Dec 10, 2009 - speed to be proportional to a moving average of the market's transaction ... average daily temperature in Tokyo and number of blog entr...

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A new field attracts the attention of physics researchers when precise quantitative observation becomes available. A good example of this is financial market price fluctuation that was not among physicists’ interests twenty years ago. However, the huge number of precise observations and numerical simulation techniques led to a new field of applied statistical physics by the name of econophysics [1]. More recently, the sales figures for popular books have been analyzed in view of the universal responses of complex systems [2]. Further, the increase in the number of registrations for a conference as the deadline nears has been studied and concluded to a result of the superposition of random processes caused by naive human behaviors [3]. At an individual level, any human behavior may be recognized as intentional and not random. However, by observing a large number of people simultaneously, we can expect to observe the random nature of the gross properties owing to mutual independence. The first step in the data analysis of human behaviors is to clarify how real human system shows randomness by comparing it with predictions, based on the assumption of independence. The next step is to establish an empirical relation about the gap between the simple theory and observation. The third step is to develop a new theory that captures a deeper human action or hidden social interaction. For example, paying attention to the time intervals of deals in financial markets, an exponential distribution of intervals is naturally expected as the occurrence of a deal is caused by a collision of buy and sell orders at the same price, which can be modeled by a Poisson process. However, in reality we generally observe a time interval distribution characterized by a fat-tail. It is clarified from data analysis that the underlying Poisson parameter is changing in the time scale of a few minutes and the observed fat-tailed distribution is given by the superposition of exponential distributions with various characteristic scales [4]. The human behavior underlying this phenomenon is considered to be a general tendency of market participants to modulate their own clock speed to be proportional to a moving average of the market’s transaction intervals. This effect, named self-modulation, is known to induce long-term corre-

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INTRODUCTION

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Temperature (Celsius) FIG. 1: Relation between temperature and number of blog entries. (a) Time series of tempetature and number of blogs including word “hot” and “cold”. (b) Scatter plot of temperature and number of blogs.

lation and make the power spectrum of events follow a 1/f spectrum [5]. An agent-based numerical model can reproduce this effect and we can find that the resulting transaction intervals are compatible with real data [6]. In this paper, we focus on the data analysis of blogs to observe collective human behaviors. A blog is a new type of social communication medium where personal impression can be easily uploaded on the web. A blog first appeared in the late 1990s and rapidly gained popularity. The number of blogs has been increased every year since then. A typical blog is maintained by an individual or a small group. It consists of entries posted by users that contain text and/or images and are frequently accompanied by links to other web pages. One of the characteristics of a blog is that readers can easily

Temperature (Celsius)

arXiv:0912.1911v1 [physics.soc-ph] 10 Dec 2009

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II.

BASIC STATISTICS

A challenging blog analysis is applied for early detection of epidemics. Ginsberg et al. observed keywords such as “flu” by using the search engine query data on the Google homepage and confirmed a positive correlation between the actual spread and word appearances [12]. Here, we focus on the more basic properties of word frequency. A pioneering basic study of blog word appearance has been done by Lambiotte et al. who analyzed appearance intervals of common words [13]. It was reported that a naive Poissonian assumption fails in general even for low-frequency words. However, two of the authors (Y.S. and M.T.) recently examined similar time series of blog word appearances carefully and found that such deviation from the Poisson process is caused by the following three effects: 1. Repeated spamming. 2. Non-stationary changes caused by server breakdowns or the growth in blog population. 3. Periodic fluctuations during specific days of the week.

Standard Deviation σ

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leave their comments in an interactive format. Consequently, a blog has evolved as a new type of social communication tool for individuals. Search engine technologies have been developed to observe the details of blog entries automatically at high speeds [7, 8]. We focus on the temporal changes in the frequency of a word on the web. For a given keyword, the search engine automatically lists all blog entries that include the keyword along with the time of posting. According to Technorati [9], an internet search engine company for blogs, the number of blogs in the world is now more than 70 million and this figure is still increasing. Categorizing blogs in terms of language, we find that in 2007, 37% of the blogs were in Japanese, 36% in English, 8% in Chinese, etc. [10]. Here, we observe Japanese blogs from January 1st, 2007 to December 31st, 2008 using a search engine provided by Dentsu Buzz Research [11]. This search engine covers 20 major blog providers in Japan covering more than 10 million users. The total number of observed entries exceeds 600 million; in other words, on average, about 800 thousand new entries are uploaded every day. Here, we show an example that blogs are often reflect peoples reaction to the social and natural phenomena. In Fig.1, we show relationship between the average daily temperature in Tokyo and number of blog entries including the keywords “hot” and “cold”. In summer time, in the middle of Fig.1(a), it is getting hot and people tend to post blog entries with keyword “hot” more frequently. We confirm that there is a nonlinear relation between real temperature and number of entries by the scatter plot [Fig.1(b)].

