Master’s Thesis

Title

Dynamic Network Slicing Control Method
for Beyond 5G Mobile Network
using Quality-Diversity Algorithms

Supervisor
Professor Masayuki Murata

AuthorAmato Otsuki

February 2nd, 2024

Department of Information Networking
Graduate School of Information Science and Technology
Osaka University

Master’s Thesis

Dynamic Network Slicing Control Method for Beyond 5G Mobile Network using
Quality-Diversity Algorithms
Amato Otsuki

Abstract
In hosting diverse applications with various service requirements in 5G network, network slicing technology is critical factor, which provides multiple isolated and dedicatedvirtual network. It is important for the satisfaction of the requests of clients and the profit
of providers of the substrate network to allocate the network slices requested by clientseﬀiciently into the substrate network. The optimization problem of substrate resource
allocation to network slices is closely related to the virtual network embedding (VNE)
problem known as NP-hard problem, thus it is nearly impossible to derive the optimal
solution within practical time in a large scale network. In addition, considering a practicaluse case, it is required to allocate multiple network slices simultaneously, and optimize
the allocation of all network slices while new requests arrive sequentially. Therefore,
the method which can find a feasible solution immediately in such dynamic environmentis essential for network slicing in Beyond 5G network. For the implementation of themechanism of deriving a new solution adaptively for dynamic environment utilizing theprevious solutions, genetic algorithm which is an optimization algorithm based on a modelof organism evolution can be applied. As genetic algorithm can deal with general combinatorial optimization problems, we modeled the problem of deciding the mapping between
the nodes or links of the substrate network and those of network slices as a combinatorial optimization problem, and realize dynamic network slice embedding applying genetic
algorithm to this problem and finding feasible solutions. However, it is known that conventional genetic algorithm tends to lose solution diversity, and Quality-Diversity (QD)
algorithms are proposed as methods overcoming this problem, which output solutions
with diverse characteristics. It seems to be desirable to maintain solution diversity for the1

immediate adaptation to dynamic environment, so we proposed a dynamic network slice
control method using QD algorithm. When multiple network slices are controlled integratedly, allocation of elements of all network slices must be considered simultaneously,
and the scale of problem space increase exponentially for the number of network slices.
Thus, for the scalability, it is necessary to assign a separate controller to each network sliceand control them separately. On the other hand, it is unable to achieve total optimization
only optimizing each slice separately because it may lead to the degradation of quality ofservice (QoS) of other slices in B5G network there are the diverse service requirements.
Hence, introducing unselfish evaluation values additionally as a multi-objective optimization, we attempt to realize the cooperative control considering other slices while controllingslices separately. We model temporal arrival of network slice requests as Poisson process
and evaluated the proposed methods through computer simulation. We assess the degreeof total optimization through the acceptance rate of network slice requests and compared
our methods with methods using conventional genetic algorithm and heuristic proposed
in previous work. As a result, it is shown the acceptance rate of our methods was higher
than other methods. Moreover, we reveal the effects of unselfish evaluation values throughsimulation under static environment.

2

Keywords
Beyond 5G network
Network slicing
SDN (Software defined network)
VNE (Virtual network embedding)
Genetic algorithm
Quality-Diversity algorithm
MAP-Elites

3

Contents
1 Introduction

7

2 Related work

11

2.1

Network Slice Embedding Problem . . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Quality-Diversity Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1

MAP-Elites Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Formulation of Dynamic Network Slice Embedding Problem

15

3.1

Substrate Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2

Network Slice Request Arrival Model . . . . . . . . . . . . . . . . . . . . . . 19

4 Dynamic Network Slice Control Method using QD algorithm

20

4.1

Encoding of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2

Definition of Fitness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3

4.2.1

Unselfish Behavior: Load Balancing . . . . . . . . . . . . . . . . . . 24

4.2.2

Unselfish Behavior: Giving Way . . . . . . . . . . . . . . . . . . . . 24

4.2.3

Definition of Fitness for Cooperative Control . . . . . . . . . . . . . 25

Design of Feature Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Evaluation

27

5.1

Comparison under the Dynamic Environment . . . . . . . . . . . . . . . . . 29

5.2

Comparison under the Static Environment . . . . . . . . . . . . . . . . . . . 33

6 Conclusion

35

Acknowledgments

36

References

37

4

List of Figures
1

Diagram of E2E network slicing in 5G network . . . . . . . . . . . . . . . .

