Interacting Branching Process as a Simple Model of Innovation

We describe innovation in terms of a generalized branching process. Each new invention pairs with any existing one to produce a number of offspring, which is Poisson distributed with mean $p$. Existing inventions die with probability $p/\tau$ at each generation. In contrast with mean field results, no phase transition occurs; the chance for survival is finite for all $p>0$. For $\tau =\infty$, surviving processes exhibit a bottleneck before exploding superexponentially—a growth consistent with a law of accelerating returns. This behavior persists for finite $\tau$. We analyze, in detail, the asymptotic behavior as $p\to 0$.

Figure 1

Death and survival probabilities. Diamonds: Rescaled rate $\mathcal{D}(t)$ at which surviving populations die versus scaled time $t=pg$, for two different values of $p\ll 1$, ${10}^{-6}$ (outer blue lines), and ${10}^{-7}$ (solid red). As $p\to 0$, the death probability converges to a limiting function that vanishes at $t=t_e\approx 0.67$. Circles: $-p\log Z(t/p)$ versus $t$, for the same two values of $p$.

Figure 2

Mean generation size as a function of branching ratio. The solid black curve is the numerical prediction for the mean generation size $s$ versus the mean branching ratio $u$ (both conditioned on growth) as obtained from Eq. (5). The diamonds are results of numerical simulations of the IBP. We used $p={10}^{-6}$ and $\tau =\infty$, and averaged over ${10}^5$ conditioned-to-grow populations. Agreement is excellent. Also plotted are three lines showing $u/\tau$ on the $y$ axis, for three values of $\tau$ from top to bottom: 0.16, subcritical $\tau <{\tau }_c$, critical $\tau ={\tau }_c=0.391$, and 0.5, supercritical $\tau >{\tau }_c$, respectively.

Figure 3

Mean time $t_e$ to enter the superexplosive phase. The red circles (blue diamonds) are the results of numerical simulations at $p={10}^{-6}$ (${10}^{-5}$) for the mean time to enter the superexplosive phase. The black curve going through them is a fit $\propto (\tau -{\tau }_c{)}^{-1/2}$.

Figure 4

Mean first passage (generation) times (MFPT) to reach a generation of size $m$, for $\tau =\infty$. MFPTs for $m=2,3,4,5$ are shown from bottom to top, respectively, along with the predicted values from Eq. (7). For the range of $p$ studied, numerical results obtained by averaging over ${10}^6$ surviving processes agree with predictions for $m=2,3$. For $m=4,5$, they converge to the predicted values as $p\to 0$.