  Recently, coded distributed computing (CDC), with advantages in intensive
computation and reduced latency, has attracted a lot of research interest for
edge computing, in particular, IoT applications, including IoT data
pre-processing and data analytics. Nevertheless, it can be challenging for edge
infrastructure providers (EIPs) with limited edge resources to support IoT
applications performed in a CDC approach in edge networks, given the additional
computational resources required by CDC. In this paper, we propose coded edge
federation, in which different EIPs collaboratively provide edge resources for
CDC tasks. To study the Nash equilibrium, when no EIP has an incentive to
unilaterally alter its decision on edge resource allocation, we model the coded
edge federation based on evolutionary game theory. Since the replicator
dynamics of the classical evolutionary game are unable to model economic-aware
EIPs which memorize past decisions and utilities, we propose fractional
replicator dynamics with a power-law fading memory via Caputo fractional
derivatives. The proposed dynamics allow us to study a broad spectrum of EIP
dynamic behaviors, such as EIP sensitivity and aggressiveness in strategy
adaptation, which classical replicator dynamics cannot capture. Theoretical
analysis and extensive numerical results justify the existence, uniqueness, and
stability of the equilibrium in the fractional evolutionary game. The influence
of the content and the length of the memory on the rate of convergence is also
investigated.
