Cooperation in spatial Prisoner's Dilemma with two types of players for increasing number of neighbors

We study a spatial two-strategy (cooperation and defection) Prisoner's Dilemma game with two types ($A$ and $B$) of players located on the sites of a square lattice. The evolution of strategy distribution is governed by iterated strategy adoption from a randomly selected neighbor with a probability depending on the payoff difference and also on the type of the neighbor. The strategy adoption probability is reduced by a pre-factor ($w < 1$) from the players of type $B$. We consider the competition between two opposite effects when increasing the number of neighbors ($k=4$, 8, and 24). Within a range of the portion of influential players (type $A$) the inhomogeneous activity in strategy transfer yields a relevant increase (dependent on $w$) in the density of cooperators. The noise-dependence of this phenomenon is also discussed by evaluating phase diagrams.