# Dynamics in Coded Edge Computing for IoT: A Fractional Evolutionary Game
Approach

Yue Han, Dusit Niyato, , Cyril Leung, Chunyan Miao, Dong In Kim  Y. Han is
with Alibaba Group and the Alibaba-NTU Joint Research Institute (JRI), Nanyang
Technological University (NTU), Singapore. E-mail<EMAIL_ADDRESS>Niyato is with the School of Computer Science and Engineering (SCSE), NTU,
Singapore. E-mail<EMAIL_ADDRESS>Leung is with The University of
British Columbia (UBC) and the Joint NTU-UBC Research Centre of Excellence in
Active Living for the Elderly (LILY). E-mail<EMAIL_ADDRESS>Miao is
with SCSE, NTU, Singapore, the Alibaba-NTU JRI, and LILY, Singapore. E-mail:
<EMAIL_ADDRESS>In Kim is with the Department of Electrical and
Computer Engineering, Sungkyunkwan University, Suwon, South Korea. E-mail:
<EMAIL_ADDRESS>
###### Abstract

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.

###### Index Terms:

Edge computing, coded distributed computing, game theory, fractional calculus,
fractional replicator dynamics, long-term memory, edge federation.

##  I Introduction

Coded distributed computing (CDC) [1, 2, 3, 4, 5, 6, 7, 8, 9] has recently
received much attention in both machine learning and information theory
domains for its advantages in speeding up computations for large parallelable
tasks. In parallel computing, a task is divided into several sub-tasks, each
of which is assigned to a worker. Thus, the overall computation time is
determined by the slowest worker. If some workers take significantly longer to
complete their sub-tasks, the overall performance of parallel computing is
degraded. To combat such a straggler effect, CDC treats delayed workers as
erasures and leverages coding techniques to effectively create redundant
computational tasks. In this way, the results of a subset of workers whose
completion times are relatively shorter can be used to recover the original
results. The idea of CDC has been extensively studied when resources are
plentiful, e.g., in centralized data centers. Examples of CDC used to mitigate
straggler effects include gradient descent computation in large-scale machine
learning platforms [4, 5], matrix multiplication [1, 2, 3], and data sorting
[7]. It is also applied to address security and privacy issues [6] and to
improve communication bottlenecks [8]. More details are provided in [9].

Recently, there has been a growing interest in applying CDC to edge computing
scenarios, such as federated learning [10, 11] and task offloading for
latency-sensitive applications[12, 13]. This is due to the fact that prevalent
edge computing tasks such as data pre-processing, data analytics, object
detection, and $3$D rendering are parallelable, computation-intensive, and
latency-sensitive [12]. In addition, computation results transmitted over
noisy wireless networks are error-prone.

However, a direct implementation of CDC at the edge is challenging. First, the
objective of using CDC is to improve the Quality-of-Service (QoS) of latency-
critical applications, e.g., IoT applications, in which data are collected and
processed at the edge. Therefore, the placement of the computing resources
(e.g., CPU, memory, and storage disk) is critical, and a shorter distance to
the cloud user with lower transmission delays and bandwidth costs is
preferable.A cloud computing approach [14], in which computation is
transferred to centralized data centers, incurs high transmission delays.
Alternatively, geographically diversified edge clouds, or sometimes called
distributed clouds [15, 16, 17, 18], consisting of a large number of small
data centers, geographically spread at the edge and interconnected by medium
to high speed links [15], are closer to the cloud users and provide more
location diversity. The close proximity benefits real-time applications in
which latency is important (e.g., virtual reality, gaming, machine-to-machine
communication, industrial applications, interactive collaboration, smart grid
control) and laws require user data to be stored in a specific locality, such
as the same county, city, or office [19]. A challenge with this approach is
that the computation capacity of a single edge cloud is considerably smaller
compared to a centralized cloud. This will degrade the performance of CDC, as
it trades additional computational resources for a shorter completion time.
Meanwhile, multiple edge infrastructures providers (EIPs) may place edge
resources at the same location (Fig. 1), or relatively close to each other in
some regions [20]. Thus, an opportunity for federation among edge clouds
controlled by different EIPs is promising.

Figure 1: System model of the coded edge federation (CEF)

To address the aforementioned challenge, we propose a framework called “coded
edge federation” (CEF), in which multiple EIPs cooperate and share their edge
resources. When edge resources from one EIP are insufficient, nearby edge
clouds belonging to different EIPs can collaboratively deliver the required
CDC services. The benefits of the CEF are as follows. First, CEF’s close
proximity feature helps to avoid the long transmission delays caused by
offloading to far-away centralized clouds or distant clusters located in other
regions. Second, CEF can unify the heterogeneous edge resources from various
EIPs by enforcing EIPs to contribute standardized resources measured by a set
of workers, each corresponding to a standard set of resources, e.g., CPU,
memory, and storage disk. Third, to ensure the performance of CDC,
particularly when normal service requests are heavy at the edge, a CDC task is
given a higher priority and served ahead of normal services. Fourth, with more
artificial intelligence (AI) and IoT services deployed at the edge, CEF allows
an EIP to rapidly respond to the increasing QoS requirements of demanding
applications by enabling an EIP to supplement its limited edge capacity from
its federation partner EIPs, before its new edge clouds are in place.

Existing studies of resource management for CDC [21, 22, 23] focus on either
the resource trade-off between computation and communication or the incentive
mechanism for edge devices (e.g., smart phones and drones) to cooperate as a
coalition and deliver latency-sensitive IoT services. There have been few
studies on a federation among EIPs to provide CDC services in edge networks.
We aim to address this problem using the framework of a non-cooperative game
in which EIPs, each with a large number of geo-diversified edge clouds, behave
as selfish players. EIPs need to decide on the amount of resources (number of
workers) to contribute from their edge clouds to the CEF. Because priority is
given to CDC tasks by the CEF, the service quality for other regular edge
services offered by the same workers will be affected. Thus, the EIP needs to
balance the resources allocated by it and other players to the federation.
After many rounds of strategic interactions, a balanced and stable strategy
profile is achieved whereby no EIP has an incentive to unilaterally change its
decision. This equilibrium may be a desirable solution for the EIPs that
participate in the CEF.

We adopt an evolutionary game approach, a particular type of non-cooperative
game characterized by bounded rationality and a revision protocol [24], to
identify such an equilibrium point. In addition, we consider incorporating a
power-law fading memory [25] of past decisions into a well-known type of
revision protocol, namely replicator dynamics, via the left-sided Caputo
fractional derivatives [26]. The proposed new dynamics are thus referred to as
fractional replicator dynamics, and the associated evolutionary game is called
fractional evolutionary game. Fractional replicator dynamics can overcome the
myopic nature of classical replicator dynamics [25] and capture the
economically rational nature of EIPs who act as business owners and use past
memory to improve their decision-making in the market. The memory parameter
$\alpha$ in Caputo fractional derivatives can capture additional and important
aspects of EIP decision-making process, e.g., the content and the length of
the memory, which reflect an EIP’s sensitivity and aggressiveness in the
strategy adaptation.

We summarize the major contributions of the paper as follows:

  * •

We highlight the edge resource insufficiency problem in implementing CDC at
the edge and propose the coded edge federation (CEF) framework based on
evolutionary game to address the challenge.

  * •

We propose fractional replicator dynamics with a power-law fading memory via
fractional calculus to model strategy adaptation of economic-aware EIPs, which
overcome the myopic limitation of classical replicator dynamics used in an
evolutionary game approach.

  * •

We theoretically show and experimentally verify the validity of the proposed
dynamics, by establishing the existence, uniqueness, and stability of the
solution. In addition, the proposed dynamics demonstrate faster convergence
rates compared to the classical ones.

The rest of this paper is organized as follows. Section II reviews related
works. Section III presents the system model and the problem formulation as a
classical evolutionary game. Section IV proposes the fractional replicator
dynamics and formulates a fractional evolutionary game. Section V provides a
theoretical study of the game equilibrium in terms of its existence,
uniqueness, and stability. Section VI investigates the dynamic behavior of
EIPs experimentally. Finally, Section VII summarizes the main findings and
lists potential future work.

##  II Related Work

###  II-A Coded Distributed Computing (CDC)

CDC is a promising technique for addressing the straggler problem in
distributed computing [2, 1, 3]. The study in [1] proposed using erasure codes
(e.g., repetition and MDS code) to speed up distributed computing in
homogeneous settings. In [2], the study was extended to heterogeneous settings
where different workers receive different amounts of local data. A
hierarchical coded computation scheme is proposed in [27] to utilize the
partial results of stragglers. Privacy aspects of coded computing are
investigated in [28]. Other studies [4, 5] propose methods to improve gradient
computing for a large-scale machine learning platform. All these studies
assume ample computing resources are available, e.g., in a large data center.
They do not consider the limited computational resources typically available
in edge computing scenarios. Recent studies [22, 23] focus on applying CDC in
edge computing. However, they mainly focus on incentive mechanisms for
heterogeneous edge devices. In contrast, we study the dynamics of edge
resource federation among EIPs which provide coded edge computing for IoT
services. In particular, we adopt a fractional evolutionary game approach to
model the dynamic behavior of an EIP with memory.

###  II-B Edge Federation

For edge federation, studies on cross-cloud cooperation architectures, such as
Joint Cloud [29] and Hybrid Cloud [30], propose to horizontally integrate
public and private cloud resources allowing EIPs to elastically handle short-
term spikes, e.g., Black Friday for Amazon and Double $11$ for TaoBao. They
provide insight into the horizontal edge federation for coded distributed
computing. On the other hand, vertical integration in terms of content caching
and computation offloading are proposed in [31, 32], by which frequent content
duplication can be reduced [33] and the power and computation limitation of an
edge device can be mitigated. In [20], the integration of edge cloud from both
horizontal and vertical perspectives is studied. The key differences between
our work and existing studies on the edge (or cloud) federation are as
follows. First, we mainly consider a horizontal integration among independent
EIPs, as the vertical offloading to the cloud may incur potential transmission
delay in the back-haul network, which negates the benefits of using CDC to
reduce latency for IoT services. Second, instead of using a centralized
method, which is static and has a high communication overhead, we adopt a
dynamic and decentralized resource allocation strategy based on evolutionary
game theory. In particular, the rewards competition among multiple EIPs is
modeled by fractional replicator dynamics with awareness of past decisions.

###  II-C Power-law Fading Memory and Fractional Calculus

The concept of memory is widely used in mathematical modeling in various
fields, e.g., physical sciences [34, 35], economics [36, 37, 25], and control
[38]. In many systems, the dynamics depend not only on the information at the
present time but also the past. Fractional calculus has been widely used to
represent past memory [26, 36]. In [25], a Volterra operator and power-law
fading memory are used to aggregate past information via the left-sided Caputo
derivatives. The extension to memory-aware elasticity via fractional calculus
is investigated in [37]. The authors in [39] extends the economic growth model
under the effect of dynamic memory with a constant pace. In coded edge
federation, EIPs as market players are economically rational in the sense that
they can recall past information and use it in future decision making. It is
therefore necessary to model EIP memory effects in the strategy adjustment.
However, classical replicator dynamics are known to be myopic and memoryless
[25]. One of our main contributions is to study the strategies adaptation of
memory-aware EIPs in the coded edge federation.

##  III System Description

In this section, we first introduce the system model, notation, and
assumptions in Section III-A. Then, we formulate the evolution of the CEF as
an evolutionary game in Section III-B.

TABLE I: Notation used in the system model Notation |  Explanation
---|---
$\mathcal{I},i,I$ |  set of EIPs, an EIP, and the number of EIPs in the system model
$\mathcal{E}_{i},e_{i},E_{i}$ |  set of edge clouds, an edge clouds, and the total number of clouds for EIP $i$
$\mathcal{P},p,P$ |  set of types of coded tasks, a type of coded task, and the total number of types of tasks
$(n_{p},k_{p},D_{p},\lambda_{p})$ |  code configuration, task size, and task frequency for a type $p$ task
$\mathcal{L}_{i},l_{i},L_{i}$. |  set of pure strategies (possible contributions) for EIP $i$, a pure strategy, and the maximum contribution
$l,\bm{l},L$ |  total number of contribution to an CEF by all EIPs, a pure-strategy profile, and the dimension of the pure-strategy profile. $l=\sum_{i\in\mathcal{I}}l_{i}$; $\bm{l}=[l_{i}]_{i\in\mathcal{I}}$; $L=\sum_{i\in\mathcal{I}}L_{i}+I$
$N_{i,l_{i}}$ |  estimated number of edge clouds of EIP $i$ contributing $l_{i}$ workers to the coded edge federation
$\bm{x}_{i}=[x_{i,l_{i}}]_{l_{i}\in\mathcal{L}_{i}}$ |  mixed strategy for an edge cloud of EIP $i$
$\bm{x}=[\bm{x}_{i}]_{i\in\mathcal{I}}$ |  mixed strategy profile of all EIPs
$\Delta_{i},\Theta$ |  space of $\bm{x}_{i}$ and the space of $\bm{x}$. $\Delta_{i}\in\mathbb{R}^{L_{i}+1},\Theta=\times_{i\in\mathcal{I}}\Delta_{i}$
$\tilde{n}_{i},\tilde{k}_{i},i\in\mathcal{I}$ |  number of workers of EIP $i$ among the set of $n_{p}$ workers requested by a type $p$ task and among the set of $k_{p}$ workers used to recover the result $\sum_{i\in\mathcal{I}}\tilde{n}_{i}=n_{p}$;$\sum_{i\in\mathcal{I}}\tilde{k}_{i}=k_{p}$
$p(\tilde{n}_{i};n_{p},\bm{l},\bm{\tilde{n}})$ |  probability that $\tilde{n}_{i}$ workers of EIP $i$ are among the set of $n_{p}$ workers requested by a type $p$ task
$p(\tilde{k}_{i};k_{p},\bm{\tilde{n}},\bm{\tilde{k}})$ |  probability that $\tilde{k}_{i}$ workers of EIP $i$ are among the set of $k_{p}$ workers used to recover the original result
$R^{(0)}_{i},R^{(1)}_{i},R^{(2)}_{i}$ |  base reward, additional reward, and fixed reward for EIP $i$, respectively
$r_{0}$, $r_{1}$, $r_{2}$ |  unit reward for base reward, additional reward, and fixed reward, respectively
$C_{i}^{(1),p}$ |  energy cost for EIP $i$ for a type $p$ task
$C_{i,l_{i}}^{(2)}$ |  resource utilization cost for an edge cloud of EIP $i$ contributing $l_{i}$ workers to an CEF
$w_{i}$ |  resource utilization, i.e., the fraction of edge resources used by the coded edge federation for EIP $i$
$W_{i}$, $C_{i}$, $c_{i}$ |  maximum edge capacity, fixed cost to be calibrated for EIP $i$, and cost per CPU cycle for an edge cloud of EIP $i$
$\pi_{i}^{(p)}(l_{i};\bm{l})$ |  utility that an edge cloud of EIP $i$ contributing $l_{i}$ workers to an CEF can receive given the pure-strategy profile and a type $p$ task
$\pi_{i}(l_{i};\bm{l})$ |  utility that an edge cloud of EIP $i$ contributing $l_{i}$ workers can receive given the pure-strategy profile
$u_{l_{i},\bm{x}}(l_{i};\bm{l})$ |  expected utility that an edge cloud of EIP $i$ contributing $l_{i}$ workers can receive given the mixed strategy profile
$u_{i}(\bm{x})$ |  average utility that EIP $i$ receives with the mixed strategy profile $\bm{x}$ $\bm{x}$