2

Diagram of NSE problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3

Layer substrate network model . . . . . . . . . . . . . . . . . . . . . . . . . 18

4Addition of RAN layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5

Acceptance ratio through a single simulation in scenario 1 . . . . . . . . . . 31

6Long-term revenue through a single simulation in scenario 1 . . . . . . . . . 31

7

The transition of the number of accepted network slices over time in scenario 1 32

8

The transition of R/C ratio over time in scenario 1 . . . . . . . . . . . . . . 32

9

The transition of the number of accepted requests over time in scenario 2 . 34

5

7

List of Tables
1

The methods used in this comparative evaluation . . . . . . . . . . . . . . . 28

2

Parameter settings of the substrate network . . . . . . . . . . . . . . . . . . 28

3

The network slice requirements by service types in scenario 1 . . . . . . . . 30

4Evaluation metrics in scenario 1 . . . . . . . . . . . . . . . . . . . . . . . . . 30

5The network slice requirements by service types in scenario 2 . . . . . . . . 33

6

1

Introduction

There are various use cases in the 5G network, and the network requirements vary by
the applications running on them. As new network services appear daily and their usage
conditions change, it is not realistic to build a dedicated network for each service. Therefore, one of the most important factors is network slicing technology, which virtuallydivides physical resources on a shared substrate network and constructs multiple virtual
networks on the substrate network. As multiple clients request network slices with various
requirements, network providers are required to deploy those slices within finite physicalresources. It is important for both clients and providers to eﬀiciently deploy more slices
under the resource requirements.

Figure 1: Diagram of E2E network slicing in 5G network
Although there have been studies on network slicing in 5G networks from various perspectives, few have treated it as a resource allocation problem assuming end to end (E2E)
communication between user equipment(UE) and applications deployed on cloud servers [1].
When network slicing is considered as a resource allocation problem, it is required to optimize the allocation of the substrate resources in terms of acceptance ratio and cost whilesatisfying the requirements of each slice. This problem is closely related to the virtual network embedding (VNE) problem, which has been studied extensively [2], and formulated
in [3] as network slice embedding(NSE) problem based on the VNE problem. The charac7

teristics of 5G network slicing assuming E2E communication are that the requirements for
slices vary significantly depending on the service, and that it targets the construction of
networks between UE and cloud servers. Compared to embedding multiple homogeneous
virtual networks, when the requirements of each slice are significantly different, only selfish selection is likely to degrade the QoS of the other slices and fail to meet service levelagreement (SLA) and unable to achieve total optimization.
NSE problem is a combinatorial optimization problem, and methods for solving it using
linear programming solver and heuristics based on node ranking have been proposed [2].
Linear programming solver is not scalable to the size of network, and computation time
is a trouble in practical use. In addition, in practical use cases, multiple slices need to
be optimized in parallel as new slice requests arrive over time. In order to find solutions
adaptively to such a dynamic environment in a short time, a method is required that canfind the optimal solution for the new environment by utilizing previous solutions.
The mechanism to adapt to a certain environment by modifying the solution at thattime can be regarded as a model of organism evolution, and one of the optimization
algorithms based on the model is genetic algorithm. Genetic algorithm is one of the metaheuristics that can handle general combinatorial optimization problems and has been used
in a wide range of problems because it can handle optimization problems independentlyof the properties of the objective function. The previous research [4], which applies thegenetic algorithm to the VNE problem, shows the acceptance ratio of the method is higher
than that of the typical VNE method based on mixed integer programming [5, 6]. Since
the genetic algorithm allows the objective function to be arbitrarily designed, it is usefulin NSE problems to satisfy various requirements simultaneously and to achieve the goalof total optimization.
On the other hand, it is known that the population converges to a single form inconventional genetic algorithms, unlike in nature where diverse forms can be found. For
rapid adaptation to environmental variations in the NSE problem, it is required to maintain diverse solutions in the population. In response to this, methods have been proposedthat can generate diverse and high-quality solutions by proceeding with optimization while
maintaining solution diversity, and such methods are called quality-diversity algorithms.
MAP-Elites (Multi-dimensional Archive of Phenotypic Elites) [7] and NSLC (Novelty
8Search with Local Competition) are representative examples, and our group’s paper applying them to the dynamic VNE shows [8]. Because of its lower computational cost,in this paper, we applies the MAP-Elites algorithm to the NSE problem and proposes amethod of optimizing multiple network slices in parallel.
The genetic algorithm treats the solution as if it were an individual organism, andproceeds with optimization by repeatedly generating and selecting new individuals froma randomly generated population. Each individual has a sequence of integers called the
genotype, and the genotype must be decodable into a phenotype, or solution, by someprocedure. In addition, each individual has an evaluation value that determines whether
or not it can survive in its environment, which is called its fitness. The genotypes canbe changed by genetic manipulation, such as crossover with the genotype of another individual or mutation, to produce new individuals with new genotypes. Their fitness iscalculated from the solutions obtained by decoding their genotypes, and the values determines whether they remain in the population.
In order to apply the MAP-Elites algorithm to the NSE problem, it is necessary todefine the encoding method of the solution and the fitness. In MAP-Elites, it is known theencoding method that directly encodes parameters as genotypes is effective [9], and in the
NSE problem, the correspondence of the substrate node/path to each virtual node/linkin the network slice can be used as the genotype. Since it is important to embed the
network slice in the NSE problem by satisfying the requirements and saving the substrate
resources, the fitness is defined as a value that decreases when the slice requirements are
violated and increases when the residual resources are large.
As mentioned above, in a realistic scenario of the NSE problem, it is required to beembed multiple network slices in parallel. If embedding of all network slices is controlled
by a single controller, the size of the problem space increases exponentially with respect tothe number of network slices, since combinations about elements of all network slices mustbe handled. Therefore, it is not realistic to control multiple network slices with a single
controller from the perspective of scalability. Therefore, in this method, a controller using
the MAP-Elites algorithm is assigned to each network slice separately, and optimizationof each network slice is performed in parallel. However, it is unable to achieve total optimization by simply optimizing each individual slice. Therefore, by introducing unselfish
9

