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Power-Law Distribution
in Japanese Book Sales Market
Takashi Iba
Faculty of Policy Management, Keio University
Mariko Yoshida
Faculty of Policy Management, Keio University
Yoshiaki Fukami
Keio Research Institute at SFC
Masaru Saitoh
Nippon Shuppan Hanbai, Inc.
In this paper, we analyze the real sales data of the book sales market in Japan. The
data which we analyze is the data taken from POS (Point Of Sales) system of over
2,000 bookstores in all areas of Japan. The target term is from April 2005 to March
2006. As a result of analysis, it became clear that the relation between sales volume
and sales rank follows power law in both of annual and monthly sales. This is not only
an important finding but also the big chance to understand the market mechanism
by the analogy from the other phenomena which follow power law. In this paper, we
also focus on the books in the top 1.5 percent of sales, and analyze the power index
and market share on the time series. We find that the both of index and market share
are increasing from April 2005 to March 2006, therefore it shows that the books in
top sales are getting to be sold more and more.
1 Introduction
In this paper, we verify that the relation between
sales volume and sales rank follows “power law”
in the book sales market, which can be consid-
ered as a hidden law of the market. Power law in
a market means that there are very few top sale
products while the rest of them sold few, and
surprisingly all of products in the market follow
a simple statistical law. Therefore the sales vol-
ume can be calculated by the sales rank, and vice
versa. Power-law distribution is plotted linearly-
graded in double logarithmic graph (Figure 1).
Interestingly, the power-law distributions have
been found in various fields such as natural and
social sciences[1, 2, 3]. In these studies, the power-
law distribution is deemed to be emerged in the
system which the elements interact one another
and self-organized into “critical state”[4, 1]. Based
on the hypothesis, the fact that the power-law
distribution is found in the market implies the
market is always organized into the critical state.
In this paper, we analyze the real sales data
in Japanese book sales market, in order to verify
the relation between sales volume and sales rank
follows “power law”. The result has two signifi-
cant implications. First, the macroscopic law is
emerged from the interactions despite that the
customers buy products by their own decision.
It means that the law is called as an “emergent
order” in market level, which cannot be reduced
to individuals level. Second, the fact that the
law is the “power law” implies that the funda-
mental mechanism is similar to other natural and
social phenomena. Therefore there is chance to
understand the market mechanism analogically
by knowing that of other phenomena. In the fol-
lowing sections, we start the brief introduction of
related studies, and then show our analyses.
1
Linear Graph
Sales Volume
Double Logarithmic Graph
Sales Rank Sales Rank
Figure 1: Power-law distribution in linear graph
and double logarithmic graph
2 Background
The target in this paper is book sales market.
As a product, a book has various value and util-
ity, and there are wide variations in the market.
For example, the number of books circulated in
Japan is over 1.2 billion and over 77 thousands
of new titles are published in a year. In United
States, 200 thousands of books are published in
every year. Under the situation, the large-scale
bookstore always keep over hundreds and thou-
sands of titles in order to meet various demands
of customers, however the fact is that sales are
polarized. In recent years, “Harry Potter and the
Half-Blood Prince” and “The Da Vinci Code”
had explosive sales, on the other hand, many
other books have not moved their shelves and
some books are returned to publishers. Since all
books are not equally sold and there is polariza-
tion, the book sales market can be considered as
“Winner-Take-All” market [5]. “Winner-Take-
All” market means that the top-sales products
earn overwhelming share. It is well-known that
this kind of winner exists in our life, but a whole
reality of the book sales market is not clear.
There are some studies trying to analyze book
sales market in the viewpoint of power law1. Sor-
nette and Deschatres [6] mentioned the relation
between sales volume and sales rank, and intro-
duced the presumption by Rosenthal [7]. Rosen-
thal investigated the sales rank in Amazon.com
of the books which were published from his pub-
lisher, and estimated that the relation between
sales volume and sales rank seems to be the power-
law distribution. In the analysis, however, it was
just a presumption made by some points data2,
and they are not verified using the real data.