###  III-A System Description, Notation, and Assumptions

As shown in Fig. 1, the system model includes multiple CDC task publishers
(TPs), a set of edge infrastructure providers (EIPs), and the coded edge
federation (CEF) consisting of edge clouds controlled by various EIPs. TPs,
EIPs, and the CEF communicate with a centralized controller.

####  III-A1 Task Publishers

Let $\mathcal{P}=\\{1,\ldots,p,\ldots,P\\}$ be a set of integers representing
$P$ types of tasks requested by the TPs. A type $p$ task is described by a
tuple $\\{(n_{p},k_{p},D_{p},\lambda_{p})\\}_{p\in\mathcal{P}}$, where the
code configuration, task size, and task arrival rate are denoted by
$(n_{p},k_{p})$, $D_{p}$, and $\lambda_{p}$, respectively. Various codes, such
as repetition code [1], MDS code [1], Lagrange code [6], and Polynomial
code[3], can be used for different CDC applications. With the code
configuration $(n_{p},k_{p})$ in a homogeneous setting, a type $p$ task is
divided into $k_{p}$ equal sub-tasks and encoded into $n_{p}$ sub-tasks. Here,
we consider a master-worker setting, in which a TP acting as a master sends a
CDC request with code configuration $(n_{p},k_{p})$ to its nearby CEF. Then, a
set of $n_{p}$ workers from the CEF compute $n_{p}$ sub-results in parallel.
After the computation, the sub-results are sent back to the TP. In CDC, the
first $k_{p}$ results received by the TP are sufficient to recover the
original result.

####  III-A2 EIPs and Edge Clouds

Let $\mathcal{I}=\\{1,\ldots,i,\ldots,I\\}$ denote a set of $I$ EIPs, which
provide a large amount geo-diversified edge clouds, based on which the CEF is
formed. To unify the heterogeneous resource provision from various EIPs for
executing CDC tasks, the CEF issues a standardized measure of resource
contribution to all EIPs by requesting them to contribute resources in the
form of a set of workers, each of which offers a standardized amount of
computation and storage capacity. Thus, the resources allocated to the CEF by
an EIP can be quantified by the number of workers. Let
$\mathcal{E}_{i}=\\{1,\ldots,e_{i},\ldots,E_{i}\\}$ be a set of $E_{i}$ edge
clouds controlled by EIP $i$, geographically distributed at the network edge
[40], e.g., base stations, access points, central offices, and university
campus, which can directly receive CDC task requests from TPs via wireless
links. Here, we study the region in which different EIPs co-place their edge
clouds, or more generally, the region in which the distances among the edge
clouds of various EIPs are much shorter than the distances to edge clouds of
the same EIP placed in other areas. For example, in Fig. 1, we can assume that
the distance between the shopping plaza and the campus is long, and thus EIP
$1$ edge clouds placed in the campus and the plaza are far away from each
other, whereas edge clouds of EIPs $1$ and $2$ placed in the same area, e.g.,
campus, are much closer to each other. Therefore, the CEF formed around the
campus can benefit the campus-located TPs who have latency-critical tasks.

####  III-A3 Strategies of EIPs

As introduced previously, EIPs independently decide how much resources
(measured by the number of workers) their edge clouds should contribute when
collaborating with neighboring edge clouds of other EIPs. Without loss of
generality, we assume that the allocation decision spaces for all edge clouds
of an EIP are the same111The setting can be easily extended if the edge clouds
have different decision spaces. For example, we may group the edge clouds with
the same decision space and indicate the group index in an additional
dimension. and we denote this common decision space by
$\mathcal{L}_{i}=\\{0,1,2,\ldots,l_{i},\ldots,L_{i}\\},i\in\mathcal{I}$, where
$L_{i}$ is the maximum number of workers that can be allocated to the CEF.
Here, $l_{i}$, i.e., a particular allocation chosen from the set
$\mathcal{L}_{i}$, is referred to as a pure-strategy, and the set
$\mathcal{L}_{i}$ is called the set of pure-strategies for EIP $i$. We assume
that the set $\mathcal{L}_{i}$ is fixed by the EIP in a given period, e.g.,
one week, but can be changed for different periods. Moreover, EIPs are allowed
to use probabilistic strategies, i.e., a probability distribution over
$\mathcal{L}_{i}$, with which each pure strategy is to be played with a
certain probability. Let
$\bm{x}_{i}=[x_{i,0},x_{i,1},\ldots,x_{i,l_{i}},\ldots,x_{i,L_{i}}]$ denote
the mixed strategy for EIP $i$. Therefore, the number of edge clouds that
adopt strategy $l_{i}$ can be estimated by $N_{i,l_{i}}=E_{i}x_{i,l_{i}}$, if
$E_{i}$ is large, which is common in the case of a large amount of geo-
diversified edge clouds [41]. To study the strategy profile, i.e., the
combined decisions made by all EIPs, we let
$\bm{l}=[l_{1},l_{2},\ldots,\l_{I}]$ denote the pure strategy profile and
$\bm{x}=[\bm{x}_{1},\ldots,\bm{x}_{I}]$ to denote the mixed strategy profile.
Since $\sum_{l_{i}\in\mathcal{L}_{i}}x_{i,l_{i}}=1$, we let unit simplex
$\Delta_{i}\subset\mathbb{R}^{L_{i}+1}$ denote the space of the mixed strategy
for EIP $i$ and have $\bm{x}_{i}\in\Delta_{i}$. Let
$\Theta=\times_{i\in\mathcal{I}}\Delta_{i}$ denote the Cartesian product of
$\Delta_{i}$. Then, $\Theta$ is a subspace in $\mathbb{R}^{L}$, where
$L=I+\sum_{i\in\mathcal{I}}L_{i}$. Therefore, $\bm{x}\in\Theta$. With EIPs
independently making their decisions, the probability of having a pure
strategy profile $\bm{l}=[l_{1},\ldots,l_{i},\ldots,l_{I}]$ is
$\prod_{i\in\mathcal{I}}x_{i,l_{i}}$.

####  III-A4 Composition of Workers in the CEF

We can quantify the capacity of an CEF, e.g., the one formulated around the
campus in Fig. 1, as $l=\sum_{i\in\mathcal{I}}l_{i}$, i.e., the sum of
allocation by all EIPs from their edge clouds to this CEF. Thus, with the
request of $n_{p}$-workers from type $p$ tasks sent to this CEF, we consider a
random assignment from the request to the $l$ homogeneous workers in the CEF
for simplicity. Let $\tilde{n}_{i},i\in\mathcal{I}$ denote the number of
workers contributed by EIP $i$ among the $n_{p}$ workers, subject to
$0\leq\tilde{n}_{i}\leq l_{i}$. Let
$\bm{\tilde{n}}=[\tilde{n}_{1},\tilde{n}_{2},\ldots,\tilde{n}_{I}]$ denote the
joint placement by all EIPs, which is subjected to
$\sum_{i\in\mathcal{I}}\tilde{n}_{i}=n_{p}$. Then, the probability that
$\tilde{n}_{i}$ workers of EIP $i$ in performing the type $p$ CDC task is

$p(\tilde{n}_{i};n_{p},\bm{l},\bm{\tilde{n}})=\frac{\binom{l_{1}}{\tilde{n}_{1}}\ldots\binom{l_{i-1}}{\tilde{n}_{i-1}}\binom{l_{i+1}}{\tilde{n}_{i+1}}\ldots\binom{l_{I}}{\tilde{n}_{I}}}{\binom{l}{n_{p}}}\\\
\prod_{i\in\mathcal{I}}\mathbb{I}\\{\tilde{n}_{i}\geq
0\\}\prod_{i\in\mathcal{I}}\mathbb{I}\\{l_{i}\geq\tilde{n}_{i}\\}\mathbb{I}\\{\sum_{i\in\mathcal{I}}\tilde{n}_{i}=n_{p}\\},$
(1)

where $\mathbb{I}\\{A\\}$ is the indicator function such that
$\mathbb{I}\\{A\\}=1$ if the event $A$ is true and $\mathbb{I}\\{A\\}=0$
otherwise. Note that Eq. 1 is based on that the pure-strategy profile $\bm{l}$
and the joint placement $\bm{\tilde{n}}$ are given. When $I=2$, Eq. 1 reduces
to a hyper-geometric distribution

$p(\tilde{n}_{i};n_{p},\bm{l},\bm{\tilde{n}})=\frac{\binom{l_{1}}{\tilde{n}_{1}}\binom{l-l_{1}}{n_{p}-\tilde{n}_{1}}}{\binom{l}{n_{p}}}\mathbb{I}\\{\tilde{n}_{1}\geq
0,n_{p}-\tilde{n}_{1}\geq 0\\}\\\
\mathbb{I}\\{l_{1}\geq\tilde{n}_{1},l-l_{1}\geq n_{p}-\tilde{n}_{1}\\}.$ (2)

As soon as TP receives the results from $k_{p}$ workers, its task is
completed. Due to the homogeneous setting, each of the $n_{p}$ workers is
equally likely to be among the first $k_{p}$ workers to return the results.
Let $\tilde{k}_{i},i\in\mathcal{I}$ denote the number of workers belonging to
EIP $i$ that are in the set of $k_{p}$ workers, subjected to
$0\leq\tilde{k}_{i}\leq\tilde{n}_{i}$, and
$\bm{\tilde{k}}=[\tilde{k}_{1},\ldots,\tilde{k}_{I}]$ denote the joint results
of all EIPs. Therefore, the probability that $\tilde{k}_{i}$ workers of EIP
$i$ are in the set of first $k_{p}$ workers is

$p(\tilde{k}_{i};k_{p},\bm{\tilde{n}},\bm{\tilde{k}})=\frac{\binom{\tilde{n}_{1}}{\tilde{k}_{1}}\ldots\binom{\tilde{n}_{i-1}}{\tilde{k}_{i-1}}\binom{\tilde{n}_{i+1}}{\tilde{k}_{i+1}}\ldots\binom{\tilde{n}_{I}}{\tilde{k}_{I}}}{\binom{n_{p}}{k_{p}}}\\\
\prod_{i\in\mathcal{I}}\mathbb{I}\\{\tilde{k}_{i}\geq
0\\}\prod_{i\in\mathcal{I}}\mathbb{I}\\{\tilde{n}_{i}\geq\tilde{k}_{i}\\}\mathbb{I}\\{\sum_{i\in\mathcal{I}}\tilde{k}_{i}=k_{p}\\}.$
(3)

When $I=2$, Eq. 3 reduces to a hypergeometric distribution

$p(\tilde{k}_{i};k_{p},\bm{\tilde{n}},\bm{\tilde{k}})=\frac{\binom{\tilde{n}_{1}}{\tilde{k}_{1}}\binom{n_{p}-\tilde{n}_{1}}{k_{p}-\tilde{k}_{1}}}{\binom{n_{p}}{k_{p}}}\mathbb{I}\\{\tilde{k}_{1}\geq
0,k_{p}-\tilde{k}_{1}\geq 0\\}\\\
\mathbb{I}\\{\tilde{n}_{1}\geq\tilde{k}_{1},n_{p}-\tilde{n}_{1}\geq
k_{p}-\tilde{k}_{1}\\}.$ (4)

####  III-A5 Utilities of EIPs

The utility or payoff of an EIP participating in the CEF is the difference
between the rewards it receives and the costs it incurs in delivering the CDC
services.

##### Rewards

Given a TP sends a type $p$ task to a nearby CEF, an EIP $i\in\mathcal{I}$
that participates in this CEF can receive one or more of the following
rewards.