evaluation values into the fitness, we aim to realize cooperative behavior and achieve total
optimization.
We evaluated the effectiveness of this method through computer simulation. We modeled the arrival of network slice requests as a Poisson process, quantified the degree of
total optimization through the acceptance ratio, and compared the methods introducing
unselfish evaluation values and not in a simulation program in which multiple network
slices generated over time are controlled by individual controllers in parallel.
The remainder of this paper is organized as follows. Section 2 introduces the overview
of NSE problem and QD algorithms, and the detail of MAP-Elites, the QD algorithm usedin this research. Section 3 formulates the NSE problem in detail and describes the models
used in the simulation. Section 4 explains and formulates our method. Section 5 shows
the simulation settings and the results of computer simulations. Section 6 summarizes
this paper.

10

2

Related work

2.1

Network Slice Embedding Problem

As the use cases in 5G networks have diversified in recent years, so have the requirements
for the networks. Specifically, there are such use cases as URLLC (Ultra-reliable and
Low Latency Communications), which requires high reliability and low latency for applications such as automated driving and remote medical robots; eMBB (enhanced Mobile
Broadband), which requires high data rates for such applications as video streaming; andmMTC (Massive Machine Technology), which requires simultaneous communications inlarge-scale networks such as sensor networks and IoT [10]. Cooperative control is essential
to eﬀiciently deploy slices with different requirements simultaneously. On the other hand,
integrated control of all slices for total optimization leads to an increase in the problem
scale and makes it diﬀicult to solve the problem in a practical time. Therefore, while eachslice is controlled individually, it is required to cooperate with each other to improve theoverall acceptance ratio of network slice requests.
When the control of network slicing in 5G networks is regarded as an optimization
problem, it has various algorithmic aspects, one of which is as a resource allocation problem [1]. In [11], network slice optimization is defined as a Virtual Network Embedding(VNE) problem, but the target of control is from the base station to the core cloud, not
E2E communication including UE. In addition, Chien et al. [12] assumes E2E communication on 5G networks and implements resource allocation control for network slices, but
the slice embedding is not subject to control, and the resource allocation algorithm does
not consider optimization. Ludwig et al. [2] defines the optimization of network slice embedding for E2E communication including UE as the NSE problem, which is an extensionof the VNE problem.
In the NSE problem, the substrate network and network slices are modeled as undirected graphs, and the nodes composing the substrate network are classified as UE andcloud nodes, while the nodes comprosing the network slice are classified as UE and applications. Cloud nodes and substrate links have finite resources, which are consumed to hostapplications and communications between UE and applications. The NSE problem is todetermine which requests to accept from multiple network slices and the way to allocate
11resources to those slices under this resource constraint. In [3], the NSE problem including
application chains is formulated, and the definition in this paper is mainly based on thedescription in this work. However, we exclude the terms of availability and reliability thatare defined in this work.
In [2], an extension of the NodeRank method is proposed, which is a heuristics for
the VNE problem, and shows high scalability in terms of computation time, while having
similar acceptance ratio of slice requests compared to an existing method based on integer
linear programming. On the other hand, even though this work includes latency in theslice requirements, the proposed method does not consider latency. In addition, although
the proposed method assumes the parallel embedding of multiple network slices, it does
not incorporate the network resource condition varying over time and information fromother slices into the ranking, and does not consider the total optimization.
The aim of this research is to realize a parallel optimization method that controls network slices individually to ensure scalability, while cooperating with each other to achieve
total optimization. Taking advantage of the flexibility in designing the fitness (objectivefunction) and search space of the quality diversity algorithm, we incorporate the terms
about latency requirements and necessary for cooperation with the controller managing
other slices into the fitness and design a search space effective for improving the fitness toachieve eﬀicient total optimization.

