Influence of long-range interactions on strategy selection in crowd

An order--disorder phase transition is observed for Ising-like systems even for arbitrarily chosen probabilities of spins flips [K. Malarz et al, Int. J. Mod. Phys. C 22, 719 (2011)]. For such athermal dynamics one must define $(z+1)$ spin flips probabilities $w(n)$, where $z$ is a number of the nearest-neighbours for given regular lattice and $n=0,\cdots,z$ indicates the number of nearest spins with the same value as the considered spin. Recently, such dynamics has been successfully applied for the simulation of a cooperative and competitive strategy selection by pedestrians in crowd [P. Gawro\'nski et al, Acta Phys. Pol. A 123, 522 (2013)]. For the triangular lattice ($z=6$) and flips probabilities dependence on a single control parameter $x$ chosen as $w(0)=1$, $w(1)=3x$, $w(2)=2x$, $w(3)=x$, $w(4)=x/2$, $w(5)=x/4$, $w(6)=x/6$ the ordered phase (where most of pedestrians adopt the same strategy) vanishes for $x>x_C\approx 0.429$. In order to introduce long-range interactions between pedestrians the bonds of triangular lattice are randomly rewired with the probability $p$. The amount of rewired bonds can be interpreted as the probability of communicating by mobile phones. The critical value of control parameter $x_C$ increases monotonically with the number of rewired links $M=pzN/2$ from $x_C(p=0)\approx 0.429$ to $x_C(p=1)\approx 0.81$.