  Configurational arrangement of network architecture and interaction character
of individuals are two most influential factors on the mechanisms underlying
the evolutionary outcome of cooperation, which is explained by the
well-established framework of evolutionary game theory. In the current study,
not only qualitatively but also quantitatively, we measure Moran-evolution of
cooperation to support an analytical agreement based on the consequences of the
replicator equation in a finite population. The validity of the measurement has
been double-checked in the well-mixed network by the Langevin stochastic
differential equation and the Gillespie-algorithmic version of Moran-evolution,
while in a structured network, the measurement of accuracy is verified by the
standard numerical simulation. Considering the Birth-Death and Death-Birth
updating rules through diffusion of individuals, the investigation is carried
out in the wide range of game environments those relate to the various social
dilemmas where we are able to draw a new rigorous mathematical track to tackle
the heterogeneity of complex networks. The set of modified criteria reveals the
exact fact about the emergence and maintenance of cooperation in the structured
population. We find that in general, nature promotes the environment of
coexistent traits.