2.2

Quality-Diversity Algorithm

Conventional optimization algorithms search for a single global optimal solution that maximizes the objective function, and multi-objective optimization algorithms also search forvertices in the search space. Quality diversity algorithm is evolutionary algorithmic framework that differ from them in that it exhaustively searches for qualitatively different andsuperior solutions distributed throughout the search space [13]. The main difference frommulti-objective optimization algorithms is that they search within the feature space rather
than the genotype space, and they also search in regions of the feature space where there
are no vertices of the objective function. This concept originated in the novelty search [14],and algorithms that divergently search for superior solutions in the search space, such as
NSLC [15] and MAP-Elites [7], were developed, and the quality diversity algorithm has
12been proposed as the framework of such algorithms [16]�
In optimization with the QD algorithms, it is necessary and important to define thefitness which is the evaluation value of the solution and the feature space. Since thepopulation evolves to improve the fitness, it is necessary to define the fitness appropriately
in order to achieve the objective. It is also important to design the feature space so thatvarious solutions are widely distributed, since solutions are divergently searched in thefeature space.
2.2.1 MAP-Elites Algorithm
MAP-Elites [7] is a sort of QD algorithm that aims to compute the maximum fitness
distribution in a low-dimensional feature space for a high-dimensional search space. Usersof MAP-Elites need to select several features of interest from the problem object anddesign a feature space. The feature space has dimensions whose variables are those features
and they are divided at a certain granularity, which makes cells in the space; MAP-Elites
outputs the solution with the highest fitness in each cell on the feature space. It is reported
that MAP-Elites is unlikely to fall into a local optimum compared to conventional genetic
algorithms, as it calculates individuals with high fitness in a larger region of the feature
space in parallel and generates a new solution based on them.
MAP-Elites is inspired by novelty-search [15] with local competition: since MAP-Elites
seeks only a single optimal solution for each cell (this solution is called an occupant), if
the newly generated solution belongs to a cell that already has an occupant, the solution
with the higher fitness of them remains as the occupant. This property reflects the factthat selection occurs through local competition with neighboring individuals in nature.
The pseudocode of MAP-Elites is shown in Algorithm 1. F and X are sets, returning
fitness and individuals with a cell identifier (ID) as index, respectively. A unique ID
for each cell is returned by f eature_descriptor where the same value is always returned
for features contained in the same cell. Each cell contains the individual with the highest
fitness among the individuals explored in it. In each iteration, random variation (mutationand crossover) is applied to a randomly selected individual from the population, and theindividual is compared with the current occupant of the cell corresponding to its features,and if it has higher fitness, the occupant is replaced with it, leading to evolution.
13

Algorithm 1 Pseudocode of MAP-Elites [7]procedure MAP-Elites
F ← ∅, X ← ∅for iter ← 1, I do
if iter < G thenx′ ← random_solution()elsex ← random_selection(X )x′ ← random_variation(x)b′ ← f eature_descriptor(x′ )f ′ ← f itness(x′ )if F(b′ ) = null or F(b′ ) < f ′ thenF(b′ ) ← f ′X (b′ ) ← x′
return f eature-f itness map (F and X )14