  * •

fixed reward $R_{i}^{(2)}$, when an CEF is formed, based simply on EIP $i$’s
participation in the CEF,

  * •

base reward $R_{i}^{(0)}$, when the CEF satisfies the requirement of $n_{p}$
workers, based on the number $\tilde{n}_{i}$ of workers from EIP $i$ in
executing the task,

  * •

and additional reward $R_{i}^{(1)}$, when TP receives the $k_{p}$ sub-results,
based on the number $\tilde{k}_{i}$ of EIP $i$ workers among the first $k_{p}$
workers to return the results to the TP.

With the reward scheme described above, we have

$R^{(0)}_{i}=r_{0}n_{p}\tilde{n}_{i},\quad
R^{(1)}_{i}=r_{1}k_{p}\tilde{k}_{i}\text{, and}\quad R^{(2)}_{i}=r_{2},$ (5)

where $r_{0}$, $r_{1}$, and $r_{2}$ are the unit reward for the base,
additional, and fixed reward, respectively. Note that $R^{(0)}_{i}$ and
$R^{(1)}_{i}$ are proportional to $n_{p}$ and $k_{p}$, due to the higher
communication, storage, and computation costs incurred by an EIP when the
values of $n_{p}$ and $k_{p}$ are higher.

##### Costs

Given a TP sends type $p$ task to a nearby CEF and the edge cloud of EIP
$i,i\in\mathcal{I}$ that participates in this CEF contributes $l_{i}$ workers,
the associated costs for the EIP are energy cost $C_{i}^{(1),p}$ and edge
resource utilization cost $C_{i,l_{i}}^{(2)}$. First, energy cost to EIP $i$
when running a type $p$ task is proportional to the number $\tilde{n}_{i}$ of
workers running the task,

$C_{i}^{(1),p}=c_{i}\frac{D_{p}}{k_{p}}\tilde{n}_{i},$ (6)

where $c_{i}$ is the cost per CPU cycle for EIP $i$. Second, edge resource
utilization cost is to take into account the scarcity of the computing
resources and thus amortize the fixed cost for an EIP, denoted by
$C_{i},i\in\mathcal{I}$, to its different service provisions [42], namely the
CEF or the normal edge services in this paper. Let $W_{i},i\in\mathcal{I}$
denote the total edge capacity for EIP $i$, measured by the number of workers,
and $w_{i}$ denote the resource utilization, i.e., the fraction of EIP edge
resources allocated to the CEF. Therefore, we have

$w_{i}=\frac{E_{i}\sum_{l_{i}\in\mathcal{L}_{i}}l_{i}x_{i,l_{i}}}{W_{i}}$ (7)

Here, we have assumed that edge clouds of EIP $i$ can simultaneously provide
$L_{i}$ workers, i.e., $E_{i}L_{i}\leq W_{i}$. The resource utilization cost
function [42] for EIP $i$ is

$f(w_{i})=-C_{i}\rho_{i}\left(1-\frac{1}{1-w_{i}}\right),$ (8)

where $\rho_{i}$ is a parameter between $0$ and $1$, representing the ratio of
the fixed cost to be calibrated. The edge resource utilization cost for an
edge cloud contribute $l_{i}$ workers to an CEF is defined as follows:

$C_{i,l_{i}}^{(2)}=\frac{l_{i}x_{i,l_{i}}}{\sum_{j\in\mathcal{L}_{i}}{jx_{i,j}}}\frac{f(w_{i})}{E_{i}}$
(9)

##### Net Utility

We first consider the homogeneous case when the tasks assigned to the CEF are
all of the same type, e.g., $p$. Given an CEF formed among edge clouds of $I$
EIPs whose joint contribution of workers is $\bm{l}$, the expected utility
that EIP $i$ can receive is

$\pi_{i}^{(p)}(l_{i};\bm{l})=R_{i}^{(2)}+\mathbb{E}\left[R_{i}^{(0)}-C^{(1),p}_{i}+\mathbb{E}[R^{(1)}_{i}|\tilde{n}_{i}]\right]-C_{i,l_{i}}^{(2)}\\\
=r_{2}+\sum_{\tilde{n}_{i}\in\mathcal{L}_{i}}p(\tilde{n}_{i};n_{p},\bm{l},\bm{\tilde{n}})\left(r_{0}n_{p}\tilde{n}_{i}-c_{i}\frac{D_{p}}{k_{p}}\tilde{n}_{i}+\sum_{0\leq\tilde{k}_{i}\leq\tilde{n}_{i}}\right.\\\
\left.\sum_{\forall 0\leq\tilde{k}_{j}\leq\tilde{n}_{j},j\neq
i\in\mathcal{I}}p(\tilde{k}_{i};k_{p};\bm{\tilde{n}},\bm{\tilde{k}})r_{1}k_{p}\tilde{k}_{i}\right)-\frac{l_{i}x_{i,l_{i}}}{\sum_{j\in\mathcal{L}_{i}}{jx_{i,j}}}{f(w_{i})}.$
(10)

In the heterogeneous settings, when tasks assigned to the federation are of
multiple types, i.e.,
$\\{(n_{p},k_{p},D_{p},\lambda_{p})\\}_{p\in\mathcal{P}}$, the expected
utility in Eq. 10 is extended to

$\pi_{i}(l_{i};\bm{l})=\sum_{p\in\mathcal{P}}\frac{\lambda_{p}}{\sum_{q\in\mathcal{P}}\lambda_{q}}\pi_{i}^{(p)}(l_{i};\bm{l}),$
(11)

where $\frac{\lambda_{p}}{\sum_{q\in\mathcal{P}}\lambda_{q}}$ indicates the
probability that the type $p$ task is requested.

Recall that the mixed strategy profile $\bm{x}$ defines the probability of
having the pure-strategy profile $\bm{l}$. Let $\bm{l}_{-i}$ denote the pure
strategy played by EIPs in $\mathcal{I}$ other than $i$. Therefore, given EIP
$i$ edge cloud contribute $l_{i}$ workers to an CEF, EIP $i$ expected payoff
is

$u_{i,l_{i}}(\bm{x})=\sum_{\forall l_{j}\in\mathcal{L}_{j},j\neq
i\in\mathcal{I}}\pi_{i}(l_{i};\bm{l})\prod_{j\neq
i,j\in\mathcal{I}}x_{j,l_{j}},$ (12)

where ${\forall l_{j}\in\mathcal{L}_{j},j\neq i\in\mathcal{I}}$ enumerate all
the instances for $\bm{l}_{i}$ given fixed $l_{i}$. When $I=2$, Eq. 12 becomes

$u_{i,l_{i}}(\bm{x})=\sum_{l_{h}\in\mathcal{L}_{h}}\pi_{i}(l_{i};\bm{l})x_{h,l_{h}},$
(13)

where $h$ is the other EIP in the game. Finally, the average payoff that EIP
$i$ receives from an CEF is

$u_{i}(\mathbf{x})=\sum_{l_{i}\in\mathcal{L}_{i}}u_{i,l_{i}}(\bm{x})x_{i,l_{i}}.$
(14)

In summary, Section III-A presents the system model. In particular, we define
the mixed strategy profile $\bm{x}$ and the payoff of each strategy for EIP
$i$, i.e., $u_{i,j}(\mathbf{x}),j\in\mathcal{L}_{i},i\in\mathcal{I}$. Both
play key roles later in the formulation of the evolutionary game. Note that
when there is no confusion, we replace $l_{i}$ by $j$ for convenience, e.g.
$u_{i,j}(\mathbf{x}),j\in\mathcal{L}_{i}$ is the same as that in Eq. 12.
Moreover, Eq. 12 indicates that the payoff for EIP $i$ is based on $\bm{x}$
and thus depends on all EIPs’ decisions. The existence of an equilibrium point
where no EIP can benefit by unilaterally altering its decision is therefore of
interest. Next, we illustrate how to use game theory to identify such an
equilibrium point.

###  III-B Coded Edge Federation Formed as a Classical Evolutionary Game

Game theory (GT) is a well-studied framework for analyzing the decision making
process of various agents that have similar or conflicting objectives [24].
The most common solution for a game is the Nash Equilibrium (NE), an
equilibrium point at which no EIP can benefit by unilaterally changing its
decision. To identify the NE, we adopt an evolutionary game approach due to
its important property of bounded rationality. With the replicator dynamics,
the dynamics behavior of EIPs can be represented by a system of differential
equations, the solution of which is the equilibrium point in the game. In the
following, we present the formulation of the evolutionary game (Section
III-B1), an introduction of replicator dynamics (Section III-B2), and the
limitation of classical replicator dynamics (Section III-B3).

####  III-B1 Game Formulation

A game can be represented by a set of players, a set of strategies of each
player, and the utility of each strategy. In particular, they are the set of
EIPs $\mathcal{I}$, the set of strategies $\mathcal{L}_{i},\forall
i\in\mathcal{I}$, and the payoffs $u_{i,j}(\bm{x}),\forall
i\in\mathcal{I},j\in\mathcal{L}_{i}$, where $\bm{x}$ is the mixed strategy
profile, defined in the system model.

In the context of the evolutionary game, edge clouds of EIP $i$ can be
considered to be a population. Thus, there are $I$ populations and the size of
population $i$ is $E_{i}$. In addition, for an EIP $i$, its mixed strategy
$\bm{x}$ is referred to as the population states in evolutionary game [24]. A
strategy that leads to a higher payoff is more widely spread in the
population, reflected by a large population states, after several rounds of
strategic interactions among the EIPs. Therefore, if we randomly pick an edge
cloud of EIP $i$, it is of higher chance, reflected by the mixed strategy,
that the edge cloud adopts the successful strategy. In the following, we
interchangeably use the terms of population states and mixed strategy. Next,
we present the revision protocol in the evolutionary game.

####  III-B2 Revision Protocols and the Classical Replicator Dynamics

According to [24], the evolutionary game allows a strategy adjustment as
follows: at each time instant, each cloud in the CEF has a revision
opportunity to switch to some other strategy. Let $\rho^{(i)}_{j,k}$ denote
the switch rate of a edge cloud of EIP $i$ in the CEF from strategy $j$ to
$k$. Then the switch probability from $j$ to $k$ is proportional to
$\rho^{(i)}_{j,k}$. The number of devices of EIP $i$ contributing $j$ workers
can be modeled with the dynamics as follows:

$\dot{x}_{i,j}=\gamma\sum_{k\in\mathcal{L}_{i}}x_{i,k}\rho^{(i)}_{k,j}(\bm{x})-x_{i,j}\sum_{k\in\mathcal{L}_{i}}\rho^{(i)}_{j,k}(\bm{x}),\quad
i\in\mathcal{I},j\in\mathcal{L}_{i},$ (15)

which also corresponds to the mean dynamics [43]. In Eq. 15,
$\dot{x}_{i,j}=\frac{d}{dt}x_{i,j}(t)$ is the first order time derivative. The
first term in the right-hand side of Eq. 15 models the inflow of players
switched from other strategies to strategy $j$, while the second term models
the outflow of players from the strategy $j$ to the other strategies.
Therefore, given a revision protocol $\rho$, Eq. 15 models the dynamics of a
population involved in a strategic game.

Although there are many types of protocols, a commonly used protocol is the
proportional imitation protocol,
$\rho^{(i)}_{j,k}(\bm{x})=x_{i,k}[u_{i,k}(\bm{x})-u_{i,j}(\bm{x})]_{+}$, where
$[x]_{+}=\max(0,x)$. In this protocol, when two random players with strategies
$j$ and $k$ play against each other, a player with a lower payoff may learn
from the other player by imitating his strategy, while the player with higher
payoff does not change his strategy. The probability of imitation is
proportional to the difference between the payoffs. Due to the wide
application of this protocol [44, 45, 46, 47], we also adopt it in the
classical evolutionary game for the CEF. Together with Eq. 15, the
proportional imitation protocol leads to the so-called replicator dynamics

$\dot{x}_{i,j}=\gamma x_{i,j}\left[u_{i,j}(\bm{x})-u_{i}(\bm{x})\right],\quad
i\in\mathcal{I},j\in\mathcal{L}_{i},$ (16)

where $x_{i,j}(t)\,i\in\mathcal{I},j\in\mathcal{L}_{i}$ is the population
state, $u_{i,j}(\bm{x})$ is defined in Eq. 12, and $\gamma$ is adaptation
speed in the evolutionary process. With the replicator dynamics defined in Eq.
16, we can present the strategy adaptation process of an EIP as a system of
ordinary differential equations, the solution to which is the stable mixed
strategy profile of all EIPs in the CEF.

####  III-B3 Limitation of the Classical Replicator Dynamics

Although replicator dynamics are frequently used in engineering applications,
it is myopic or memoryless [25]. In other words, in the CEF, when an EIP
adjusts its mixed strategy at the current time, it forgets all the mixed
strategies it used in the past. However, this assumption is not realistic, as
the EIPs are economically rational market players who can use the information
from the past to assist in future decision-making. The reason for the myopic
feature is that integer-order derivatives are limit values, defined only in an
infinitesimal neighborhood of a given time. Thus, the dynamics only capture
the information in this small neighborhood and do not take into account the
historical information before the present time [24]. Therefore, the strategy
adaptation of an EIP modeled by the replicator dynamics is memoryless, which
restricts the application of the classical evolutionary game to the CEF in the
real life.

Next, we show how to overcome the memoryless limitation of classical
replicator dynamics using fractional calculus.

##  IV Fractional Evolutionary Game

To incorporate memory in the classical replicator dynamics, we use the left-
sided Caputo derivatives, a particular type of fractional derivatives [26,
38], which allows past information to be aggregated in a logical way. We
present the basics of fractional calculus in Section IV-A, the concept of
power-law fading memory in Section IV-B, and the formation of the fractional
evolutionary game in Section IV-C.

