How the fittest compete for leadership: A tale of tails

We investigate how leaders emerge as a consequence of the competitive dynamics between coupled papers in a model citation network. Every paper is allocated an initial fitness depending on its intrinsic quality. Its fitness then involves dynamically as a consequence of the competition between itself and all the other papers in the field. It picks up citations as a result of this adaptive dynamics, becoming a leader if it has the highest citation count at a given time. Extensive analytical and numerical investigations of this model suggest the existence of a universal phase diagram, divided into regions of weak and strong coupling. In the former, we find an “extended” and rather structureless distribution of citation counts among many fit papers; leaders are not necessarily those with the maximal fitness at any given time. By contrast, the strong-coupling region is characterized by a strongly hierarchical distribution of citation counts, that are “localized” among only a few extremely fit papers, and exhibit strong history-to-history fluctuations, as a result of the complex dynamics among papers in the tail of the fitness distribution.

Figure 1

Mean-field phase diagram of the model in the situation where the damping rate $\delta$ is very small. WCR, weak-coupling regime ($g<g_c$). SCR, strong-coupling regime ($g>g_c$).

Figure 2

Mean-field prediction for the reduced mean activity $A/A_{\infty }$ against $g/g_c$ for damping rates $\delta$ ranging from 0.01 to 0.5 (bottom to top). Black line, SCR prediction Eq. (35). Symbol, location of the threshold coupling $g_{\mathrm{eff}}\approx 0.45$ [see Eq. (44)] for $\delta =0.1$ and $\omega =1$, so that $g_c=0.2$.

Figure 3

Total activity $A\left(t\right)$ against time $t$ during a single history of $N=1000$ papers with $\omega =1, \delta =0.1$, and $g=0.5$.

Figure 4

Mean total activity $\langle A\left(t\right)\rangle$ against $1/\left(\omega t\right)$. Black (lower) dataset: $g=0.5$ and $\omega =1$. Red (upper) dataset: $g=1$ and $\omega =0.5$. Each dataset is averaged over 10 000 independent histories of $N=500$ papers. Blue lines, two-parameter fits with common intercept 3.01.

Figure 5

Fitness-resolved gain $G(\varepsilon ,t)$ against $\varepsilon$, for (a) $t=100$ and (b) $t=200$, and values of $g$ ranging from 0.2 to 1 (bottom to top in each panel).

Figure 6

Mean number $L_t$ of successive most cited papers (leaders) up to time $t$ against $lnt$ for coupling constants $g$ ranging from 0 to 1.2 (top to bottom). The dashed lines have slopes $C_0=1$ and $C_{\mathrm{SCR}}=0.57$.

Figure 7

(a) Probability $P_t$ that the most cited paper at time $t$ is the current fittest paper; (b) probability ${\mathrm{\Pi }}_t$ that the most cited paper at time $t$ belongs to the sequence of fittest papers. Both probabilities are plotted against $g$ for several times $t$ ranging from 100 to 1000 (top to bottom in each panel).

Figure 8

Log-log plots of (a) the average $\langle C_{\max }\left(t\right)\rangle$ of the largest citation count in units of $\lambda$ and (b) the product $t\langle Y\left(t\right)\rangle$, against $t$, for coupling constants $g$ ranging from 0 to 1.2 (bottom to top in each panel). Dashed lines with unit slope are guides to the eye.

Figure 9

Probability distribution of the moment ratio $Y$ for $g=1.2$ and $t=500$ and $t=1000$.

Figure 10

Log-log plot of the citation counts $C_k\left(t\right)$ in units of $\lambda$ for a single history of the effective model with $\mathrm{\Omega }=1$, against time $t$ up to the observation time $t=e^{10}\approx 22026$. Colors alternate for a clearer reading.

Figure 11

The first three probabilities $p_k$ and the mean moment ratio $\langle Y\rangle$ of the effective model, against the effective rate $\mathrm{\Omega }$.

Figure 12

Logarithmic plots of the probabilities $p_k$ against $k$ for four values of the effective rate $\mathrm{\Omega }$ ranging from 0.5 to 4 (top to bottom). Full lines: fits of the form $p_k=A{[{\mu }^k/\mathrm{\Gamma }(k+b)]}^2$, with $\mu \approx 2.48$ for $\mathrm{\Omega }=0.5, \mu \approx 1.82$ for $\mathrm{\Omega }=1, \mu \approx 1.26$ for $\mathrm{\Omega }=2, \mu \approx 0.92$ for $\mathrm{\Omega }=4$. The product $\mathrm{\Omega }{\mu }^2$ is observed to be slightly above 3 in all cases.

Figure 13

Probability distribution $f\left(Y\right)$ of the moment ratio $Y$ for the same values of the effective rate $\mathrm{\Omega }$ as in Fig. 12.