3

Formulation of Dynamic Network Slice Embedding Problem

In this research, the NSE problem is formulated mainly based on the previous work [2] [3].
The substrate network and network slices are modeled as undirected graphs. A undirected
graph is expressed generally as a ordered pair G = (V, E), where V is the set of vertices, andE is the set of edges. The edge between two vertices vi , vj ∈ V is denoted by ei,j = (vi , vj ).
Then, the substrate network is represented by NS = (VS , ES ). The vertices of substrate
network is classified into UE and cloud nodes, they are denoted by VU , VC , respectively, andVS = VU ∩VC . ES represents the wired or wireless links, and there is no connection between
UE, which means ∀ei,j ∈ ES (vi ∈ VC ∨ vj ∈ VC ). The cloud nodes and the wired/wirelesslinks have finite resources. CPU resource C(v) and memory capacity M (v) of a cloud nodev ∈ VC , and throughput T (e) and latency L(e) of a link e ∈ ES are considered in thisresearch. In addition, all cloud nodes have a loop with infinite throuput and zero latency.
This is because the situation is assumed that each cloud node consists of multiple nodes
and each of them can host application individually.
Similarly, the requested network slice is represented by NR = (VR , ER ). The network slice consists of some of the UE in the substate network and applications hosted
by cloud nodes. Thus, let VU′ ⊂ VU and VA denote the set of the UE used in thisslice and the applications, respectively, and VR = VU′ ∪ VA . ER is the set of virtual
links between UE and applications, and the is no virtual link between UE, which means∀ei,j ∈ ER (vi ∈ VA ∨ vj ∈ VA ). The applications and the virtual links have attributes of
the resource requirements corresponding to the substrate resources(C, M, T, L).

Figure 2: Diagram of NSE problem
15

In the NSE problem, for multiple network slice requests, it is required to decide whether
each of them is accepted or not and how to allocate resources. Let R denote the set of thek ) = (V ′k ∪ V k , E k )(1 ≤N requested network slices, and for each element NRk = (VRk , ERRAUk ≤ N ), whether it is accepted or not is represented by the mapping MA : R → {0, 1}.
In addition, the set of the numbers of accepted network slices is denoted by NA = {k |MA (NRk ) = 1}.

MA (NRk ) :=
1 if N k is acceptedR0otherwise

For each element NRk of the set of accepted network slices RA = {NRk ∈ R | MA (NRk ) =k : Vk →1}, the way to allocate its applications to cloud nodes is represented the mapping MNA′k : V k → V is defined additionally.
VC . To extend the domain to VRk , the mapping MNSRk does not have to be injective while the node
It should be noted that the mapping MN

allocation in VNE problem does so.
′kMN(n) :=

M k (n) (n ∈ V k )
N
A

n

(n ∈ VU′k )The way to allocate virtual links to substrate paths is represented by the mappingk → P , where P is the set of all possible path on the substrate network. A path
MLk : ERS
S

is a finite sequence of vertices, the kth term(vertex) of a path p is denoted by pk , and its
inlength is denoted by |p|. In addition, the notation ei,j ∼ p means that the edge ei,j isincluded in the path p, and the latency of the whole path p is denoted by LP (p), which isformulated byindef

ei,j ∼ p ⇔ ∃q ((pq = vi ∧ pq+1 = vj ) ∨ (pq = vj ∧ pq+1 = vi ))|p|−1LP (p) :=∑L ((pk , pk+1 ))k=1

The allocated network slice must have the same topology as the requested network
slice, and this constraint is defined as the logical formula 1.
(
)
k
′k
′k
∀k ∈ NA ∀ei,j ∈ ER
MLk (ei,j )1 = MN(vi ) ∧ MLk (ei,j )|M k (ei,j )| = MN
(vj )
L

16

(1)

Moreover, each cloud node must satisfy the resource requirements of applications it
hosts, and this constraint is defined as the logical formula 2.
∀c ∈ VC ∀Attr ∈ {C, M } ∑∑Attr(a) < Attr(c)

(2)k∈NA a∈V k (c)
A

where
k
VAk (c) = {a ∈ VAk | MN(a) = c}Similarly, each substrate link must satisfy the resource requirements of the virtual links
established on it, and this constraint is defined as the logical formulae 3,4.