In this paper, the data we used in analysis is
the real data which was collected by one of the
biggest wholesales of books in Japan. The valid-
ity and amount of the data are major features of
our analysis3.
3 Sales Volume - Rank Distri-
bution in Book Sales Market
We, here, analyze the real sales data in Japanese
book sales market, in order to verify the rela-
tion between sales volume and sales rank follows
“power law”. The data we analyze is the real
sales data, which is taken from POS (Point Of
Sales) system of over 2,000 bookstores in Japan,
from April 2005 through March 2006. In this sec-
tion, we show the results of (1) Analysis of the
annual sales, (2) Analysis of monthly sales, (3)
Analysis of sales in genres, and (4) Analysis of
the tendency of top titles.
3.1 Analysis of the annual sales
The relation between sales volume and sales rank
in fiscal year 2005 is shown in Figure 2 (linear
graph) and Figure 3 (double logarithmic graph).
The vertical axis shows the sales volume, to be
exact which is normalized by dividing the sales
volume of the title by the sales volume of all
books, and the horizontal axis shows the sales
rank. These figures show that the top-sales titles
merely exist and most of the titles are not very
much sold. And we can find that the relation be-
tween sales volume and sales rank follows power
law, because the plotted values on the straight
line of approximation in the double logarithmic
graph. The approximate expression of power law
distribution is described as follows:
v=αr−β
Here, vis sales volume and ris sales rank.
Note that βis a power index, which shows the
gradient of approximate line in the double loga-
rithmic graph. In Figure 3, it can be considered
that the part which does not fit the approxima-
tion is the area of “cut-off” due to the capacity
of bookstores.
2
‣†‧↝᥄
Sales Volume
Sales Rank
Figure 2: The relation between sales volume and
sales rank (from April 2005 to March 2006: linear
graph)
Sales Volume
Sales Rank
Figure 3: The relation between sales volume and
sales rank (from April 2005 to March 2006; dou-
ble logarithmic graph)
3.2 Analysis of monthly sales
The relation between sales volume and sales rank
in the monthly sales from April 2005 to March
2006 is shown in from Figure 4 to Figure 15 re-
spectively. From these figures, it can be said
that the power-law distribution in the book sales
market is the fractal phenomena, because the
monthly distribution shows the almost same dis-
tribution of the annual distribution. It also can
be said that the power-law distribution is “emer-
gent order” and is kept every time, despite that
the customers buy products by their own deci-
sion. The customers never imagine that their
purchases contribute to such a macroscopic order,
because they think that they buying the products
by their own decision.
Figure 4: The relation between sales volume and
sales rank (April 2005; double logarithmic graph)
Figure 5: The relation between sales volume and
sales rank (May 2005; double logarithmic graph)
Figure 6: The relation between sales volume and
sales rank (June 2005; double logarithmic graph)
Figure 7: The relation between sales volume and
sales rank (July 2005; double logarithmic graph)
3
Figure 8: The relation between sales volume
and sales rank (August 2005; double logarithmic
graph)
Figure 9: The relation between sales volume and
sales rank (September 2005; double logarithmic
graph)
Figure 10: The relation between sales volume
and sales rank (October 2005; double logarith-
mic graph)
Figure 11: The relation between sales volume and
sales rank (November 2005; double logarithmic
graph)
Figure 12: The relation between sales volume and
sales rank (December 2005; double logarithmic
graph)
Figure 13: The relation between sales volume
and sales rank (January 2006; double logarith-
mic graph)
Figure 14: The relation between sales volume
and sales rank (February 2006; double logarith-
mic graph)
Figure 15: The relation between sales volume
and sales rank (March 2006; double logarithmic
graph)
4
Table 1: Power index and market share of upper
1.5% titles
target the number power index market share
period of upper of upper of upper
1.5% titles 1.5% titles 1.5% titles
2005s 8324 0.648 48.7465
2005/04 4450 0.684 47.3896
2005/05 4487 0.688 47.2258
2005/06 4411 0.714 48.2682
2005/07 4490 0.706 50.2240
2005/08 4469 0.719 51.5646
2005/09 4397 0.733 51.2183
2005/10 4512 0.718 51.4517
2005/11 4413 0.706 51.0998
2005/12 4461 0.744 53.5941
2006/01 4542 0.725 52.0992
2006/02 4684 0.739 52.5805
2006/03 4536 0.735 52.7173
3.3 Analysis of sales in genres
As a result of analysis in each genres, Figure 16 to
26 show that the relations between sales volume
and sales rank in each genres also follow power
law. We, here, would like to focus on two genres,
Chemistry (Figure 21) and Physics (Figure 22).