###  IV-A Basics of Fractional Calculus

We use the Caputo fractional derivative, since the Caputo fractional
derivative of a constant is zero, and the initial conditions of the systems of
Caputo fractional differential equations are integer-order derivatives [26].
These two features make the application of the Caputo derivative convenient,
compared to other types of the fractional derivatives. However, the definition
and the properties of the Caputo derivative still rely on the Riemann-
Liouville fractional integrals and derivatives. For the convenience of the
reader, we present the key concepts adapted from [26, 38]. Let
$\mathbb{N}=\\{1,2,3,\ldots\\}$ denote the set of positive integers,
$\mathbb{N}_{0}=\\{0\\}\cup\mathbb{N}$ denote the set of non-negative
integers, $[a,b]$ denote a closed interval of the real line $\mathbb{R}$,
$AC^{n}[a,b]$ denote the space of real-valued functions $f(x)$ which have
continuous derivatives up to order $n-1$ on $[a,b]$ such that $f^{(n-1)}(x)$
is absolutely continuous on $[a,b]$, and $[x]$ denote the largest integer that
is smaller than or equal to $x$, e.g., $[3.1]=3$.

######  Definition 1 (Riemann-Liouville Fractional Integrals).

The (left-sided) Riemann-Liouville (RL) fractional integrals $I^{\alpha}_{a+}$
of order $\alpha>0$ is defined as

$\left(I_{a+}^{\alpha}f\right)(x):=\frac{1}{\Gamma(\alpha)}\int_{a}^{x}\frac{f(t)dt}{(x-t)^{1-\alpha}}\quad(x>a;\alpha>0).$
(17)

In Eq. 17, $\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt\quad(\Re(z)>0)$ is the
Euler Gamma function.

Note that for $\alpha\in\mathbb{N}$, Eq. 17 reduces to $n$th integrals of the
form

$\left(I_{a+}^{n}f\right)(x)=\int_{a}^{x}dt_{1}\int_{a}^{t_{1}}dt_{2}\cdots\int_{a}^{t_{n-1}}f\left(t_{n}\right)dt_{n}\\\
=\frac{1}{(n-1)!}\int_{a}^{x}(x-t)^{n-1}f(t)dt\quad(n\in\mathbb{N}).$ (18)

######  Definition 2 (Riemann-Liouville Fractional Derivatives).

The Riemann-Liouville (RL) fractional derivatives $D_{a+}^{\alpha}$ of order
$\alpha>0$ is defined as

$\left(D_{a+}^{\alpha}y\right)(x):=\left(\frac{d}{dx}\right)^{n}\left(I_{a+}^{n-\alpha}y\right)(x)=\frac{1}{\Gamma(n-\alpha)}\\\
\left(\frac{d}{dx}\right)^{n}\int_{a}^{x}\frac{y(t)dt}{(x-t)^{\alpha-n+1}}\quad(n=[\alpha]+1;x>a),$
(19)

where $I_{a+}^{n-\alpha}$ is the RL fractional integrals of order $n-\alpha$
given in Eq. 17. Note that for $\alpha=n\in\mathbb{N}_{0}$, Eq. 19 reduces to
the normal $n$th order derivatives, i.e.,
$\left(D_{a+}^{n}y\right)(x)=y^{(n)}(x)\quad(n\in\mathbb{N})$.

######  Definition 3 (Caputo Fractional Derivatives).

Let $D_{a+}^{\alpha}[y(t)](x)\equiv\left(D_{a+}^{\alpha}y\right)(x)$ be the RL
fractional derivatives of order $\alpha\geq 0$. Then the left-sided Caputo
fractional derivative of order $\alpha$ is defined as

$\left({}^{C}D_{a+}^{\alpha}y\right)(x):=\left(D_{a+}^{\alpha}\left[y(t)-\sum_{k=0}^{n-1}\frac{y^{(k)}(a)}{k!}(t-a)^{k}\right]\right)(x),$
(20)

where $n=[\alpha]+1$ for $\alpha\notin\mathbb{N}_{0}$ and $n=\alpha$ for
$\alpha\in\mathbb{N}_{0}$. Note that for $\alpha=n\in\mathbb{N}_{0}$, the
Caputo derivatives coincide with the normal integer-order derivatives, i.e.,
$({}^{C}D_{a+}^{\alpha}y)(x)=y^{(n)}(x)$.

######  Theorem 1.

If $y(x)\in AC^{n}[a,b]$, then the left-sided Caputo fractional derivatives
${}^{C}D_{a+}^{\alpha}$ exist almost everywhere on $[a,b]$. In particular, for
$\alpha\notin\mathbb{N}_{0}$,

$\left({}^{C}D_{a+}^{\alpha}y\right)(x)=\left(I_{a+}^{n-\alpha}D^{n}y\right)(x)\\\
=\frac{1}{\Gamma(n-\alpha)}\int_{a}^{x}\frac{y^{(n)}(t)dt}{(x-t)^{\alpha-n+1}},$
(21)

where $D=d/dx$ and $n=[\alpha]+1$.

Note that Eq. 21 will be used in the theoretical analysis in Section V and Eq.
20 in the experiments, due to its efficiency in implementation. A proof of
Theorem 1 can be found in [26].

###  IV-B Power-law Fading Memory via Caputo Derivatives

The memory effect has been intensively studied in economic processes [36, 37,
25], which comprise two types of time-dependent variables: exogenous
(independent) variables $X(t)$ and endogenous (dependent) variables $Y(t)$. As
endogenous variables describe the reaction to the exogenous variables, the
most widely used dynamics to capture their relationship are the first-order
time derivatives $Y(t)=X^{\prime}(t)$. However, this expression only depends
on the present time $t$ and is memoryless [36].

To take into account the memory effect, the author in [25] proposes the
dynamic memory, represented by a general form of
$Y(t)=F^{t}_{0}(X^{\prime}(\tau)),\,\forall\tau\in[0,t]$. Here, $F^{t}_{0}$ is
an operator or a functional, which defines a certain method that identifies
the value of $Y$ at time $t$ given the past information of $X^{\prime}(\tau)$
for all $\tau\in[0,t]$. In particular, the author proposes to use the Volterra
operator on $X^{\prime}(\tau)$, i.e.,
$F^{t}_{0}(X^{\prime}(\tau)):=\int_{0}^{t}M(t-\tau)X^{\prime}(\tau)d\tau$. In
this way, the value of $Y(t)$ is an aggregated amount of the past information
$X^{\prime}(\tau),\tau\in[0,t]$ and depends not only the value of
$X^{\prime}(t)$ at the present time but also its previous states
$X^{\prime}(\tau),\tau\in[0,t)$. The kernel of the integral operator, i.e.,
the function $M(t-\tau)$, plays a key role. It is also called the memory
function, which represents the weight of the historical information in the
memory. It is worth noting that the weight $M(t-\tau)$ depends on the time
distance between $\tau$ and $t$, and therefore it provides different weights
to the different historical information in the memory. Besides
$X^{\prime}(t)$, the Volterra operator can be applied to the case of any
integer-order derivatives $\frac{d^{n}}{d\tau^{n}}X(\tau)$ of $X(\tau)$ with
respect to time $\tau$, i.e.,
$Y(t)=F^{t}_{0}(X^{(n)}(\tau))=\int_{0}^{t}M(t-\tau)X^{(n)}(\tau)d\tau\quad(n\in\mathbb{N}_{0}).$

To ensure that the dynamic memory is fading, i.e., satisfying
$M(t-\tau)\rightarrow 0$ when $t\rightarrow+\infty$ with fixed $\tau$, the
author in [25] specifies the memory with power-law fading, a concept that was
first proposed by Volterra in [48] and later played a key role in modern
physics [34, 35]. In addition, the power-law fading memory has been
demonstrated in experiments intensively in [49], e.g., power-law human
learning [50] and power-law forgetting [51]. The memory function with the
power-law fading of order $\alpha$ is defined as [25]

$M(t-\tau)=M_{n-\alpha}(t-\tau)=\frac{1}{\Gamma(n-\alpha)}\frac{B}{(t-\tau)^{\alpha-n+1}},$
(22)

where $n=[\alpha]+1$. With Eq. 22, the economic process with power-law fading
memory function is $Y(t)=\left({}^{C}D_{0+}^{\alpha}X\right)(t)$, where
${}^{C}D_{0+}$ is the left-sided Caputo fractional derivative of the order
$\alpha\geq 0$ defined in Eq. 21. Note that the Caputo fractional derivative
of the order $\alpha=1$ coincides with the first order derivative (see Eq.
20). Thus, the dynamics of power-law fading memory function generalize the
standard dynamics of first-order derivatives to the fractional-order
derivatives. Next, we formulate the evolutionary game based on the replicator
dynamics that incorporate the power-law fading memory.

###  IV-C Fractional Replicator Dynamics and the Fractional Evolutionary Game

To reflect the economic-aware nature of the EIPs, we incorporate the power-law
fading memory in Eq. 22 to the replicator dynamics Eq. 16 given by

$(\prescript{C}{}{{D}_{0+}^{\alpha}}x_{i,j})(t)=\gamma
x_{i,j}(t)\left[u_{i,j}(\mathbf{x}(t))-u_{i}(\mathbf{x}(t))\right],\\\ \quad
i\in\mathcal{I},j\in\mathcal{L}_{i},0<\alpha<2.$ (23)

The new dynamics is referred to as fractional replicator dynamics, which
depends not only on the population states at the present time $t$ but also
their previous states in the evolutionary process. We refer to the
evolutionary game with the fractional replicator dynamics in Eq. 23 as
fractional evolutionary game. In particular, when $\alpha=1$, the fractional
evolutionary game is equivalent to the classical evolutionary game; when
$0<\alpha<1$, fractional replicator dynamics in Eq. 23 capture the change of
the strategies $\bm{x}$ in the evolutionary process, and is sometimes referred
to as a subdiffusion process; when $1<\alpha<2$, fractional replicator
dynamics in Eq. 23 capture the speed of change of the strategies $\bm{x}$ in
the evolutionary process. This is sometimes referred to as a superdiffusion
process. Next, we theoretically examine the validity of the fractional
replicator dynamics.

##  V Equilibrium Analysis

Some properties of fractional replicator dynamics are examined in this
section. We present the system of fractional ordinary equations in Section
V-A, the existence and uniqueness of the solution in Section V-B, and the
evolutionary stability in Section V-C.

###  V-A The System of Fractional Differential Equations

Let mapping $\phi=[\phi_{i,j}]_{i\in\mathcal{I},j\in\mathcal{L}_{i}}$ with
$\phi_{i,j}(x(t))=\gamma
x_{i,j}\left[u_{i,j}(t)(\mathbf{x}(t))-u_{i}(\mathbf{x}(t))\right]\,(t\in J)$
denote the right-hand side of the fractional replicator dynamics in Eq. 23 and
$J=[0,T]$ denote the time domain in the evolutionary process: Thus, Eq. 23 can
be represented by a system of fractional differential equations with initial
values, also referred to as an initial value problem (IVP), defined by

$\displaystyle{}^{C}D^{\alpha}_{0+}x_{i,j}(t)=\phi_{i,j}\left(\bm{x}(t)\right),$
(24) $\displaystyle
x_{i,j}^{(k)}(0)=b_{i,j,k}\quad(i\in\mathcal{I},j\in\mathcal{L}_{i},k=0,1\ldots,[\alpha]+1),$
(25)

where $b_{i,j,k}$ is the initial value of the corresponding $k$th integer-
order differentiation for the population states $x_{i,j}$. With Eq. 24 and Eq.
25, the fractional evolutionary game becomes a system of fractional
differential equations, the solution to which becomes the equilibrium points
in the fractional evolutionary game.

The next lemma establishes an equivalence relation between the initial value
problem and the integral problem. Further details can be found in [52, 53].

######  Lemma 1.

If the function $\phi_{i,j},i\in\mathcal{I},j\in\mathcal{L}_{i}$ is continuous
on its domain, then the initial value problem defined in Eqs. 24 and 25 is
equivalent to the nonlinear Volterra integral equation of the second kind,

$x_{i,j}(t)=\sum_{k=0}^{n-1}\frac{t^{k}}{k!}x_{i,j}^{(k)}(0)+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\phi_{i,j}(\bm{x}(s))ds\\\
\quad(i\in\mathcal{I},j\in\mathcal{L}_{i},n=[\alpha]+1).$ (26)

In other words, every solution of the Volterra equation Eq. 26 is also a
solution of our original initial value problem defined by Eqs. 24 and 25, and
vice versa. Thus, instead of solving initial value problem of Eqs. 24 and 25,
we therefore focus on Eq. 26, which is more amenable to theoretical analysis.
Note that Eq. 26 is weakly singular if $0<\alpha<1$ and regular for
$\alpha>1$. We focus on $0<\alpha<1$ in the following; the proof for
$1<\alpha<2$ is similar.

###  V-B Existence and Uniqueness of the Solution

To prove the existence and uniqueness of the solution to the fractional
evolutionary game, we construct a complete metric space with
$\left\lVert\cdot\right\rVert^{*}$ by Lemma 26 and then prove the key result
in Theorem 3.

The solution trajectory $x_{i,j}(t)\,(i\in\mathcal{I},j\in\mathcal{L}_{i})$ is
easily verified as a continuous mapping defined on the domain $J$. Let
$C(J,\mathbb{R}^{L})$ be the class of all continuous column vector functions
$\bm{x}(t)=[x_{i,j}(t)]_{i\in\mathcal{I},j\in\mathcal{L}_{i}}$ defined on $J$.
Let
$\|\bm{x}\|^{*}=\sum_{i\in\mathcal{I},j\in\mathcal{L}_{i}}\sup_{t}e^{-Nt}\left|x_{i,j}(t)\right|\,(N>0)$
be a new norm on $C(J,\mathbb{R}^{L})$ and the associated metric is defined by
$d(\bm{x}_{1},\bm{x}_{2})=\left\lVert\bm{x}_{1}-\bm{x}_{2}\right\rVert^{*}$.

