Emergence and Decline of Scientific Paradigms

Scientific paradigms have a tendency to rise fast and decline slowly. This asymmetry reflects the difficulty in developing a truly original idea, compared to the ease at which a concept can be eroded by numerous modifications. Here we formulate a model for the emergence and spread of ideas which deals with this asymmetry by constraining the ability of agents to return to already abandoned concepts. The model exhibits a fairly regular pattern of global paradigm shifts, where older paradigms are eroded and subsequently replaced by new ones. The model sets the theme for a new class of pattern formation models, where local dynamics breaks the detailed balance in a way that prevents old states from defending themselves against new nucleating or invading states. The model allows for frozen events in terms of the coexistence of multiple metastable states.

Figure 1

Twelve consecutive snapshots of a $N=128\times 128$ system with $\alpha =25\times {10}^{-6}$. The time intervals between the pictures are not equidistant, as can be seen from the times given for each panel [times correspond to Fig. 2b]. Time is measured in units of full sweeps, i.e., one update for each agent.

Figure 2

Three time series of the sizes of the dominant states of the system, at $\alpha =0.4\times {10}^{-6}$, $\alpha =25\times {10}^{-6}$, and $\alpha =400\times {10}^{-6}$, respectively. Time is measured in units of sweeps (updates per agent). Notice that the length of each period does not change substantially with $\alpha$, and in fact becomes more regular with larger $\alpha$.

Figure 3

Distribution of waiting times between shifts of the dominant state for a $N=128\times 128$ system at different values of innovation rate $\alpha$. Inset: Average waiting time vs $\alpha N$ for different system sizes $N=162$ ($+$), $N=642$ ($\times$), and $N={256}^2$ (*). The diagonal line measures the time between individual innovations $t=1/\alpha N$. For small innovation rates, waiting time is close to this line, independently of system size $N$. For large innovation rates, waiting times become insensitive to $\alpha N$, while strongly increasing with $N$.

Figure 4

Distribution of the number of sites $s$ that a particular idea visits during its life span for different values of $\alpha$. Scaling $\sim 1/s^{2.5}$ for comparison (blue line). Note the gap between the main part of the distribution and the bins counting the ideas with system-wide sweeps (crosses). System size is $N=128\times 128$.