In the genre of Chemistry, most of the top titles
seem to be purchased as textbooks. On the con-
trary, in the genre of Physics, the top titles are
the books about the introduction to Einstein the-
ory or quantum theory. This characteristic may
make the difference between the sales volume of
the top titles in Chemistry and Physics.
3.4 Analysis of the tendency of top ti-
tles
Power law distribution is known to be much dif-
ferent from the normal distribution, therefore sta-
tistical values of “average” and “variance” do not
make sense under the power-law distribution. For
that reason, the other way of capturing the char-
acteristics is required, and the power index βis
used as one of the indicator of the distribution.
Strictly speaking, it should be noted that the
power index varies depending on the range of ap-
proximation, so it is necessary to make a specific
baseline for approximation. In order to make the
baseline, we investigated the part that along to
the vertical axis in the graph. After some explo-
1e-007
1e-006
1e-005
0.0001
0.001
0.01
0.1
1 10 100 1000 10000 100000
4CVKQQHDQQMUCNGURQKPV
4CPMKPI
Figure 16: The relation between sales volume and
sales rank of “Literature” (May 2006; double log-
arithmic graph)
1e-006
1e-005
0.0001
0.001
0.01
0.1
1 10 100 1000 10000 100000
4CVKQQHDQQMUCNGURQKPV
4CPMKPI
Figure 17: The relation between sales volume and
sales rank of “Japanese Literature” (May 2006;
double logarithmic graph)
1e-007
1e-006
1e-005
0.0001
0.001
0.01
0.1
1 10 100 1000 10000 100000
4CVKQQHDQQMUCNGURQKPV
4CPMKPI
Figure 18: The relation between sales volume
and sales rank of “Paperback Novels” (May 2006;
double logarithmic graph)
1e-006
1e-005
0.0001
0.001
0.01
0.1
1 10 100 1000 10000
4CVKQQHDQQMUCNGURQKPV
4CPMKPI
Figure 19: The relation between sales volume
and sales rank of “Economics and finance” (May
2006; double logarithmic graph)
5
1e-006
1e-005
0.0001
0.001
0.01
0.1
1 10 100 1000 10000
4CVKQQHDQQMUCNGURQKPV
4CPMKPI
Figure 20: The relation between sales volume and
sales rank of “Management” (May 2006; double
logarithmic graph)
0.0001
0.001
0.01
0.1
1
1 10 100 1000
4CVKQQHDQQMUCNGURQKPV
4CPMKPI
Figure 21: The relation between sales volume and
sales rank of “Chemistry” (May 2006; double log-
arithmic graph)
1e-005
0.0001
0.001
0.01
0.1
1
1 10 100 1000
4CVKQQHDQQMUCNGURQKPV
4CPMKPI
Figure 22: The relation between sales volume and
sales rank of “Physics” (May 2006; double loga-
rithmic graph)
1e-006
1e-005
0.0001
0.001
0.01
0.1
1 10 100 1000 10000
4CVKQQHDQQMUCNGURQKPV
4CPMKPI
Figure 23: The relation between sales volume and
sales rank of “Information and electronics” (May
2006; double logarithmic graph)
1e-005
0.0001
0.001
0.01
0.1
1
1 10 100 1000 10000
4CVKQQHDQQMUCNGURQKPV
4CPMKPI
Figure 24: The relation between sales volume and
sales rank of “Drawings and scalpture”
1e-007
1e-006
1e-005
0.0001
0.001
0.01
0.1
1
1 10 100 1000 10000 100000
4CVKQQHDQQMUCNGURQKPV
4CPMKPI
Figure 25: The relation between sales volume and
sales rank of Hardcover (May 2006; double loga-
rithmic graph)
1e-007
1e-006
1e-005
0.0001
0.001
0.01
0.1
1 10 100 1000 10000 100000
4CVKQQHDQQMUCNGURQKPV
4CPMKPI
Figure 26: The relation between sales volume and
sales rank of Softcover (May 2006; double loga-
rithmic graph)
6
ration, we found that the range takes over upper
1.5% titles (Figure 27), and these titles make up
the share of 50% in book sales market. Thus we
decided to set a baseline to 1.5% in this paper).