######  Lemma 2.

The norm $\left\lVert\cdot\right\rVert^{*}$ is equivalent to the supremum norm
on the space $C(J,\mathbb{R}^{L})$.

The proof is given in Appendix A. Lemma 2 shows that the function space
$\left\langle C(J,\mathbb{R}^{L}),d\right\rangle$ is a metric complete space.
Note that the constructed equivalent norms and the respective metrics are the
main tools in the proof of existence and uniqueness for the fractional
evolutionary game. The method of equivalent norms is widely used in the theory
of differential equations [54, 55]. Next, we show that the mapping $\bm{x}$ is
a contraction on $\left\langle C(J,\mathbb{R}^{L}),d\right\rangle$ in Eq. 27
and Theorem 3.

######  Theorem 2.

The right-hand functions of the fractional replicator dynamics $\phi$ satisfy
the Lipschizian conditions, i.e., for any $\bm{x},\bm{y}\in
C(J,\mathbb{R}^{L})$,

$\left|\phi_{i,j}\left(\bm{x}(t)\right)-\phi_{i,j}\left(\bm{y}(t)\right)\right|\leq\sum_{i\in\mathcal{I},j\in\mathcal{L}_{i}}A_{i,j}\left|x_{i,j}(t)-y_{i,j}(t)\right|\\\
\quad A_{i,j}>0,i\in\mathcal{I},j\in\mathcal{L}_{i}$ (27)

The proof is given in Appendix B. With Eq. 27, we can obtain the key result
that there exists a unique equilibrium point in the fractional evolutionary
game as follows.

######  Theorem 3.

With Eq. 27, Eq. 26 has a unique solution $\bm{x}\in C(J,\mathbb{R}^{L})$.

The proof is given in Appendix C.

###  V-C Stability of the Solution

After verifying the existence and uniqueness of the equilibrium point in the
fractional evolutionary game, we investigate the stability of the solution.

######  Definition 4 (Uniformly stable (See [56])).

The solution to the fractional evolutionary game defined by Eqs. 24 and 25 is
said to be uniformly stable if for any $\epsilon>0$ there exists
$\delta(\epsilon)>0$ such that for every initial point
$\bm{y}_{o}\in\mathbb{R}^{L}$ satisfying
$\left\lVert\bm{x}_{o}-\bm{y}_{o}\right\rVert_{L}<\delta$, we have
$\left\lVert\bm{x}(t)-\bm{y}(t)\right\rVert^{*}<\epsilon$. Here, $\bm{y}(t)$
is the solution of initial value problem defined by Eq. 24 and the initial
value given by Eq. 28

$\bm{y}_{o}=[\tilde{b}_{i,j,1}]_{i\in\mathcal{I},j\in\mathcal{L}_{i}}.\vspace{-0.5cm}$
(28)

######  Theorem 4.

The solution of the fractional evolutionary game given by Eqs. 24 and 25 is
uniformly stable.

The proof is given in Appendix D.

Next, we verify the above theoretical findings as well as the dynamic
behaviors of the EIPs in the CEF through experiments.

TABLE II: Parameter Values for EIPs EIP $i$ | $E_{i}$ | $L_{i}$ | $C_{i}$ | $\rho_{i}$ | $c_{i}$ | $W_{i}$
---|---|---|---|---|---|---
1 | 100 | 4 | 1800 | 1 | $10^{-5}$ | 500
2 | 120 | 8 | 2800 | 1 | $10^{-5}$ | 1100
TABLE III: Parameter Values for the Task Publishers task | id | $n$ | $k$ | $r_{0}$ | $r_{1}$ | $r_{2}$ | $D$ | $\lambda$
---|---|---|---|---|---|---|---|---
homogeneous | 1 | 6 | 4 | 30 | 30 | 10 | $10^{6}$ | 1

##  VI Performance Evaluation

In this section, we use numerical simulations to investigate the dynamic
behavior of EIPs in the CEF. We consider the fractional replicator dynamics of
three types, including the classical replicator dynamics ($\alpha=1$),
superdiffusion processes ($\alpha=[1.2,1.4]$), and subdiffusion processes
($\alpha=[0.65,0.8]$). Unless otherwise stated, the simulated system consists
of $2$ EIPs, i.e., EIP $1$ and EIP $2$, with parameter values as listed in
Table II. The domain of the trajectory is $[0,T]$, where $T=1$, and the
maximum number of time steps is $10^{4}$. Table III summarizes the task
parameters and we omit the subscript $p$ due to the homogeneous tasks assumed.
In the following, we investigate the properties of the power-law fading memory
function (Section VI-A) and equilibrium existence and convergence rate
(Section VI-B2). Subsequently, we examine the stability of the equilibrium
point (Section VI-C) and the adaptation of the equilibrium when certain hyper-
parameter values are varied (Section VI-D).

###  VI-A The Memory Weight in Caputo Derivatives

(a) Memory weight for short-term memory

(b) Memory weight for long-term memory

Figure 2: Power-law fading memory function vs. the time difference $(t-s)$

Figure 3: The trajectory of the last strategy of EIP $1$ in different types of
the evolutionary process. Here, the code configuration is $n=6,k=4$.

Figure 4: Time to converge of two types for different evolutionary processes.
Here, the code configuration is $n=6,k=4$.

Figure 5: Change of the last strategy used by EIP $1$ as time evolves.

The memory weight, $(t-s)^{\alpha-n+1}$, utilized in the Caputo derivatives in
Eq. (21) is plotted against the time lag, $t-s$, in Fig. 2. Here, $t$
represents the current timestamp, and $s$ is some historical timestamp before
$t$. Thus, $t-s$ in fact represents how distant the historical information is.
The larger the value of $t-s$ is, the more distant the historical information
is. In addition, in Fig. 2, we draw the memory function of the classical
evolutionary game ($\alpha=1$) and a horizontal line with memory weight being
$40$ as two reference lines.

It can be observed from Fig. 2 that for both superdiffusion $(\alpha>1)$ and
subdiffusion processes $(\alpha<1)$, a certain weight is placed on the memory
about the historical information, since $(t-s)^{\alpha-n+1}$ is always
positive. Additionally, Fig. 2(a) shows that in the subdiffusion processes
($\alpha\in[0.65,0.8]$) the memory weight decreases rapidly with time lag.
This indicates that the memory is in fact short-term, since the very recent
memory dominates the aggregated memory. In contrast, Fig. 2(b) shows that in
the superdiffusion process ($\alpha\in[1.2,1.4]$), the memory is rather long.
For instance, the memory function of $\alpha=1.2$ in Fig. 2(b) is nearly flat,
and thus there is little discrimination between the distant and recent
historical information. Therefore, the memory, i.e., to aggregate the
historical information, is long. For $\alpha=1.4$, the weight discrimination
between the recent and past information is not significant either, compared to
those in Fig. 2(a). The range of the memory weight is around $[0,40]$ in Fig.
2(b) for $\alpha\in[1.2,1.4]$, compared to $[0,1500]$ for
$\alpha\in[0.65,0.8]$ in Fig. 2(a). Thus, with
$\alpha\in[0.65,0.8,1,1.2,1.4]$, fractional replicator dynamics are short-term
memory-aware in subdiffusion processes and long-term memory-dependent in the
superdiffusion process in the CEF game.

Note that the value of $(t-s)^{\alpha-n+1}$ is the same for both $\alpha=0.6$
and $\alpha=1.6$ since $n=[\alpha]+1$. However, based on the definition of the
Caputo derivatives in Eq. (21), the contents of the memory differ. In the
subdiffusion process, it is the change of past decisions, $y^{(1)}(t)$, to be
stored in the memory kernel, while the rate of change or the acceleration
speed, $y^{(2)}(t)$, is to be memorized in the superdiffusion process. Hence,
even though the memory weights are the same for the values of $\alpha=0.6$ and
$\alpha=1.6$, their evolutionary patterns can be very different.

In conclusion, $\alpha$ determines what information is stored in the memory
and the weight of recent past information relative to the distant one in the
memory.

###  VI-B Existence of Equilibrium and the Convergence Rate

We now analyze the equilibrium point in terms of its existence and uniqueness
for different values of $\alpha$. In particular, we consider the case where
homogeneous tasks assigned to the CEF. The task parameter values are shown in
Table III. First, we review and clarify the concept of equilibrium and
convergence of two types in Section VI-B1. Second, we illustrate the existence
and uniqueness of equilibrium by plots of trajectories in Section VI-B2.
Finally, we analyze the convergence pattern for different values of $\alpha$
in Section VI-B3.

####  VI-B1 Equilibrium and Convergence

To better describe how processes reach the equilibrium points, we introduce
the following two definitions:

  * •

Adjacency Type of Convergence: This concept is the same as the condition where
the equilibrium point is reached. It gives the timestamp after which the
difference between adjacent population states is within some given threshold.
We define the adjacency threshold as $0.0001$. Unless otherwise stated, we
simply use convergence to refer to this type in our paper.

  * •

Neighborhood Type of Convergence: This concept gives the timestamp after which
the population states fall into the equilibrium neighborhood, i.e., some
points within a certain threshold of the equilibrium point. We set the
neighborhood threshold at $0.01$.

The convergence describes an evolutionary process reaching the equilibrium in
an exact sense, while the neighborhood type can provide an additional aspect
to understand the pattern of convergence in a loose manner.

####  VI-B2 Existence of Equilibrium

To examine the existence of the equilibrium, we plot the evolutionary
trajectories of the last strategy used by EIP $1$ in Fig. 5 for all
evolutionary processes with $\alpha\in\\{0.65,0.8,1,1.2,1.4\\}$. The last
strategy is to contribute $4$ workers, which is the maximum number of workers
for EIP $1$, from its edge clouds to the CEF. As shown in Fig. 5, the
algorithm of all evolutionary processes stops before reaching the maximum
number of iterations, and the trajectory curves converge to a point around
$0.37$. This illustrates the existence and uniqueness of the equilibrium in
the fractional evolutionary game.

####  VI-B3 Impact of $\alpha$ on the Rate of Convergence

Besides the existence of equilibrium, Fig. 5 shows different convergence rates
for different evolutionary processes. For example, the process with
$\alpha=0.65$ seems to converge fastest to the equilibrium neighborhood, and
the process with $\alpha=1.4$ appears to converge slowest and fluctuates most.

To better understand how different processes converge, we plot the time to
converge for both the neighborhood type and adjacency type in Fig. 5. The
figure shows that the time to converge of the neighborhood type decreases as
$\alpha$ decreases. In other words, it takes longer for the superdiffusion
process to reach the equilibrium neighborhood and shorter for the subdiffusion
process compared to the classical evolutionary game. A similar pattern can be
observed for the adjacency type of convergence, except for the case with
$\alpha=0.65$. Particularly, the convergence for the subdiffusion process with
$\alpha=0.65$ is dramatically longer, though it reaches the equilibrium
neighborhood in the shortest time. Detailed explanations are provided in
Section VI-B3 and Section VI-B3.

##### Faster Convergence as Value of $\alpha$ Decreases

Recall that an equilibrium is defined as certain population states, if the
trajectories start with them are stationary. In other words, there is no
change in decisions if both the EIPs’ starting strategies are at equilibrium
points in the CEF game.

For the superdiffusion process, the replicators memorize the rate of change of
past information. Thus, to reach the equilibrium, the trajectory curves must
oscillate around $0$ with a decreasing range. In other words, the rate of
change, i.e., the acceleration velocity, fluctuates around $0$ with a
decreasing range. Otherwise, the acceleration velocity is either continuously
positive or continuously negative. As a result, the trajectory curve is either
convex upwards or concave downward. In neither case, the trajectory can be
stationary, indicated by a horizontal line. Due to this fluctuation nature of
the superdiffusion process, a process where the impact from the recent past
information is reduced, can help speed up the convergence. This effect happens
when a longer memory of the past exists, represented by the decreasing value
of $\alpha$.

The above abstract analysis is consistent with the experiment findings as
shown in Fig. 5 and Fig. 5, i.e., the superprocess with $\alpha=1.2$ converges
faster than the one with $\alpha=1.4$. This also has real-life implications
when strategy adjustments for the EIP are driven by the rate of change of the
information in the past ($\alpha>1$). Such EIPs are sensitive to the strategy
adaptation and therefore, are harder to stabilize their strategy in a shorter
time. However, if the EIP average past oscillation information over a longer
time span, indicated by a smaller $\alpha$ in $(1,2)$, the impact from the
recent oscillation is reduced, and the convergence velocity can be much
improved.

As for the subdiffusion process with $\alpha\in[0.65,0.8]$, we plot the change
in population state in Fig. 5. Unlike the superdiffusion process, the rate of
change for the subdiffusion process can be continuously positive, continuously
negative (e.g., $\alpha=0.8$ in Fig. 5), and fluctuated ($\alpha=0.8$ in Fig.
5). Additionally, in comparison to the superdiffusion process ($\alpha=1.4$),
the early changes in population states in the subdiffusion process are more
significant. This is due to the amplification of the recent past information
in the memory kernel as introduced in Section VI-A. The lower the value of
$\alpha$ is, the longer range of the amplified past information. Thus, with
the recent past information, the EIP quickly identify the right direction to
adjust its strategy, which accelerates the convergence of the evolutionary
process.

