Phase transitions in Paradigm models

In this letter we propose two general models for paradigm shift, deterministic propagation model (DM) and stochastic propagation model (SM). By defining the order parameter $m$ based on the diversity of ideas, $\Delta$, we study when and how the transition occurs as a cost $C$ in DM or an innovation probability $\alpha$ in SM increases. In addition, we also investigate how the propagation processes affect on the transition nature. From the analytical calculations and numerical simulations $m$ is shown to satisfy the scaling relation $m=1-f(C/N)$ for DM with the number of agents $N$. In contrast, $m$ in SM scales as $m=1-f(\alpha^a N)$.Comment: 5 pages, 3 figure

Analytic and simulation results of SM.

<p>(<b>A</b>) Plot of against of SM on the complete graph. The data both from the exact enumerations and simulations are shown. The curve represents the analytic result . (<b>B</b>-<b>D</b>) Scaling plots of the simulation data for of SM against on the scale-free network (<b>B</b>), on the random network (<b>C</b>) and on the square lattice (<b>D</b>). Inset of <b>D</b> shows the plots of for various against .</p

Analytic and simulation results of DM.

<p>Scaling plots of against of DM (<b>A</b>) on the complete graph with , (<b>B</b>) on a scale-free network (<b>C</b>) on a random network and (<b>D</b>) on a square lattice. Curves in the figures show the analytic results <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0070928#pone.0070928.e160" target="_blank">Equation (6</a>) and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0070928#pone.0070928.e220" target="_blank">Equation (9</a>). All the simulation data in <b>B</b>, <b>C</b> and <b>D</b> are obtained by use of . Inset of <b>D</b> shows the plots of for various against .</p

Scaling plot of and a snapshot in DM.

<p>(<b>A</b>) Scaling plot of against of DM on a square lattice with and . Inset: plot of against . (<b>B</b>) A snapshot of a steady state configuration of DM on the square lattice with the size . Black dots denote agents with a dominant idea . White dots denotes those with ideas different from .</p

Schematic diagram for the evolution of configurations in SM on CG.

<p><b>A</b> is a configuration with and . The next propagation process (a), (or the -th propagation) at , changes <b>A</b> into <b>B</b> with all and . Successive innovation processes (b) change <b>B</b> into <b>C</b> with with and . The propagation process (c), which cannot be executed by the probability , leaves <b>C</b> as it is. The propagation process (d) at initiated from an agent with drives <b>C</b> into <b>D</b> with all and . An innovation process (e) drives <b>D</b> into <b>E</b> with ( and ).</p