The dotted line in the figures shows the boundary
of upper 1.5% titles. Note that upper 1.5% titles
indicate about 8300 titles in a year, and about
4400 titles in a month4.
The power index of upper 1.5% titles and
market share in each months are shown in Ta-
ble 1. First of all, we find that the number of
power index increases gradually (Figure 30). It
means that the power index is getting to be big-
ger, the bigger difference is made among the top
titles. Second, we find that the market share of
top titles increase more (Figure 31). It means
that the gap between the top titles and the other
titles are widen.
Combining the results above, it can be drawn
as the relation between the change of power in-
dex and that of market share (Figure 32). We
conclude that the upper 1.5% titles have the ten-
dency to be sold more and more by the analysis
of sales from April 2005 to March 2006.
4 Discussion: New Approach to
Understand Market and Con-
sumers Behavior
We have investigated the book sales market and
verified that the relation between sales volume
and sales rank follows power law by analyzing
real sales data. From the analysis, now we have
chance to understand the mechanism of market
and consumers behavior based on power-law dis-
tribution. In this section, we would like to discuss
the future perspective of the study on market and
consumers behavior.
When the consumers purchase products, they
are mostly influenced by friends’ opinions, win-
dow advertisements, or mass media (Figure 33).
The choice of products by each consumers is “con-
tingent”, which means that the individual result
of selection is not inevitable and it can be oth-
erwise. However, the order in the market is gen-
erated by accumulating of the result of the con-
tingency. It is because the exposure of top sales
affects the choice of other consumers. The kind
‣†‧↝᥄
Sales Volume
Sales Rank
Upper 1.5% Titles
Figure 27: Upper 1.5% titles as a baseline for
approximation (from April 2005 to March 2006:
linear graph)
Sales Volume
Sales Rank
Upper 1.5% line
Figure 28: Upper 1.5% titles as a baseline for
approximation (from April 2005 to March 2006:
double logarithmic graph)
Sales Volume
Sales Rank
Upper 1.5% line
Figure 29: Upper 1.5% titles as a baseline for
approximation (April 2005: double logarithmic
graph)
7
0.64
0.66
0.68
0.7
0.72
0.74
0.76
2005
April
2005
August
2005
December
2006
April
Power Index
Figure 30: The transition of power index of upper
1.5% titles (from April 2005 to March 2006)
44
46
48
50
52
54
56
2005
April
2005
August
2005
December
2006
April
Market Share of Upper 1.5% Titles
Figure 31: The transition of market share of up-
per 1.5% titles (from April 2005 to March 2006)
46
48
50
52
54
0.66 0.68 0.7 0.72 0.74 0. 76
’April
’05
’05
’
’05
’05
05
’06
’06
06
’05
05
05
05 June
05
05
05
’
05
’
06
’06
05
Power Index
Market Share (%)
July
May
November
October
September
August
February
January
March
December
Figure 32: The relation between power index and
market share of upper 1.5% titles (from April
2005 to March 2006)
macroscopic level
EQPUWOGT
Choosing Products by their own decision
Microscopic
Communication
Emergence
Top Ranking
╙㧝
╙㧞
╙㧟
Power-Law Distribution
Mesoscopic
Feedback
Macroscopic
Feedback
mesoscopic level
microscopic level
Figure 33: Multilayered interactions in market
of multilayered interactions make the market self-
organize to critical state.