∀ei,j ∈ ES ∑∑
T (er ) < T (ei,j )

k∈NA er ∈E k (ei,j )(Rk∀k ∈ NA ∀er ∈ ER
LP (MLk (er )) < L(er )

)

(3)
(4)

where
in

k
k
ER
(ei,j ) = {er ∈ ER
| ei,j ∼ MLk (er )}k , M k satisfying these
The NSE problem is defined as deciding the mappings MA , MN
Llogical formula.
In the following subsections, the models about the NSE problem is explained assumed
in the evaluation of this research.

3.1

Substrate Network ModelIn this research, the substrate network architecture is assumed to be the layer substrateconsisting of four layers: UE, Node B, edge cloud, and central cloud, proposed in [2], [17].
The layers except for UE are classified as cloud nodes, the resource amount increases, andthe number of nodes decrease in the above order. There are only links connecting between
adjacent layers. Let VN B , VEC , VCC denote the sets of nodes of Node B, edge cloud, central
cloud, respectively, this is formulated by∀ei,j ∈ ES ((i ∈ VU ∧ j ∈ VN B ) ∨ (i ∈ VN B ∧ j ∈ VEC ) ∨ (i ∈ VEC ∧ j ∈ VCC ))17

Figure 3: Layer substrate network model
Although the wireless links between UE and Node B are modeled similarly to otherwired links in [2], the available bandwidth is shared among UE connecting to the same
Node B in actual, thus they should be modeled such that the throughput is shared. In thispaper, we added the layer representing radio access network(RAN) between UE layer andNode B layer, and UE connecting to the same Node B is made through the same RAN toshare the available throughput. Here, |VN B | = |VRAN |∧∀vi ∈ VN B ∃!vj ∈ VRAN (ei,j ∈ ES ).
In addition, the links between Node B and RAN have certain throughput and zero latency,
and the links between UE and RAN have infinite throughput and certain latency.

Figure 4: Addition of RAN layer

18

3.2

Network Slice Request Arrival Model

Network slice requests is supposed to be issued individually by each client, the issues are
events independent of each other, and such process can be modeled as Poisson process. The
occurrence probability of events in Poisson process follows Poisson distribution, which is adiscrete probability distribution that express the probability of the case the event, which
occurs λ times per time unit on average, occurs k times within a time unit. Let X be therandom variable representing the number of events that occurs within a time unit, theprobability is
P (X = k) =λk e−λk!

Network slices are deactivated after a certain interval from the time they are hosted,
and the interval is called lifetime. The interval t between occurrence of events in Poissonprocess follows the exponential distribution with the below probability density function.
f (t) = λe−λt
Therefore, the model in which n network slice requests is alive simultaneously can be
implemented by setting the lifetime of slices as the random number following the same
distribution as the interval of event which occurs λ/n times per time unit on average.
Thus, the lifetime of each network slice is generated as random number following the
exponential distribution with the below probability density function.
f (t) =

λ −λt
e n
n

19

4

Dynamic Network Slice Control Method using QD algorithm

In the following explanation, the formulae defined in section 3 are used. In this method, thebehavior of the controller of embedding individual network slice is described with MAPElites algorithm, and the controllers assigned to network slices optimize their own embedk , Mkding separately in parallel. In short, the controllers optimize the mappings MA , MN
L

that is the solution of the NSE problem.
The controller of the network slice NRk is defined as C k .
C k = (P k , F k , Ak )where P k is the population in MAP-Elites algorithm, which is the set of the genotype of anindividual in each cell(occupants) of the feature space, and F k , Ak are the mappings fromthe genotype to the fitness and the value representing whether it is feasible solution ornot, respectively. A feasible solution mentioned here is the solution satisfying the logicalformulae 1–4, and Ak (g) = 1 if the genotype g is a feasible solution, otherwise Ak (g) = 0.
The details of the mappings F k , Ak are described in section 4.2. P k varies every iterationin Algorithm 1, so the state of the controller at the time t is denoted by Ctk = (Ptk , F k , Ak ).
Here, the mapping MA at the time t is defined as1 if ∃i ≤ t ∃g ∈ P k (Ak (g, i) = 1)ikMA (NR | t) :=0 otherwise
which means the network slice NRk is accepted if the controller C k has found a feasible
solution by the time t.
The genotypes which are elements in the population Ptk can be translated into the pairk , M k by certain procedure, and the details is mentionedof mappings corresponding to MN
L

in section 4.1. Let us define this procedure as the mapping deck here, and the mappings20

k , M k at the time t are decided asMNLkk(a | t) := MN
MN
t (a)
k
(l)
MLk (l | t) := MLt




k
k
k
(MN
argmax F k (g, t′ )
t , MLt ) = decg∈Akt′

wheret′ = max{i ≤ t | ∃g ∈ Pik (Ak (g, i) = 1)}, Akt′ = {g ∈ Ptk′ | Ak (g, t′ ) = 1}
These formulae means the solution at the time t is made by decoding the genotype of theindividual with the highest fitness of the feasible solutions found last by the time t.

