Empirical analysis of collective human behavior for extraordinary events in the blogosphere

To uncover an underlying mechanism of collective human dynamics, we survey more than 1.8 billion blog entries and observe the statistical properties of word appearances. We focus on words that show dynamic growth and decay with a tendency to diverge on a certain day. After careful pretreatment and the use of a fitting method, we found power laws generally approximate the functional forms of growth and decay with various exponents values between $-$0.1 and $-$2.5. We also observe news words whose frequencies increase suddenly and decay following power laws. In order to explain these dynamics, we propose a simple model of posting blogs involving a keyword, and its validity is checked directly from the data. The model suggests that bloggers are not only responding to the latest number of blogs but also suffering deadline pressure from the divergence day. Our empirical results can be used for predicting the number of blogs in advance and for estimating the period to return to the normal fluctuation level.

Figure 1

Temporal change of the word frequency of “April fool” per week. The results are from Google Trends, which is targeted worldwide, and our blog data “Kuchikomi@kakaricho,” which is targeted only in Japan. The number of blogs is normalized by the whole number.

Figure 2

Probability density distribution of circadian activity of blogs posted by 10 000 bloggers. The solid line is calculated from all entries and the red bar is from the entries excepted that have a time stamp of 00:00:00. In both, 04:00 is the smallest ratio in a day.

Figure 3

Typical example of time series of the peaked word Marine Day in 2008. $w=1$ corresponds to no revision and $w=0.9$ corresponds to modified time series introduced in Sec. 3a. Because of the circadian effect, the data of the day after the peak is always higher than that before the peak without modification.

Figure 4

Examples of data fitting by power laws of Marine Day in 2008 for fore-slope (a) and after-slope (b) plotted in log-log scale. For fore-slope, models are fitted by Eq. (2) with ${\alpha }_j=1.27$ and $A_j=3100$ ($q=0.239$, $n=58$). For after-slope, ${\alpha }_j=1.42$ and $A_j=3171$ ($q=0.108,n=18$). The shaded area shows the interval $n$ in which the power-law model is accepted.

Figure 5

Distribution of power exponent ${\alpha }_j$ of the fore-slopes (a) and the after-slopes (b). The mean value of ${\alpha }_j$ of the fore-slope is 0.85 $\pm$ 0.30 and that of the after-slope is 1.13 $\pm$ 0.21. Bars are colored by three categories of peaked words.

Figure 6

Decay of tsunami observed for blogs for 417 days since March 1, 2011, (a) and tweets for 648 h (27 days) since March 9, 2011 (b). Horizontal time step size is per day for blogs and per hour for tweets. Inset: Log-log plots of the time series. Red dashed lines show the slope of power law with the exponent ${\alpha }_j=0.67$ for blogs and ${\alpha }_j=0.75$ for tweets.

Figure 7

Relationship between difference rates $|\Delta x_j\left(t\right)|/x_j\left(t\right)$ and the absolute inverse number of time from the peak calculated by Eq. (8) for after-slopes, which summarized 1341 samples, shown in Table . The solid line shows the theoretical relation, Eq. (8), with the averaged empirical value, ${\alpha }_j=1.13$.

Figure 8

Number of blog entries including Marine Day per week which is calculated from randomly selected 30 000 bloggers (bottom) and the ratio of the number of newly posted bloggers in the same week (top). The dashed line shows the ratio 0.5.

Figure 9

Typical examples of prediction for (a) 20 days and (b) 5 days before the peak day for Marine Day in 2008 in semilogarithmic scale. The red line indicates the prediction line, the blue dashed line indicates median $\overline{x_j}=17$, and the solid line shows the real values. Open circles indicate the future values and colored circles are the known values used for prediction. The estimation is started 85 days before $t_c$. Estimated values are ${\alpha }_j=1.78$ and $A_j=20222$ for 20 days before the peak day and ${\alpha }_j=1.38$ and $A_j=4725$ for 5 days before $t_c$.

Figure 10

Examples of data fitting without time-shift pretreatment by power laws of Marine Day in 2008 for fore-slope (a) and after-slope (b) plotted in log-log scale. For fore-slope, models are fitted by Eq. (2) with ${\alpha }_j=1.10$ and $A_j=2273$ ($q=0.103$, $n=26$). For after-slope, ${\alpha }_j=1.44$ and $A_j=3369$ ($q=0.104$, $n=18$). The shaded area shows the interval $n$ in which the power-law model is accepted.

Figure 11

Distribution of power exponent ${\alpha }_j$ of fore-slopes (a) and after-slopes (b) without a time shift. Mean value of ${\alpha }_j$ of fore-slope is 0.73 $\pm$ 0.35 and after-slope is 1.16 $\pm$ 0.21. Bars are colored by three categories of peaked words.

Figure 12

Relationship between mean and standard deviation of word frequency in the blogosphere. Empirical results of 1771 adjectives and theoretical result of Eq. (B12) are duplicated in the figure.