The fact that the law we found in the mar-
ket is “power law” implies that the fundamental
principle of self-organization is similar to other
natural and social phenomena that follows the
power law. From the principle of “universality”
in non-equilibrium physics, the phenomena which
belong to the same class of universality work in
the same mechanism to form the order. It means
that there is chance to understand the market
mechanism analogically by knowing that of other
phenomena, like avalanches on sandpile, earth-
quakes, or growth of cities. Here is a new way
to understand market and consumers behavior
(Figure 34).
This approach can lead us to understand the
mesoscopic mechanism, which it is difficult to in-
vestigate in the study of consumers behavior due
to the difficulty of observation. As a conclusion,
finding the power-law distribution in the market
is important not only because it is the “emergent
order”, but also because it opens up new chance
to study market and consumers behavior with us-
ing the analogy from systems belonging to same
class.
5 Conclusion
In this paper, we verified the power-law distribu-
tion between sales volume and sales rank in book
sales market. From the results, we can suggest
that the consumers behavior are deeply affected
by the market’s organizing principle. Therefore,
now we have started to develop the multi-agent
8
Consumer
Consumers’ Interaction Sands’ Interaction Plates’ Interaction
New Approach
to Understand
Consumer
Behaviors
Conventional Approach
to Understand Consumer
Behaviors
Macroscopic
Understanding
with behavior analysis
Mesoscopic
Understanding
with analogy
Macroscopic Understanding
with aggregation
Emergence Emergence Emergence
Power-Law Distribution
Self-organizing Criticality
of Sales in Product Market
Self-organizing Criticality
of Avalanche in Sandpile
Self-organizing Criticality
of Earthquake
Power-Law Distribution Power-Law Distribution
macroscopic level
mesoscopic level
microscopic level
Figure 34: New approach to understand market and consumers behavior
simulation model to understand the mechanism
of generating power-law distribution in the mar-
ket. It is assumed that the consumers’ choices
are influenced by the information from mass me-
dia and their friends. In further studies, we are
going to investigate this hypothesis.
Notes
1 The related researches which focus on the other
viewpoints are done by Sornette and Deschatres
[6, 8], there are some analysis about the tip-tops
of books sales, and its forerunner and aftershocks,
using the data of ranking system in Amazon.
Lambiotte and Ausloos [9] also analyzed the peak
of book sales and endogenous peak using the data
from Amazon’s ranking system. This analysis is
addressing the distinction of the peak in the long
time scale, so we take this for related research.
Furthermore, Brynjolfsson et. al.[10] estimated
that the online bookstore has wider variety than
the real bookstore . But it is suggested that this
estimation does not show the real data, because
the supposition of algorithm differ from the ac-
tual condition.
2 Rosenthal pointed out that the graph which
shows the relation between sales volume and sales
rank is his “personal guesstimate,” and any offi-
cial recognition or authorization is not given by
Amazon [7].
3 The analyses in this paper were originally re-
ported in the papers[11, 12, 13] in Japanese.
4 Upper 10% titles make up the share of 85%
(Figure 28), and upper 20% titles make up that
of 95% in the market (Figure 29). It means that
the book sales market is really “Winners-Take-
All” market.
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*This paper is for Fourth Joint Japan-North Amer-
ica Mathematical Sociology Conference, May, 2008.
(Contact Us: E-mail to iba@sfc.keio.ac.jp)
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