Growth fluctuation in preferential attachment dynamics

In the Yule-Simon process, creation and selection of words follows the preferential attachment mechanism, resulting in a power-law growth in the cumulative number of individual word occurrences as well as the power-law population distribution of the vocabulary. This is derived using mean-field approximation, assuming a continuum limit of both the time and number of word occurrences. However, time and word occurrences are inherently discrete in the process, and it is natural to assume that the cumulative number of word occurrences has a certain fluctuation around the average behavior predicted by the mean-field approximation. We derive the exact and approximate forms of the probability distribution of such fluctuation analytically, and confirm that those probability distributions are well supported by the numerical experiments.

Figure 1

A diagram of the relationship between the variables depicting the growth of the cumulative number of word occurrences.

Figure 2

The rank-frequency distribution in the case of (a) $\alpha =0.01$, (b) 0.1, and (c) 0.5. Solid and dotted lines show, respectively, the simulation results and theoretical curves as an eye guide, which is given by Simon's rate equation approach [4] and proportional to $[\text{word rank}{]}^{\overline{\alpha }}$.

Figure 3

The growth of the cumulative number of occurrences of three sampled words created at the (a) 89th, (b) 90th, and (c) 91st orders in the case of $\alpha =0.1$. Solid and dotted lines show, respectively, the simulation results and the corresponding expected growth curves given by the mean-field estimation Eq. (5).

Figure 4

Numerical solutions of the general form Eq. (12) for $t_i={10}^1$ (black circles), ${10}^2$ (dark gray), ${10}^3$ (light gray), and ${10}^4$ (white) under the condition of different $\lambda$ and $\alpha$ values: $\lambda =2$, 5, 10 and $\alpha =0.01$, $0.1$, $0.5$. For example, (a) shows the case of $\alpha =0.01$ and $\lambda =2$. The horizontal and vertical axes are the value of $x$ and $P\left(x\right)$, respectively. Note that the vertical axis is in logarithmic scale, whereas the horizontal axis is in linear, so that exponential decay draws a descending straight line.

Figure 5

Comparison between the numerical results of the simulation (white circles), the numerical solution of the general form for $t_i={10}^4$ (solid lines), and analytic curves drawn from the particular solution for a sufficiently small $\alpha$ (dotted lines), for different $\alpha$ and $\lambda$ values. In the case of small values of $\alpha$, as shown in (a), (d), and (g), all results exhibit a good match. At the same time, the mismatch between the particular solution and the others increases for large values of $\alpha$, as shown in (c), (f), and (i).

Figure 6

The relationship between $\lambda$ (the observation timescale), and fitted $x_c$ (the characteristic scale of the deviation). White, light gray, and black circles show the qualified results that have the standard error less than 1%, for $\alpha =0.01$, $0.1$ and $0.5$, respectively.