##### The Minimum Value of $\alpha$

Although smaller values of $\alpha$ can lead to faster convergence for both
superdiffusion and subdiffusion processes, a small $\alpha$ value may over-
amplify the recent past information, causing the EIP to over-adjust the
strategy in the early stage. Consequently, the EIP needs to reverse some of
its adjustments in the next step, which is represented as fluctuating
population states in the early stage. Given the longer memory effect from the
smaller value of $\alpha$, the impact of such fluctuation lasts longer. As a
result, more iterations are required to dilute the impact of the oscillation
to reach equilibrium. In conclusion, there is some value of $\alpha$, below
which the subdiffusion process start to show the a pattern of fluctuations.

This also explains the oscillation curve for $\alpha=0.65$ and the smooth
curve for $\alpha=0.8$ in Fig. 5 and Fig. 5. Specifically, for $\alpha=0.65$,
the evolution of the strategy for EIP $1$ demonstrates an oscillating pattern
with a small range. With more iterations, the range of the fluctuation
decreases, and the process eventually converges. The small fluctuations
increase the convergence time.

Our experiments demonstrate that $0.65$ is already near the minimum point at
which such oscillations begin to appear. We observe fluctuations with a
broader range for $\alpha=0.5$, and the trajectory does not converge within
$10^{4}$ iterations. A longer iterative process is needed, which is beyond the
settings of our experiments. Unless otherwise stated, we focus on the values
of $\alpha$ from $0.7$ to $1.4$.

To summarize Section VI-B3, i.e., the impact of $\alpha$ to the pattern of
convergence, we can conclude that a smaller value of $\alpha$ can accelerate
the evolutionary process of the EIP’s strategy adaptation. However, the value
of $\alpha$ should not be too small for the subdiffusion process. Otherwise,
the fluctuations around the equilibrium make the process take longer or even
fail to converge in an exact sense.

###  VI-C Evolutionary Stability of the Equilibrium

(a)

(b)

(c)

Figure 6: The direction of evolutionary processes with different values of
$\alpha$ and initial strategies.

Next, we examine the evolutionary stability of the equilibrium point in the
fractional evolutionary game for different values of $\alpha$. Fig. 6 shows
the direction fields of the fractional replicator dynamics in steps of $200$
iterations for different initial population states. We select
$\alpha=0.8,1,1.3$ to represent the subdiffusion process, classical
evolutionary game, and the superdiffusion process, respectively. In addition,
we vary the initial states of the last strategy for each EIP from $0.2$ to
$0.6$ and keep the other strategies unchanged.

As shown in Fig. 6, all equilibrium points obtained in the fractional
evolutionarily game, marked by the red circle, are evolutionarily stable. In
particular, strategies that are not in equilibrium can follow the arrows
indicated by the direction field and eventually reach equilibrium points.
Strategies in equilibrium are thus robust to mutation and deviation within a
certain threshold. This is consistent with the convergence and stability of
the equilibrium point derived in Theorem 4.

(a) population states of the last strategy

(b) average utility

Figure 7: Impact of edge capacity.

(c) population states of the last strategy

(d) average utility

Figure 8: Impact of the number of edge clouds.

(a) population states of the last strategy

(b) average utility

Figure 9: Impact of unit of reward.

(a) population states of the last strategy with $k=4$ and $n\in[4,14]$

(b) population states of the last strategy with $n=12$ and $k\in[4,12]$

Figure 10: Impact of code configuration.

###  VI-D Equilibrium Adaptation

Having verified the properties of equilibrium in terms of its existence,
uniqueness, and stability, we now examine the equilibrium adaptation when
certain system parameter values are changed. In particular, we are interested
in the edge capacity of each EIP (Section VI-D1), the number of edge clouds of
each EIP (Section VI-D2), the reward (Section VI-D3), and the code
configuration of the task publisher (Section VI-D4).

####  VI-D1 Impact of the Edge Capacity

Recall that edge capacity is defined as the EIP’s maximum total number of
workers that it can contribute to the CEF. Here, we vary the edge capacity of
EIP $1$ while keeping that of EIP $2$ fixed.

Figure 107(a) shows that when EIP $1$ edge capacity increases, the probability
of contributing $4$ workers from its edge clouds increases. The reason for
this is that as EIP $1$ has more resources, it is more likely to fulfill the
TP’s requirements and earn a reward. On the other hand, we can see that the
population states of the last strategy of EIP $2$ decline because given the
same task requirement, EIP $2$ can contribute fewer workers, but overall the
federation can still meet the requirements of the TP and thus earn a payoff.

Figure 107(b) shows the average utility received by each EIP as the edge
capacity of EIP $1$ increases. We observe that the average utility of each EIP
increases if the edge capacity of EIP $1$ increases. Interpreting the upward
trend for the average utility of EIP $1$ is straightforward. Given a larger
probability of adopting the last strategy, workers of EIP $1$ on average have
a greater presence in the federation and hence have a greater chance of being
selected by the TP. As a result, the average utility received by EIP $1$
increases. On the other hand, the increasing contribution of EIP $1$ also
benefits EIP $2$, as EIP $2$ can have a similar chance of meeting the TP’ s
requirement with fewer workers and correspondingly lower the cost for EIP $2$.

In summary, Fig. 10 reveals several benefits of the CEF. First, when one
party, EIP $1$, increases its edge resource contribution, the other party, EIP
$2$, can also enjoy a benefit, as reflected by a higher average utility.
Second, the reward allocation scheme is fair, given that EIP $1$ enjoys a
larger increase in the payoff relative to EIP $2$. Third, the scheme deters
EIP $2$ from being a free-rider, as its amount of free benefit decreases as
the share of EIP $1$’s workers in the CEF increases. The scheme thus
incentivizes EIP $2$ to increase its edge resources contribution to maintain
its market share in the CEF, thereby helping the CEF ecosystem to flourish.
All the benefits help the federation draw and sustain EIPs to participate,
incentivize EIPs to scale up edge resources investment, and ultimately drive
the CEF ecosystem.

####  VI-D2 Impact of the Number of Edge Clouds

This section examines the impact of the number of geo-diversified edge clouds
in each EIP, given the edge capacity is fixed. With more geo-diversified
clouds spread at the edge, the EIP coverage area becomes larger and receives
more service requests from the TPs. In particular, we fix the number of edge
clouds of EIP $2$ and vary the number of edge clouds of EIP $1$.

Figure 107(c) shows that as the number of edge clouds of EIP $1$ increases,
the percentage of edge clouds configured to contribute $4$ workers is
dramatically decreased. This is due to the fact that more services requests
sent to the EIPs incurs higher edge resources utilization costs, i.e., a
higher cost of allocating the scarce resources to the CEF. Therefore, from
Fig. 7(d), we see the average utility of joining the CEF decreases for EIP $1$
as the number of edge clouds increases. Meanwhile, Fig. 7(c) shows that while
EIP $1$ reduces its probability of contributing $4$ workers in its mixed
strategy (reflected by the population states) to the federation, EIP $2$
increases the probability of investing $8$ workers in the federation. The
reason is that investing more workers can help EIP $2$ improve the chance of
fulfilling the requirement of the TP and receiving the reward. Thus, we see a
trend of increasing population state for EIP $2$.

####  VI-D3 Impact of Rewards

We examine the impact of reward on the equilibrium points in the CEF by
analyzing the effect of $r_{1}$ on the payoff defined in Eq. (10). As shown in
Fig. 8(a), the probability that EIP $1$ contributes $4$ workers increases with
$r_{1}$. Similarly, EIP $2$ increases the probability of contributing $8$
workers to the CEF. This can be explained by the increased incentive provided
by the CEF to the EIPs.

Figure 8(b) shows that the average utility received by each EIP increases as
the reward issued by the CEF increases. In addition, the increase to EIP $2$
is larger since on average EIP $2$ with $L_{2}=8$ invests more workers than
EIP $1$ with $L_{1}=4$ to the CEF.

####  VI-D4 Impact of the Code Configuration

This section examines the impact of the code configuration on the equilibrium
points in the CEF. Figure 5 shows that all fractional evolutionary games with
different values of $\alpha$ can converge to the same equilibrium point. Thus,
we only plot the case when $\alpha=1$. Figure 9(a) shows that the probability
of choosing the last strategy, i.e., investing $4$ workers, for EIP $1$
increases as $n$ increases. A similar trend is observed for the EIP $2$. The
reason is that as the number of workers required by the TP increases, it is
more likely that EIP $1$ must invest its maximum number of workers to meet the
TP’s requirement and to earn the reward. Thus, we see an increasing trend of
population states as $n$ increases. Note that the maximum total contribution
by both EIPs is $12$ workers. Consequently, when the value of $n$, which is
part of the TP’s request, is greater than $12$, the federation begins to fail
to meet the TP’s requirement and receives no reward. Consequently, we see that
both EIPs begin to reduce their contributions.

On the other hand, Fig. 9(b) shows that the population states of the last
strategy in equilibrium increases as $k$ increases, because with more workers
contributed to the CEF, the EIPs are more likely to return results in the
first $k$ positions and thus receive higher additional rewards.

##  VII Conclusion

In this paper, we proposed coded edge federation (CEF) to address the problem
of insufficient edge resources when implementing CDC services in edge
networks. In particular, we adopted an evolutionary game approach to model the
dynamic behaviors of EIPs. To enhance the classical replicator dynamics, we
proposed fractional replicator dynamics that incorporates a power-law fading
memory via the left-sided Caputo derivatives. We theoretically showed and
experimentally verified the validity of the fractional replicator dynamics in
the sense that the equilibrium point of the game exists and is evolutionarily
stable. In addition, the experiments demonstrate that the fractional
replicator dynamics can model EIPs with additional features, e.g., the
sensitivity or aggressiveness of the strategy adaptation and can provide a
faster convergence rate than the classical one. The equilibrium adaptation for
different hyper-parameter values is also investigated. The incorporation of
pricing models for the EIP resources will be considered in future work.

