  While actors in a population can interact with anyone else freely, social
relations significantly influence our inclination towards particular
individuals. The consequence of such interactions, however, may also form the
intensity of our relations established earlier. These dynamical processes are
captured via a coevolutionary model staged in multiplex networks with two
distinct layers. In a so-called relationship layer the weights of edges among
players may change in time as a consequence of games played in the alternative
interaction layer. As an reasonable assumption, bilateral cooperation confirms
while mutual defection weakens these weight factors. Importantly, the fitness
of a player, which basically determines the success of a strategy imitation,
depends not only on the payoff collected from interactions, but also on the
individual relationship index calculated from the mentioned weight factors of
related edges. Within the framework of weak prisoner's dilemma situation we
explore the potential outcomes of the mentioned coevolutionary process where we
assume different topologies for relationship layer. We find that higher average
degree of the relationship graph is more beneficial to maintain cooperation in
regular graphs, but the randomness of links could be a decisive factor in harsh
situations. Surprisingly, a stronger coupling between relationship index and
fitness discourage the evolution of cooperation by weakening the direct
consequence of a strategy change. To complete our study we also monitor how the
distribution of relationship index vary and detect a strong relation between
its polarization and the general cooperation level.
