Theory of fads: Traveling-wave solution of evolutionary dynamics in a one-dimensional trait space

We consider an infinite-sized population where an infinite number of traits compete simultaneously. The replicator equation with a diffusive term describes time evolution of the probability distribution over the traits due to selection and mutation on a mean-field level. We argue that this dynamics can be expressed as a variant of the Fisher equation with high-order correction terms. The equation has a traveling-wave solution, and the phase-space method shows how the wave shape depends on the correction. We compare this solution with empirical time-series data of given names in Quebec, treating it as a descriptive model for the observed patterns. Our model explains the reason that many names exhibit a similar pattern of the rise and fall as time goes by. At the same time, we have found that their dissimilarities are also statistically significant.

Figure 1

(a) Fractions of a hundred male given names and (b) those of the same number of female given names in Quebec, from year 1900 to 2000. The names are sorted in an ascending order of the mean time of prevalence [Eq. (1)].

Figure 2

$P$ versus $C$, where $P$ is scaled by its maximum amplitude $P_{\max }$. The speed is chosen as $v=2$. Note that the flow of Eq. (15) goes from $(C,P)=(0,0)$ to $(1,0)$ as $\eta \equiv x-vt$ increases. (a) The solid line with $g=+0.4$ has $\frac{dP}{dC}>0$ at $C=\frac{1}{2}$ in this plot, and $P\left(\eta \right)$ has negative skewness. If we fix $x$ and observe its time series as $t$ varies, therefore, the height grows rapidly and then decreases slowly. (b) The dotted line with $g=-0.4$ shows the opposite case, as explained in the main text.

Figure 3

(a) More than two-thirds of the names show left-skewed $P\left(\tilde{\eta }\right)$, and one such example is plotted here. It means that they quickly become popular and then slowly disappear, because $\tilde{\eta }\propto -t$. The noisy curve represents the corpus data, and the smooth one is obtained from our model. This example is fitted with $v=1.7$ and $g=-0.06$ (see the main text for details of the fitting). (b) A dozen of names are equally well or better fitted to right-skewed distribution. The fitting parameters are $v=2.4$ and $g=-0.43$ in this example. (c) Recurrence is observed for a couple of names like Gabrielle and Marie-Anne, roughly with a centennial period. (d) There are a few ‘steady sellers’, including Rachel and Élisabeth, with no particular patterns.

Figure 4

Monte Carlo fitting results with the Metropolis algorithm. The energy function is identified with ${\chi }^2$ and the temperature is set to be $T=1$. (a) The distribution of ${\chi }^2$ [Eq. (20)] for the hundred female names after the fitting. The vertical line means ${\chi }^2=130$, below which the statistical significance is greater than 0.01. (b) Distribution of $v$ and $g$ for the names fitted with ${\chi }^2<130$. The error bars express the ranges of the parameters during ${10}^5$ Monte Carlo steps. Inset: A zoomed view. The dotted lines represent $v_{\min }=\sqrt{4g+2}$. (c) Estimated $v$ and $g$ against the mean time of prevalence [Eq. (1)] for the names in the inset of (b). (d) Actual data and fitting curves for two outliers on the $(g,v)$ plane, indicated by the arrows in (b).