## Acknowledgment

The authors would like to thank…

## References

  * [1] K. Lee, M. Lam, R. Pedarsani, D. Papailiopoulos, and K. Ramchandran, “Speeding Up Distributed Machine Learning Using Codes,” _IEEE Transactions on Information Theory_ , vol. 64, no. 3, pp. 1514–1529, Mar. 2018.
  * [2] A. Reisizadeh, S. Prakash, R. Pedarsani, and S. Avestimehr, “Coded computation over heterogeneous clusters,” in _IEEE International Symposium on Information Theory_ , Jun. 2017, pp. 2408–2412.
  * [3] Q. Yu, M. A. Maddah-Ali, and A. S. Avestimehr, “Polynomial codes: An optimal design for high-dimensional coded matrix multiplication,” in _Proceedings of the 31st International Conference on Neural Information Processing Systems_ , ser. NIPS’17. Curran Associates Inc., Dec. 2017, pp. 4406–4416.
  * [4] R. Tandon, Q. Lei, A. G. Dimakis, and N. Karampatziakis, “Gradient Coding: Avoiding Stragglers in Distributed Learning,” in _International Conference on Machine Learning_. PMLR, Jul. 2017, pp. 3368–3376.
  * [5] N. Raviv, I. Tamo, R. Tandon, and A. G. Dimakis, “Gradient Coding From Cyclic MDS Codes and Expander Graphs,” _IEEE Transactions on Information Theory_ , vol. 66, no. 12, pp. 7475–7489, Dec. 2020.
  * [6] Q. Yu, S. Li, N. Raviv, S. M. M. Kalan, M. Soltanolkotabi, and S. A. Avestimehr, “Lagrange Coded Computing: Optimal Design for Resiliency, Security, and Privacy,” in _The 22nd International Conference on Artificial Intelligence and Statistics_. PMLR, Apr. 2019, pp. 1215–1225.
  * [7] S. Li, S. Supittayapornpong, M. A. Maddah-Ali, and S. Avestimehr, “Coded TeraSort,” in _IEEE International Parallel and Distributed Processing Symposium: Workshops (IPDPSW)_. IEEE Computer Society, May 2017, pp. 389–398.
  * [8] S. Li and S. Avestimehr, “Coded Computing: Mitigating Fundamental Bottlenecks in Large-Scale Distributed Computing and Machine Learning,” _Now Foundations and Trends_ , 2020.
  * [9] J. S. Ng, W. Y. B. Lim, N. C. Luong, Z. Xiong, A. Asheralieva, D. Niyato, C. Leung, and C. Miao, “A Comprehensive Survey on Coded Distributed Computing: Fundamentals, Challenges, and Networking Applications,” _IEEE Communications Surveys & Tutorials_, vol. 23, no. 3, pp. 1800–1837, 2021.
  * [10] S. Prakash, S. Dhakal, M. Akdeniz, Y. Yona, S. Talwar, S. Avestimehr, and N. Himayat, “Coded Computing for Low-Latency Federated Learning over Wireless Edge Networks,” _IEEE Journal on Selected Areas in Communications_ , vol. 39, no. 1, pp. 233–250, Jan. 2021.
  * [11] S. Ha, J. Zhang, O. Simeone, and J. Kang, “Coded Federated Computing in Wireless Networks with Straggling Devices and Imperfect CSI,” in _IEEE International Symposium on Information Theory_ , Jul. 2019, pp. 2649–2653.
  * [12] K. T. Kim, C. Joe-Wong, and M. Chiang, “Coded Edge Computing,” in _IEEE INFOCOM_ , Jul. 2020, pp. 237–246.
  * [13] Y. Han, D. Niyato, C. Leung, and D. I. Kim, “Opportunistic Coded Distributed Computing: An Evolutionary Game Approach,” in _International Wireless Communications and Mobile Computing (IWCMC)_ , Jun. 2021, pp. 1430–1435.
  * [14] L. Wang, G. Von Laszewski, A. Younge, X. He, M. Kunze, J. Tao, and C. Fu, “Cloud computing: A perspective study,” _New generation computing_ , vol. 28, no. 2, pp. 137–146, 2010.
  * [15] Z. N. Rashid, S. R. Zebari, K. H. Sharif, and K. Jacksi, “Distributed cloud computing and distributed parallel computing: A review,” in _International Conference on Advanced Science and Engineering (ICOASE)_. IEEE, 2018, pp. 167–172.
  * [16] S. Agarwal, J. Dunagan, N. Jain, S. Saroiu, A. Wolman, and H. Bhogan, “Volley: Automated data placement for geo-distributed cloud services,” in _Proceedings of the 7th USENIX Conference on Networked Systems Design and Implementation_ , ser. NSDI’10. USA: USENIX Association, Apr. 2010, p. 2.
  * [17] M. Alicherry and T. V. Lakshman, “Network aware resource allocation in distributed clouds,” in _Proceedings IEEE INFOCOM_. IEEE, 2012, pp. 963–971.
  * [18] P. T. Endo, A. V. de Almeida Palhares, N. N. Pereira, G. E. Goncalves, D. Sadok, J. Kelner, B. Melander, and J.-E. Mangs, “Resource allocation for distributed cloud: Concepts and research challenges,” _IEEE Network_ , vol. 25, no. 4, pp. 42–46, 2011.
  * [19] Y. Coady, O. Hohlfeld, J. Kempf, R. McGeer, and S. Schmid, “Distributed cloud computing: Applications, status quo, and challenges,” _ACM SIGCOMM Computer Communication Review_ , vol. 45, no. 2, pp. 38–43, 2015.
  * [20] X. Cao, G. Tang, D. Guo, Y. Li, and W. Zhang, “Edge Federation: Towards an Integrated Service Provisioning Model,” _IEEE/ACM Transactions on Networking_ , vol. 28, no. 3, pp. 1116–1129, Jun. 2020.
  * [21] S. Li, M. A. Maddah-Ali, Q. Yu, and A. S. Avestimehr, “A Fundamental Tradeoff Between Computation and Communication in Distributed Computing,” _IEEE Transactions on Information Theory_ , vol. 64, no. 1, pp. 109–128, Jan. 2018.
  * [22] J. S. Ng, W. Y. B. Lim, Z. Xiong, D. Niyato, C. Leung, D. I. Kim, J. Zhang, and Q. Yang, “A Game-theoretic Approach Towards Collaborative Coded Computation Offloading,” _arXiv:2102.08667 [cs]_ , Feb. 2021.
  * [23] N. Kim, D. Kim, J. Lee, D. Niyato, and J. K. Choi, “Incentive-Based Coded Distributed Computing Management for Latency Reduction in IoT Services –A Game Theoretic Approach,” _IEEE Internet of Things Journal_ , pp. 8259–8278, 2020.
  * [24] J. W. Weibull, _Evolutionary Game Theory_. MIT press, 1997.
  * [25] V. V. Tarasova and V. E. Tarasov, “Concept of dynamic memory in economics,” _Communications in Nonlinear Science and Numerical Simulation_ , vol. 55, pp. 127–145, Feb. 2018\.
  * [26] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, _Theory and Applications of Fractional Differential Equations_. Elsevier Science, 2006, vol. 204.
  * [27] H. Park, K. Lee, J.-Y. Sohn, C. Suh, and J. Moon, “Hierarchical Coding for Distributed Computing,” in _IEEE International Symposium on Information Theory_ , Jun. 2018, pp. 1630–1634.
  * [28] R. Bitar, P. Parag, and S. El Rouayheb, “Minimizing Latency for Secure Coded Computing Using Secret Sharing via Staircase Codes,” _IEEE Transactions on Communications_ , vol. 68, no. 8, pp. 4609–4619, Aug. 2020.
  * [29] H. Wang, P. Shi, and Y. Zhang, “JointCloud: A Cross-Cloud Cooperation Architecture for Integrated Internet Service Customization,” in _IEEE 37th International Conference on Distributed Computing Systems (ICDCS)_ , Jun. 2017, pp. 1846–1855.
  * [30] K. Meyler, S. Buchanan, M. Scholman, J. G. Svendsen, and J. Rangama, _Microsoft Hybrid Cloud Unleashed with Azure Stack and Azure_. Sams Publishing, 2017.
  * [31] R. A. Dziyauddin, D. Niyato, N. C. Luong, M. A. M. Izhar, M. Hadhari, and S. Daud, “Computation Offloading and Content Caching Delivery in Vehicular Edge Computing: A Survey,” _arXiv:1912.07803 [cs]_ , Dec. 2019.
  * [32] N. Shan, Y. Li, and X. Cui, “A Multilevel Optimization Framework for Computation Offloading in Mobile Edge Computing,” _Mathematical Problems in Engineering_ , vol. 2020, Jun. 2020.
  * [33] G. Dán and N. Carlsson, “Dynamic content allocation for cloud-assisted service of periodic workloads,” in _IEEE INFOCOM 2014-IEEE Conference on Computer Communications_. IEEE, 2014, pp. 853–861.
  * [34] R. C. Koeller, “Applications of Fractional Calculus to the Theory of Viscoelasticity,” _Journal of Applied Mechanics_ , vol. 51, no. 2, pp. 299–307, Jun. 1984.
  * [35] B. D. Coleman and V. J. Mizel, “On the general theory of fading memory,” _Archive for Rational Mechanics and Analysis_ , vol. 29, no. 1, pp. 18–31, 1968.
  * [36] G. Teyssière and A. P. Kirman, _Long Memory in Economics_. Springer Science & Business Media, 2006.
  * [37] V. V. Tarasova and V. E. Tarasov, “Elasticity for economic processes with memory: Fractional differential calculus approach,” _Fractional Differential Calculus_ , vol. 6, no. 2, pp. 219–232, 2016.
  * [38] I. Podlubny, _Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications_. Elsevier, 1998.
  * [39] V. Tarasova and V. Tarasov, “Economic Growth Model with Constant Pace and Dynamic Memory,” _Problems of modern science and education_ , vol. 84, Jan. 2017.
  * [40] L. Tong, Y. Li, and W. Gao, “A hierarchical edge cloud architecture for mobile computing,” in _IEEE INFOCOM_ , Apr. 2016, pp. 1–9.
  * [41] F. Hao, M. Kodialam, T. V. Lakshman, and S. Mukherjee, “Online Allocation of Virtual Machines in a Distributed Cloud,” _IEEE/ACM Transactions on Networking_ , vol. 25, no. 1, pp. 238–249, Feb. 2017.
  * [42] J. A. Chandy, “An analysis of resource costs in a public computing grid,” in _IEEE International Symposium on Parallel & Distributed Processing_. Rome, Italy: IEEE, May 2009, pp. 1–8.
  * [43] W. H. Sandholm, _Population Games and Evolutionary Dynamics_. MIT press, 2010.
  * [44] D. Niyato and E. Hossain, “Dynamics of network selection in heterogeneous wireless networks: An evolutionary game approach,” _IEEE Transactions on Vehicular Technology_ , vol. 58, no. 4, 2008.
  * [45] C. Jiang, Y. Chen, and K. R. Liu, “Graphical evolutionary game for information diffusion over social networks,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 8, no. 4, pp. 524–536, 2014.
  * [46] C. Dong and W. Wen, “Joint optimization for task offloading in edge computing: An evolutionary game approach,” _Sensors_ , vol. 19, no. 3, p. 740, 2019\.
  * [47] Y. Cui, D. Zhang, T. Zhang, L. Chen, M. Piao, and H. Zhu, “Novel method of mobile edge computation offloading based on evolutionary game strategy for IoT devices,” _AEU-International Journal of Electronics and Communications_ , vol. 118, 2020.
  * [48] P. J. Daniell, “The Theory of Functionals, and of Integral and Integro-differential Equations. By V. Volterra. Translated by M. Long. Pp. x + 226. 25s. 1930. (Blackie),” _The Mathematical Gazette_ , vol. 16, no. 217, pp. 59–60, Feb. 1932.
  * [49] M. Edelman, “Evolution of Systems with Power-Law Memory: Do We Have to Die?(Dedicated to the Memory of Valentin Afraimovich),” in _Demography of Population Health, Aging and Health Expenditures_. Springer, 2020, pp. 65–85.
  * [50] J. R. Anderson, _Learning and Memory: An Integrated Approach_. John Wiley & Sons Inc, 2000.
  * [51] J. T. Wixted, “Analyzing the empirical course of forgetting.” _Journal of Experimental Psychology: Learning, Memory, and Cognition_ , vol. 16, no. 5, pp. 927–935, 1990.
  * [52] K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” _Journal of Mathematical Analysis and Applications_ , vol. 265, no. 2, pp. 229–248, 2002.
  * [53] V. Daftardar-Gejji and H. Jafari, “Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives,” _Journal of Mathematical Analysis and Applications_ , vol. 328, no. 2, pp. 1026–1033, 2007.
  * [54] D. Băleanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential equations,” _Computers & Mathematics with Applications_, vol. 59, no. 5, pp. 1835–1841, 2010.
  * [55] Z. F. El-Raheem, “Modification of the application of a contraction mapping method on a class of fractional differential equation,” _Applied mathematics and computation_ , vol. 137, no. 2-3, pp. 371–374, 2003.
  * [56] S. A. AbdEl-Salam and A. El-Sayed, “On the stability of some fractional-order non-autonomous systems,” _Electronic Journal of Qualitative Theory of Differential Equations_ , vol. 2007, no. 6, pp. 1–14, 2007\.
  * [57] P. Sahoo and T. Riedel, _Mean Value Theorems and Functional Equations_. World Scientific, 1998.

##  Appendix A Equivalent Norm

###### Proof.

Let us observe that for the norm $\left\lVert\cdot\right\rVert^{*}$, the
following inequalities are valid for any function $\bm{x}\in
C(J,\mathbb{R}^{L})$:

$e^{-NT}\left\lVert\bm{x}\right\rVert\leq\|\bm{x}\|^{*}\leq\sum_{i\in\mathcal{I},j\in\mathcal{L}_{i}}\underset{t}{\sup}\left|x_{i,j}(t)\right|=\left\lVert\bm{x}\right\rVert,$
(29)

where we denote as $\left\lVert\cdot\right\rVert$ the supremum norm on
$C(J,\mathbb{R}^{L})$. Here, $T=\max\\{t;t\in J\\}$. Thus, norms
$\left\lVert\cdot\right\rVert$ and $\left\lVert\cdot\right\rVert^{*}$ are
equivalent and so are the metrics. ∎

##  Appendix B Lipschizian Conditions

###### Proof.

To prove Eq. 27, we can show that Eq. 30 holds. If Eq. 30 holds, then Eq. 27
can be proved straightforwardly by the triangle inequality. Note that in this
proof, for simplicity, we re-index the component by letting
$[i,j]_{i\in\mathcal{I},j\in\mathcal{L}_{i}}\equiv[k]_{1\leq k\leq L}$.

$|\phi_{i,j}(x_{1,0}(t),\ldots,x_{h,k}(t),\ldots,x_{m,l_{m}+1}(t))-\\\
\phi_{i,j}(x_{1,0}(t),\ldots,y_{h,k}(t),\ldots,x_{m,l_{m}+1}(t))|\\\ \leqq
A_{i,j}|x_{h,k}(t)-y_{h,k}(t)|\quad(A_{i,j}>0;\\\ i,h\in\mathcal{I},0\leq
j\leq L_{i},0\leq k\leq L_{h}).$ (30)

To verify Eq. 30, we can prove the partial derivative
$\frac{\partial\phi_{i,j}}{\partial x_{h,k}},(i,h\in\mathcal{I},0\leq j\leq
L_{i},0\leq k\leq L_{h})$ exists and bounded on $\Theta$, i.e., $\exists M\geq
0,s.t.,|\frac{\partial\phi_{i,j}}{\partial x_{h,k}}|\leq M$. The reason is
that with bounded partial derivatives, Eq. 30 is true following the Mean Value
Theorem [57].

The partial derivatives can be expressed as follows:

$\displaystyle\left|\frac{\partial\phi_{i,j}}{\partial x_{h,k}}\right|$
$\displaystyle=\left|\frac{\partial x_{i,j}}{\partial
x_{h,k}}+x_{i,j}\left(\frac{\partial u_{i,j}(\mathbf{x})}{\partial
x_{h,k}}-\frac{\partial{u}_{i}(\mathbf{x})}{\partial x_{h,k}}\right)\right|$
$\displaystyle\leq\left|\frac{\partial x_{i,j}}{\partial
x_{h,k}}\right|+\left|x_{i,j}\right|\left|\frac{\partial
u_{i,j}(\mathbf{x})}{\partial
x_{h,k}}\right|+\left|x_{i,j}\right|^{2}\left|\frac{\partial{u}_{i,j}(\mathbf{x})}{\partial
x_{h,k}}\right|.$

It is easy to check that $\left|\frac{\partial x_{i,j}}{\partial
x_{h,k}}\right|$ and $\left|x_{i,j}\right|$ are bounded, and therefore, it is
only left to show that $\frac{\partial{u}_{i,j}(\mathbf{x})}{\partial
x_{h,k}}$ is bounded. We only prove for the case of $2$ EIPs, which can be
extended to more numbers of EIPs straightforwardly.

1) If $h=-i$, $\left|\frac{\partial\phi_{i,j}}{\partial x_{h,k}}\right|$ is
equivalent to verify that $\pi_{i}(j,q)$ is bounded, which is equivalent to
show that $C_{i,j}^{(2)}$ is bounded as it is the only term involved the
decision variables. Note that
$\left|C_{i,j}^{(2)}\right|=\left|\frac{x_{i,j}j}{\sum_{q\in
Q_{i}}{x_{i,q}q}}\frac{f(w_{i})}{E_{i}}\right|\leq\left|\frac{x_{i,j}j}{E_{i}}f(w_{i})\right|\leq\left|\frac{j}{E_{i}}f(w_{i})\right|$.
$f(w_{i})$ is bounded as it reaches its maximum values when all the edge
instances are configured to the maximum number of workers to the edge
federation.

