  In this paper, we address dynamic network selection problems of mobile users
in an Intelligent Reflecting Surface (IRS)-enabled wireless network. In
particular, the users dynamically select different Service Providers (SPs) and
network services over time. The network services are composed of IRS resources
and transmit power resources. To formulate the SP and network service
selection, we adopt an evolutionary game in which the users are able to adapt
their network selections depending on the utilities that they achieve. For
this, the replicator dynamics is used to model the service selection adaptation
of the users. To allow the users to take their past service experiences into
account their decisions, we further adopt an enhanced version of the
evolutionary game, namely fractional evolutionary game, to study the SP and
network service selection. The fractional evolutionary game incorporates the
memory effect that captures the users' memory on their decisions. We
theoretically prove that both the game approaches have a unique equilibrium.
Finally, we provide numerical results to demonstrate the effectiveness of our
proposed game approaches. In particular, we have reveal some important finding,
for instance, with the memory effect, the users can achieve the utility higher
than that without the memory effect
