Dynamics of book sales: Endogenous versus exogenous shocks in complex networks

We present an extensive study of the foreshock and aftershock signatures accompanying peaks of book sales. The time series of book sales are derived from the ranking system of Amazon.com. We present two independent ways of classifying peaks, one based on the acceleration pattern of sales and the other based on the exponent of the relaxation. They are found to be consistent and reveal the coexistence of two types of sales peaks: exogenous peaks occur abruptly and are followed by a power law relaxation, while endogenous sale peaks occur after a progressively accelerating power law growth followed by an approximately symmetrical power law relaxation which is slower than for exogenous peaks. We develop a simple epidemic model of buyers connected within a network of acquaintances which propagates rumors and opinions on books. The comparison between the predictions of the model and the empirical data confirms the validity of the model and suggests in addition that social networks have evolved to converge very close to criticality (here in the sense of critical branching processes of opinion spreading). We test in detail the evidence for a power law distribution of book sales and confirm a previous indirect study suggesting that the fraction of books (density distribution) $P\left(S\right)$ of sales $S$ is a power law $P\left(S\right)\sim 1∕S^{1+\mu }$ with $\mu \approx 2$.

Figure 1

(Color online) Two time series of ranks showing that Amazon switches between at least two ranking schemes, as explained in the text. The first book (top) is “See No Evil” by R. Baer. The second book (bottom) is “F’d Companies” by P. Kaplan.

Figure 2

(Color online) Rank ordering (Zipf) plot of the sales $S$ per day, as a function of rank $R$. For $R$ in the range from 10 to 10 000, the sales as a function of rank can be fitted with a good approximation by the power law $S\left(R\right)\sim 1∕R^{1∕\mu }$ with exponent $\mu =2.0\pm 0.1$, as shown by the straight line with slope $-0.5$. This translates into $R\sim 1∕S^{\mu }$, which is proportional to the complementary cumulative distribution of sales. Note that the bend at large ranks (small sales) can be considered to be a finite size effect. The tail of the distribution of sales for large sales is described by the low ranks for which the distribution does not appear to bend but actually exhibits huge fluctuations as shown in Fig. 3 borrowed from [9].

Figure 3

(Color online) Rank ordering (Zipf plot) as in Fig. 2 presented in [9], extended to the smallest as well as largest ranks. Reproduced with kind permission from M. Rosenthal.

Figure 4

(Color online) Time evolution over a year of the sales per day of two books: Book A (top) is “Strong Women Stay Young” by Dr. M. Nelson and book B (bottom) is “Heaven and Earth (Three Sisters Island Trilogy)” by N. Roberts. The difference in the patterns is striking, book A (B) exhibiting an exogenous (endogenous) peak.

Figure 5

$\sigma \left(t_c\right)$ defined in (19) as a function of $t_c$.

Figure 6

(Color online) Histogram of the estimated power law exponents $p$ of the relaxations of the sales. One can clearly identify two clusters: the endogenous cluster with exponent $1-2\theta$ close to 0.4 and the exogenous cluster with exponent $1-\theta$ close to 0.7, compatible with the estimation $\theta \approx 0.3$. The peaks shown here are those used in Fig. 7 (see Sec. 4C).

Figure 7

(Color online) Relaxation after the peak (for both endogenous and exogenous cases) and precursory acceleration (for the endogenous case). We average (log average) over the peaks classified as endogenous and exogenous according to their precursory growth. The same sample of books was used for both Fig. 6 and Fig. 7.

Figure 8

(Color online) Time evolution of the book “Get with the Program” (see Table ). Each time the book is presented in the Oprah Winfrey Show, the sales jump overnight and then relax according to the exogenous response function $\kappa \left(t\right)\sim 1∕t^{1-\theta }$.

Figure 9

(Color online) Time evolution of the sales of the 10 books of Table . For each time series, $t=0$ is the time of the peak. The sales jumped just before the peak except for “Stars Wars” and “Wolves of the Calla.”

Figure 10

(Color online) Relaxation of the sales after the peak for “Star Wars” and “Wolves of the Calla.” The expected power law behavior for exogenous shocks can only be observed for $t>10–20\text{days}$. The small value of the slope for $t<10–20\text{days}$ can be explained by considering the time duration of the external shock triggered by the media as explained in the text.

Figure 11

(Color online) Histogram of the estimates of the power law exponents $p$ of the relaxations of the sales using a selection criterion different from that of Fig. 6: among the 1013 peaks of our prefiltered database, we kept those which have a correlation coefficient larger than 0.9, giving 388 peaks.

Figure 12

(Color online) Relaxation of sales after the peak following the same methodology as for Fig. 7 but with the same set as in Fig. 11 with the clustering procedure explained in the text.

Figure 13

(Color online) Average variation $⟨\Delta R⟩$ of the rank as a function of the rank itself. This measure was performed over a sample of 14 000 books. One can note spurious behaviors for ranks close to ${10}^4$ and ${10}^5$. This can be rationalized by a shift in ranking schemes for these value (see Sec. 2B1). The data can only be exploited for $R<{10}^4$ to avoid these artifacts. The most striking feature is the nonmonotonous behavior of $⟨\Delta R⟩$ for $R<{10}^4$. This enables us to reject an exponential conversion (see Sec. 5B2).

Figure 14

(Color online) Average variation of the rank $⟨\Delta R⟩$ as a function of the rank itself. We only take into account those time series that do not exhibit abrupt peaks (i.e., we discard exogenous shocks). Plus: experimental data; solid line: fit assuming $S\left(R\right)$ to be a power law; dashed line: fit assuming $S\left(R\right)$ to be $S\left(R\right)\propto \exp (C∕\sqrt{R})$.

Figure 15

(Color online) Rank-to-sales conversion function $S\left(R\right)$ estimated using expression (36) for arbitrary values of $\alpha$, $⟨S⟩$, $S_{\mathit{max}}$. Consequently, the units on the $y$ axis are arbitrary units.