2) if $h=i$, $\left|\frac{\partial\phi_{i,j}}{\partial x_{h,k}}\right|$ is
equivalent to show that $\left|\frac{\partial C_{i,j}^{(2)}}{\partial
x_{i,k}}\right|$ is bounded. Let $\zeta$ denote $\frac{f(w_{i})}{\sum_{q\in
Q_{i}}x_{i,q}q}$. Then, $\left|\frac{\partial C_{i,j}^{(2)}}{\partial
x_{i,k}}\right|\leq\left|\frac{\partial x_{i,j}}{\partial
x_{i,h}}\right|\left|\frac{j}{E_{i}}\right|\left|\zeta\right|+\left|\frac{1}{E_{i}}\right|\left|\frac{\partial\zeta}{\partial
x_{i,k}}\right|$. Here $\zeta$ is bounded, as
$\left|\zeta\right|\leq\left|f(w_{i})\right|$, and $f(w_{i})$ is proved
bounded in (1). Let $\tilde{q}=\max\\{q;q\in Q_{i}\\}$. Then
$\left|\frac{\partial*}{\partial x_{i,k}}\right|\leq\left|\frac{\partial
f(w_{i})}{\partial
x_{i,k}}\tilde{q}\right|+\left|k\right|\left|f(w_{i})\right|$. Here,
$\left|\frac{\partial f(w_{i})}{\partial
x_{i,k}}\right|=\left|\frac{1}{(1-w_{i})^{2}}\right|\left|\frac{E_{i}k}{W_{i}}\right|$,
and $\frac{1}{(1-w_{i})^{2}}$ is bounded and its maximum value reaches when
$w_{i}\in(0,1)$ reaches its maximized values, i.e., when all edge instances
configured to the maximum number of workers.

Thus, $\left|\frac{\partial\phi_{i,j}}{\partial x_{h,k}}\right|$ is proved
bounded, so Eq. 30 is bounded based on the Mean Value Theorem. ∎

##  Appendix C Uniqueness and Existence

###### Proof.

The Eq. 26 can be expressed as follows:

$\bm{x}(t)=\bm{x}_{o}+I_{0+}^{\alpha}\phi(\bm{x}),\quad 0<\alpha<1$ (31)

where $\bm{x}_{o}=[x_{i,j}(0)]_{i\in\mathcal{I},j\in\mathcal{L}_{i}}$ denotes
the vectored initial values.
$\bm{x}(t)=[x_{i,j}(t)]_{i\in\mathcal{I},j\in\mathcal{L}_{i}}$ and
$\phi=[\phi_{i,j}(\bm{x}(t))]_{i\in\mathcal{I},j\in\mathcal{L}_{i}},t\in J$
are defined previously.

Let $K=\underset{k}{\max}\\{A_{k}\\}$, where $A_{k}$ is given in the
definition of the Lipschizian condition, then we can have the below
inequalities adapted from Eq. 27,

$\displaystyle\left|\phi_{i,j}\left(\bm{x}(t)\right)-\phi_{i,j}\left(\bm{y}(t)\right)\right|\leq
K\left\lVert\bm{x}(t)-\bm{y}(t)\right\rVert_{L},$ (32)
$\displaystyle\left|\frac{\partial}{\partial x_{i,j}}\phi_{i,j}\right|\leq
K\quad(\forall t\in J,i\in\mathcal{I},j\in\mathcal{L}_{i}),$ (33)

where $|\cdot|$ is the absolute value and
$\left\lVert\bm{x}\right\rVert_{L}=\sum_{i\in\mathcal{I},j\in\mathcal{L}_{i}}|x_{i,j}|$
on $\mathbb{R}^{L}$.

Let the operator $T:C(J,\mathbb{R}^{L})\rightarrow C(J,\mathbb{R}^{L})$ be
defined by

$T\bm{x}(t)=\bm{x}_{o}+I^{\alpha}\phi(\bm{x}(t))$ (34)

with the norm $\left\lVert\bm{x}\right\rVert^{*}$. The fixed point under the
mapping $T$, i.e., $T\bm{x}=\bm{x}$, is the stationary point to the original
systems of fractional differential equations defined in Eqs. 24 and 25. Thus,
it is the equilibrium point in the fractional evolutionary game.

To show the existence and uniqueness of such a fixed point, the idea is to
apply the Banach Fixed Point Theorem (See Chapter 1 in [26] for more details)
over the space of $C(J,\mathbb{R}^{L})$ with norm
$\left\lVert\bm{x}\right\rVert^{*}$. Let $\bm{x}(t)$ and $\bm{y}(t)$ be any
two points in the vector space $C(J,\mathbb{R}^{L})$. We want to show that the
operator $T$ defined in Eq. 34 is a contractive mapping, i.e. $\left\lVert
T\bm{x}-T\bm{y}\right\rVert^{*}<\left\lVert\bm{x}-\bm{y}\right\rVert^{*}$.

First of all, we prove two inequalities as follows:

$\displaystyle A$
$\displaystyle=\sum_{i\in\mathcal{I},j\in\mathcal{L}_{i}}e^{-Ns}|\phi_{i,j}(\bm{x}(s))-\phi_{i,j}(\bm{y}(s))|$
$\displaystyle\leq\sum_{i\in\mathcal{I},j\in\mathcal{L}_{i}}e^{-Ns}K\left\lVert\bm{x}(s)-\bm{y}(s)\right\rVert_{L}\quad(\lx@cref{creftype~refnum}{cond1})$
$\displaystyle=e^{-Ns}LK\left\lVert\bm{x}(s)-\bm{y}(s)\right\rVert_{L}$
$\displaystyle=LK\sum_{i\in\mathcal{I},j\in\mathcal{L}_{i}}|x_{i}(s)-y_{i}(s)|e^{-Ns}$
$\displaystyle\leq
LK\sum_{i\in\mathcal{I},j\in\mathcal{L}_{i}}\underset{s}{\sup}|x_{i,j}(s)-y_{i,j}(s)|e^{-Ns}$
$\displaystyle=LK\left\lVert\bm{x}-\bm{y}\right\rVert^{*}.$ (35)

$\displaystyle B$ $\displaystyle=\int_{0}^{t}(t-s)^{\alpha-1}e^{-N(t-s)}ds$
$\displaystyle\stackrel{{\scriptstyle
z=(t-s)}}{{=}}\int_{0}^{t}z^{\alpha-1}e^{-Nz}dz$
$\displaystyle\stackrel{{\scriptstyle\tau=Nz}}{{=}}\int_{0}^{Nt}\frac{1}{N^{\alpha}}\tau^{\alpha-1}e^{-\tau}d\tau$
$\displaystyle<\frac{1}{N^{\alpha}}\underbrace{\int_{0}^{+\infty}\tau^{\alpha-1}e^{-\tau}d\tau}_{\Gamma{(\alpha)}}=\frac{\Gamma(\alpha)}{N^{\alpha}}.$
(36)

Then, we have

$\displaystyle\left\lVert e^{-Nt}(T\bm{x}(t)-T\bm{y}(t))\right\rVert_{L}$
$\displaystyle=\left\lVert
e^{-Nt}\\{I^{\alpha}[\phi(\bm{x}(s))-\phi(\bm{y}(s))](t)\\}\right\rVert_{L}$
$\displaystyle=\left\lVert\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}e^{-Nt}(\phi(\bm{x}(s))-\phi(\bm{y}(s)))ds\right\rVert_{L}$
$\displaystyle\leq\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}e^{-Nt}\left\lVert\phi(\bm{x}(s))-\phi(\bm{y}(s))\right\rVert_{L}ds$
$\displaystyle\leq\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}e^{-N(t-s)}\underbrace{\left\lVert
e^{-Ns}\phi(\bm{x}(s))-\phi(\bm{y}(s))\right\rVert_{L}}_{A}ds$
$\displaystyle\leq\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}e^{-N(t-s)}LK\left\lVert\bm{x}-\bm{y}\right\rVert^{*}ds$
$\displaystyle=LK\left\lVert\bm{x}-\bm{y}\right\rVert^{*}\frac{1}{\Gamma(\alpha)}\underbrace{\int_{0}^{t}(t-s)^{\alpha-1}e^{-N(t-s)}ds}_{B}$
$\displaystyle<\frac{LK}{N^{\alpha}}\left\lVert\bm{x}-\bm{y}\right\rVert^{*}\quad\text{(Choosing
$N$ s.t. $LK<N^{\alpha}$)}$ (37)
$\displaystyle<\left\lVert\bm{x}-\bm{y}\right\rVert^{*}.$ (38)

Then, we have

$\left\lVert
T\bm{x}-T\bm{y}\right\rVert^{*}=\left\lVert\underset{t}{\sup}\,e^{-Nt}(T\bm{x}(t)-T\bm{y}(t))\right\rVert_{L}<\left\lVert\bm{x}-\bm{y}\right\rVert^{*}.$
(39)

Here, the strict inequality in Eq. 39 holds since the supremum is the same as
the maximum value, since the mapping $e^{-Nt}(T\bm{x}(t)-T\bm{y}(t))$ is
continuous and defined on the closed interval $J$.

Thus, operator $T$ is a contractive mapping. With the Banach Fixed Point
Theorem, the operator $T$ has a unique fixed point. In other word, there
exists a unique equilibrium point in the fractional evolutionary game. ∎

##  Appendix D Stability

###### Proof.

Let $\bm{x}(t)$ be the solution to the initial value problem defined in Eqs.
24 and 25 and $\bm{y}(t)$ be the solutions to the initial value problem given
in Eqs. 24 and 28. Following the notation in Eqs. 25 and 28, we use
$\bm{x}_{o}\text{ and }\bm{y}_{o}$ to denote the corresponding initial points.
Note that the solution $\bm{x}(t)\text{ and }\bm{y}(t)$ are the fixed point
under its corresponding contractive mapping $T_{x}\text{ and }T_{y}$, where
$(T_{x}\bm{x})(t)={\bm{x}}_{o}+I^{\alpha}\phi(\bm{x}(t))$ and
$(T_{y}\bm{y})(t)=\bm{y}_{o}+I^{\alpha}\phi(\bm{y}(t))$. Then, we have

$\left\lVert e^{-Nt}(\bm{x}-\bm{y})\right\rVert_{L}=\left\lVert
e^{-Nt}\\{(T_{x}\bm{x})(t)-(T_{y}\bm{y})(t)\\}\right\rVert_{L}\\\ =\left\lVert
e^{-Nt}\\{\bm{x}_{o}-\bm{y}_{o}+I^{\alpha}[\phi(\bm{x}(s))-\phi(\bm{y}(s))](t)\\}\right\rVert_{L}\\\
\leq\underbrace{\left\lVert
e^{-Nt}(\bm{x}_{o}-\bm{y}_{o})\right\rVert_{L}}_{A}+\underbrace{\left\lVert
e^{-Nt}I^{\alpha}[\phi(\bm{x}(s))-\phi(\bm{y}(s))](t)\right\rVert_{L}}_{B}$
(40)

where
$A=e^{-Nt}\left\lVert\bm{x}_{o}-\bm{y}_{o}\right\rVert_{L}<\left\lVert\bm{x}_{o}-\bm{y}_{o}\right\rVert_{L},$
as $N>0,t>0\Rightarrow e^{-Nt}\in(0,1)$, and
$B=e^{-Nt\\{(T\bm{x})(t)-(T\bm{y})(t)\\}}_{L}<\frac{lK}{N^{\alpha}}\|\bm{x}-\bm{y}\|^{*},\forall
t\in J$ as Eq. 37 holds. Thus, we have

$\displaystyle\left\lVert
e^{-Nt}(\bm{x}-\bm{y})\right\rVert_{L}<\left\lVert\bm{x}_{o}-Y_{o}\right\rVert_{L}+\frac{lK}{N^{\alpha}}\|\bm{x}-\bm{y}\|^{*}\quad\forall
t\in J$
$\displaystyle\Rightarrow\left\lVert\underset{t}{\sup}e^{-Nt}(\bm{x}-\bm{y})\right\rVert_{L}<\left\lVert\bm{x}_{o}-\bm{y}_{o}\right\rVert_{L}+\frac{lK}{N^{\alpha}}\|\bm{x}-\bm{y}\|^{*}$
$\displaystyle\Rightarrow\left\lVert\bm{x}-\bm{y}\right\rVert^{*}<\left\lVert\bm{x}_{o}-\bm{y}_{o}\right\rVert_{L}+\frac{lK}{N^{\alpha}}\|\bm{x}-\bm{y}\|^{*}$
$\displaystyle\Rightarrow\left\lVert\bm{x}-\bm{y}\right\rVert^{*}<\frac{1}{1-\frac{LK}{N^{\alpha}}}\left\lVert\bm{x}_{o}-\bm{y}_{o}\right\rVert_{L}$
$\displaystyle\Rightarrow\left\lVert\bm{x}-\bm{y}\right\rVert^{*}<\left\lVert\bm{x}_{o}-\bm{y}_{o}\right\rVert_{L},$

if $N$ is chosen the same as that in the Theorem 3, which leads to
$\frac{1}{1-\frac{LK}{N^{\alpha}}}\in(0,1)$. Let $\delta=\epsilon$, then we
can deduce that the equilibrium of the fractional evolutionary game defined in
Eqs. 24 and 25 are uniformly stable based on Eq. 28. ∎