times as populous,does it have 10 times as many gas stations?
Empirically, G = c P.77 (economies of scale)
similarly for miles of roadway, length of electrical cables, …,k ranges from .7 to .9
https://opinionator.blogs.nytimes.com/2009/05/19/math-and-the-city/
Power laws more generally
E.g., consider power law distributions of the form c r−k ,describing the number of book sales versus sales-rank r of a book,or the number of Wikipedia edits made by the r th most frequentcontributor to Wikipedia.
Amazon book sales: c r−k , k ≈ .87
number of Wikipedia edits: c r−k , k ≈ 1.7
(More on power laws and the long tail here:Networks, Crowds, and Markets:
Reasoning About a Highly Connected World
by David Easley and Jon KleinbergChpt 18: http://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch18.pdf)
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Power Laws more generally
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User|Book rank r
40916 / r^{.87}
1258925 / r^{1.7}
Normalization given by the roughly1 sale/week for the200,000th ranked Amazon title:
40916r−.87
and by the10 edits/month for the1000th ranked Wikipedia editor:
1258925r−1.7
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User|Book rank r
1258925 / r^{1.7}
40916 / r^{.87}
Long tail: about a quarter ofAmazon book sales estimatedto come from the long tail,i.e., those outside the top100,000 bestselling titles
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User|Book rank r
40916 / r^{.87}
1258925 / r^{1.7}
Normalization given by the roughly1 sale/week for the200,000th ranked Amazon title:
40916r−.87
and by the10 edits/month for the1000th ranked Wikipedia editor:
1258925r−1.7
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iped
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k
User|Book rank r
1258925 / r^{1.7}
40916 / r^{.87}
Long tail: about a quarter ofAmazon book sales estimatedto come from the long tail,i.e., those outside the top100,000 bestselling titles
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iped
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User|Book rank r
40916 / r^{.87}
1258925 / r^{1.7}
Normalization given by the roughly1 sale/week for the200,000th ranked Amazon title:
40916r−.87
and by the10 edits/month for the1000th ranked Wikipedia editor:
1258925r−1.7
0.1
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1e+07
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iped
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User|Book rank r
1258925 / r^{1.7}
40916 / r^{.87}
Long tail: about a quarter ofAmazon book sales estimatedto come from the long tail,i.e., those outside the top100,000 bestselling titles
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User|Book rank r
40916 / r^{.87}
1258925 / r^{1.7}
Normalization given by the roughly1 sale/week for the200,000th ranked Amazon title:
40916r−.87
and by the10 edits/month for the1000th ranked Wikipedia editor:
1258925r−1.7
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User|Book rank r
1258925 / r^{1.7}
40916 / r^{.87}
Long tail: about a quarter ofAmazon book sales estimatedto come from the long tail,i.e., those outside the top100,000 bestselling titles
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Suppose y = cxk
then log(y) = k log(x) + log(c),and the plot of log(y) vs log(x) is a straight line with slope k (and y-intercept log(c) when x=1)
Now suppose some particular x0 value has associated y0 = cxk .Then x1 = 10x0 has y1 = cxk = c(10x0)k = 10k cxk = 10k y0 .So if x increases by a factor of 10, then y increases by a factor of 10k,and the value of k is easily determined.
More generally, if x1 = r x0 and y1 = s y0 for two points (x0, y0) and (x1, y1)on the curve y = cxk then s = y1 / y0 = (x1 / x0)k = rk
and comparison of r (ratio of x1 and x0) to s (ratio of y1 and y0)provides the value of the exponent k.
e.g. x -> 10x, y -> 10ky ; or generally x -> 102x, y-> 102ky
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k = 3 k = -2
k = .5 k = -1
k = 1/3 k = -1/2
10003.5k/2 = 10-4
so k = -8/10.5 = -.76
“moment magnitude” is a logarithmic scale,from 6 to 8 is 1000 times more powerful,
(so, e.g., from 5 to 6 is 31.6 times)
f = c Mk
(Figures on next three slides from: N. Silver, “The signal and the noise” (2012))
f(9.1) = 1/13000 = .00008f(9.1) = 1/300 = .0033
11 Mar 2011: 9.1
1000k = 10-2
so k = -2/3f = c Mk f = c Mk 1000k/2 = 10-2.1
so k = -1.4