Analytical solution Eq.(2) 1/2

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FIG. 2: Relation between average hF i and standard deviation σpin log-log scale. The dashed line corresponds to σ = hF i and the solid line, to σ = 0.11hF i. (a) Empirical result for the 271 words. (b) Comparison with analytical solution of Eq.(2) (bold solid line).

By introducing normalization processes to eliminate these effects, we can confirm that, as expected theoretically, low-frequency words actually follow the Poisson process [14]. It is interesting that we still observe deviation from the Poisson process for any highfrequency words. We focus on such high-frequency words. In order to clarify the reason for this deviation from the Poisson process, we introduce the following data analysis. We randomly chose 300 words from a dictionary of Japanese morphological analysis. We get rid of 29 words from these words in case that the time series has 5 times higher peak from the average or average is more than 104 . Then, we get 78 nouns, 36 verbs, 66 adverbs, 37 adjectives, 35 conjunctions, and 19 other types of words. For the j-th word, we observe the daily number of blogs that contain the j-th word by using the search engine xj (t). Then, we apply the above-mentioned normalization procedures to get a normalized blog number time series Fj (t). The average over the entire observation time hFj i and the q 2

standard deviation σj = h(Fj − hFj i) i is plotted for each word in a log-log scale in Fig.2(a). In the case that the appearance of a word is approximated by the Poisson process, the standard deviation is given by the square root of the average

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∆(∆ + 1) σj = hFj i 1 + hFj i 3N 2 (p 2 hFj i hFj i ≪ 3N ∆2 ≈ 2 . √∆ hFj i hFj i ≫ 3N2 ∆ 3N

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value, and a line with slope 1/2 is expected. For words with frequencies less than about 80 entries per day, this relation can be confirmed to be independent of the morphological category from the left part of Fig.2(a). For these words, the autocorrelations of intervals are confirmed to be almost 0 and the frequency distributions are checked to pass statistical tests of the Poisson distribution; namely, the appearance of low-frequency words can be approximated by the Poisson process. It should be noted that this result is so delicate that it cannot be obtained without the above normalization procedure, which might have been missing in the preceding study [13]. For more-frequently appearing words, we can find a clear deviation from the simple Poisson process. As shown in the right part of Fig.2(a), instead of a square root relation, a linear relation between standard deviation and average holds empirically. Similar linear relations between standard deviation and average are observed in various fields such as river flow fluctuations, traffic fluctuations on highways [15] and numerical fluctuations in cellular molecules [16]. In the case of traffic fluctuations, this relation is understood by a simple model named Random Diffusion (RD) model [17], where random walkers are randomly injected into a given network. By observing the flow of walkers at a node, the linear relation between average flow strength and standard deviation is confirmed for large value of flow. Here, we introduce a simple model of posting blog entries as an application. We assume the situation that there is no spam and the system is stationary with no weekly period. There are n(t) bloggers who are actively posting blogs at time t. Each blogger independently posts a blog entry containing the j-th keyword with probability cj . The number of bloggers n(t) is assumed to change randomly in the range of [N − ∆, N + ∆], where ∆ is a positive constant. Thus, the mean number of bloggers who post a blog entry containing the j-th word at time t is Fj (t) = cj n(t). Considering the case of a uniformly distributed n(t) for simplicity, we obtain the probability density of the number of keyword appearances pj (Fj ) by taking the superposition of the Poisson distribution.

