Influence of Luddism on innovation diffusion

We generalize the classical Bass model of innovation diffusion to include a new class of agents—Luddites—that oppose the spread of innovation. Our model also incorporates ignorants, susceptibles, and adopters. When an ignorant and a susceptible meet, the former is converted to a susceptible at a given rate, while a susceptible spontaneously adopts the innovation at a constant rate. In response to the rate of adoption, an ignorant may become a Luddite and permanently reject the innovation. Instead of reaching complete adoption, the final state generally consists of a population of Luddites, ignorants, and adopters. The evolution of this system is investigated analytically and by stochastic simulations. We determine the stationary distribution of adopters, the time needed to reach the final state, and the influence of the network topology on the innovation spread. Our model exhibits an important dichotomy: When the rate of adoption is low, an innovation spreads slowly but widely; in contrast, when the adoption rate is high, the innovation spreads rapidly but the extent of the adoption is severely limited by Luddites.

Figure 1

Schematic depiction of the LISA model. An ignorant $\mathcal{I}$ can become a Luddite $\mathcal{L}$ with rate $r\dot{A}$ (in a mean-field setting); an ignorant can also become a susceptible $\mathcal{S}$ by contagion with rate proportional to the susceptible density. A susceptible spontaneously becomes an adopter at rate $\gamma$.

Figure 2

Evolution of a realization of the LISA model on a complete graph of ${10}^6$ nodes with $I_0=0.8$ and Luddism parameter $r=0.9$. (a) $\gamma =0.3$ (extensive adoption); (b) $\gamma =1$ (sparse adoption). Evenly distributed samples of the stochastic simulation $(\square )$ are indistinguishable from the solution of Eq. (1) (solid line). The completion times for (a) and (b) are 60 and 17, respectively.

Figure 3

Dependencies of the final-state densities $L_{\infty },I_{\infty }$, and $A_{\infty }$ for a complete graph of ${10}^4$ nodes and $I_0=0.9$. In the top panel $r=0.9$ while $\gamma$ varies, whereas in the bottom panel $\gamma =0.3$ while $r$ varies. Simulations $(\square )$ in complete agreement with (3) with substitution (7) (solid line).

Figure 4

The evolution, averaged over 100 realizations, of the LISA model on an ER graph with $N={10}^3$ nodes, $k=10$, and $I_0=0.8$. (a) $\gamma =0.002$, such that $\gamma <(k/N)I_0$ and (b) $\gamma =0.1$ such that $\gamma >(k/N)I_0$. Shown are the evenly distributed samples of the stochastic simulation $(\square )$ and the solution of Eq. (8) (solid line). The Luddism parameter $r=0.9$.

Figure 5

Dependence of the final densities $L_{\infty },I_{\infty }$, and $A_{\infty }$ on the average degree for ER graphs with $N={10}^3$ nodes. The simulation $(\square )$ represents an average over 40 model realizations for 30 randomly generated networks. Parameters are $\gamma =0.005,r=0.9$, and $I_0=0.9$. The mean-field predictions (9) (solid line) match the simulation for $k\gtrsim 20$ (see main text).

Figure 6

The mean-field steady-state predictions (9) over the parameter space $(\gamma ,r)$ for $k/N=0.025$ and $I_0=0.9$. The contours $L_{\infty }=I_{\infty },L_{\infty }=A_{\infty }$, and $I_{\infty }=A_{\infty }$ split the domain into six regions which characterize the innovation (see main text).

Figure 7

Final simulated average proportions of adopters [red (gray) $\square$], ignorants [green (dark gray) $\square$], and Luddites [blue (black) $\square$] for varying values of $\gamma$, averaged over 100 simulations. Theoretical predictions using ignorant domain length (see Appendix pp1 for details) are overlaid (solid line). Parameters are $N=1000,r=0.5$. Initially ignorants and susceptibles are randomly distributed, with densities $I_0=0.8$ and $S_0=0.2$. The three regimes discussed in the text are separated by dashed lines corresponding to regions where $(2/N)I_0\ll \gamma$ and $(2/N)I_0\gg \gamma$. Typical realizations of the model for $N=100$ in each of the three regimes are given (bottom). On the vertical axis the iteration corresponds to a single step of the Gillespie algorithm, with one reaction taking place per iteration.

Figure 8

Time dependence of the densities in each state for a one-dimensional lattice of size $N={10}^5$ averaged over 100 realizations. The corresponding mean-field predictions from Eq. (8) with $k=2$ (solid line) deviate dramatically from the simulation samples $(\square )$. The parameters are $\gamma =0.005,r=0.9$, and $I_0=0.8$.