Fig. 4.

Efficiency of the majority rule in noiseless environments. (A) E_maj for k = 6, η = 0, and different system sizes as a function of p. When noise is not present, a system evolving according to the majority rule can perform the density classification only for large values of p. Note that the critical value p_c at which the system reaches the efficient regime increases with increasing system size. (B) A block of four locally connected units. Each unit has three connections to the other units in the block. Without noise, whenever all the units in the block attain the same state, they will not switch states even if all the rest of the system converges to the opposite state. (C) Onset of the transition to the efficient regime. The efficient regime is achieved when p is large enough so that there can be no blocks of locally connected units. From Eq. 5, one has p_c ≈ 1 - N ^{-[4/(k2 + 2k)]}. As the figure shows, when one subtracts the expected value of p_c, the onset of the transition of all curves collapse to a value close to zero.