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By fitting this theoretical estimation to the empirical relation in Fig.2(a), we can tune the model’s parameters. Since from Fig.2(a), the bending point is estimated as hF ic = 82, and from Eq.(2), it is given as hF ic = 3N 2 /∆2 , we have ∆/N = 0.19. The linear relation between standard deviation and average in Eq.(2) becomes σ = 0.11hF i, which automatically fulfills the empirical relation as shown in Fig.2(b). This result demonstrates that the non-trivial relation between standard deviation and average is a general property caused by the daily fluctuations in the number of bloggers. Although the above theoretical analysis captures the basic relation between the average number of keywords and the standard deviations, the data points in Fig.2(a) scatter largely for words with large averages. It is confirmed that this scattering cannot be explained by the above superposition of stationary Poisson processes, and we need to consider the effect of temporal change of the appearance probability cj .

III.

APPLICATION

As an application of this linear relation between standard deviation and average, here, we introduce a method of finding extraordinary events hidden in fluctuations of word appearances in blogs. We pay attention to the name of new bread, which went on the market on January 29th, 2008. The time series of the daily number of blogs containing this new name after normalization is given in Fig.3. Before December 26th, 2007, the number count of this keyword was 0. Later, the number count grew rapidly as though in a type of oscillation. In order to remove the weekly period, we introduce a 7-day moving average which is defined by

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where ∆/N = 0.19. The daily number count for this keyword is mostly within the 2σ(t) band. However, occasionally it is higher. We traced back the news and found that external factors such as new items released on those days did have an effect: 1. Jan./2nd/2008: One of the popular blogs introduced the name of this new bread. 2. Jan./9th/2008: The company put out a news release announcing the new bread. 3. Jan./21st/2008: The name of this new bread was introduced in a TV program.

IV.

DISCUSSION

In summary, we first showed the importance of noise reduction for quantitative analysis of blog-word frequencies. For low-frequency words, we confirmed the Poisson hypothesis. For more-frequent words, we found that the effect of the change in the number of contributors modifies the distribution drastically and the standard deviation of blog-number fluctuations becomes proportional to the average values. Analytical solutions successfully reproduce the above basic properties of blog-number fluctuations. In this research we only observed Japanese blogs; however, we expect that the analysis method and the numerical model is directly applicable to other languages. By establishing the basic observation methods and simulation models, blog analysis will become a powerful tool for the scientific study of collective human behaviors in the society.

4. Jan./29th/2008: The new bread went on sale. By using this empirical standard deviation, we can successfully detect the peculiar fluctuations that were apparently caused by the above external factors. Furthermore, we can evaluate quantitatively the impact of the company’s press releases to people.

[1] R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, (Cambridge University Press, Cambridge, England, 1999). [2] D. Sornette, F. Deschatres, T. Gilbert, and Y. Ageon, Phys. Rev. Lett. 93, 228701 (2004). [3] V. Alfi, G. Parisi, and L. Pietronero, Nature Physics (London) 3, 746 (2007). [4] M. Takayasu, H. Takayasu, and M. P. Okazaki, in proceedings of Empirical Science of Financial Fluctuations, Tokyo edited by H. Takayasu (Springer, New York, 2001) pp.18-26. [5] M. Takayasu and H. Takayasu, Physica A 324, 101 (2003). [6] K. Yamada, H. Takayasu, T. Ito, and M. Takayasu, Phys. Rev. E 79, 051120 (2009) [7] http://kizasi.jp/ [8] http://blogsearch.google.com/

Acknowledgements

The authors appreciate the cooperation of Dentsu, Inc. and hottolink, Inc. for providing the blog data.

[9] [10] [11] [12] [13] [14] [15] [16] [17]

http://technorati.com/ http://www.sifry.com/alerts/archives/000493.html http://www.dbuzz.jp J. Ginsberg, M. H. Mohebbi, R. S. Patel, L. Brammer, M. S. Smolinski, and L. Brilliant, Nature (London) 457, 1012 (2009). R. Lambiotte, M. Ausloos, and M. Thelwa, Joural of Informetrics 1, 277 (2007). Y. Sano and M. Takayasu, arXiv:0906.1744. (Submitted to J. Econ. Interact. Coord.) M.A. de Menezes and A.-L. Barab´ ai, Phys. Rev. Lett. 92, 28701 (2004). C. Furukawa, T. Suzuki, A. Kashiwagi, T. Yomo, and K. Kaneko, Biophysics 1, 25 (2004). S. Meloni, J. Gomez-Grdenes, V. Latora, and Y. Moreno, Phys. Rev. Lett. 100, 208701 (2008).