4.1

Encoding of Solutions

In dealing with combinatorial optimization problem using genetic algorithms, the genotype
and the way to decode it into solution as phenotype must be defined. It is known that it iseffective to encode the parameters directly into genotype with MAP-Elites, so the solution
parameters of the NSE problem are used as the genotype in this research. The solution
k , M k , and the parametersin optimization of the network slice NRk is the mappings MN
L
k . Therefore, the genotype
are the substrate elements mapped from a ∈ VAk and l ∈ ER

g is defined as the sequence of the values representing these elements. For nodes, whenk (a ) = c (a ∈ V k , c ∈ V k ), g is set to j. For edges, it is supposed that the substrate
MN
i
j i
i
A j
C

path assigned to each virtual link is selected from only the K shortest paths between thesubstrate nodes assigned to the ends of the link, and the number of path when arrangedin the ascending order of length is used as parameter. Let phi,j denote the h-th shortest
k can be denoted by l using
path between ci , cj , and it is supposed that each element of ERdk ),the unique number d ∈ N⩽|E k | . Finally, when MLk (ld ) = phM k (a ),M k (a ) (ld = (i, j) ∈ ER
R

N

i

N

j

k|
|ER
|VAk |×N, andg|VA |+d is set to h. Thus, any genotype is an element of the set G k = N⩽|V
⩽K|
C

the search space of the algorithm is the set G k .
Conversely, when the genotype g ∈ G k is decoded into the pair of mappings correk , M k , each term of the genotype g is translated to the substrate elements
sponding to MN
L

21

mapped by the mappings. Thus, the mapping deck is defined as
deck (g) := (ϕkg , ψgk )ϕkg (ai ) := cgi
g|V k |+dA
ψgk (ld ) = ψgk ((ai , aj )) := pϕk (a),ϕk (a )
g

i

j

g

Note that the pair of mappings ϕkg , ψgk decoded by deck does not satisfy the logical formulae 1–4 necessarily, and phenotypes not satisfying these constraints are not feasible
solutions.

4.2 Definition of Fitness
The fitness which is the objective function should be defined appropriately in order toreflect the requirements. In this method, which optimizes multiple network slices in parallel, when an individual is evaluated in the controller C k , the slices of the other controllers
should be embedded according to the solution made by them at that time. At the timet, the residual resource amounts RN of c ∈ VC and the residual throughput of e ∈ ES are
defined as∑

RN (c, Attr, k, t) := Attr(c) −∑Attr(a)n∈NkA a∈VAn (c|t)∑

RL (e, k, t) := T (e) −∑T (er )

n (e|t)n∈NkA er ∈ER

where
Attr ∈ {C, M }, NkA = {n | n ̸= k ∧ MA (NRn | t) = 1},
in

n
n
n
VAn (c | t) = {a ∈ VAn | MN
(a | t) = c}, ER(e | t) = {er ∈ ER
| e ∼ MLn (er | t)}Also, the requested resource amounts QN to c ∈ VC and the requested throughput
QL to e ∈ ES when the network slice NRk is embedded according to the pair of mappings
deck (g) are defined as
QN (c, Attr, k, g) :=∑Attr(a)a∈VAk (c|g)∑

QL (e, k, g) :=k (e|g)er ∈ER22

T (er )

whereinkk
VAk (c | g) = {a ∈ VAk | ϕkg (a) = c}, ER(e | g) = {er ∈ ER
| e ∼ ψgk (er )}The essential requirement in NSE problem is satisfaction of the logical formulae 1–4,
so the fitness should be defined such that it decrease as these constraints are violated,
and then the population evolves to satisfy the requirement. The logical formula 1 isalways satisfied essentially by the definition of genotype mentioned in section 4.1, thelogical formulae 2–4 only have to be taken into account. Therefore, the penalty term P krepresenting the degree of the violation of the resource requirements is defined as below.
In this term, the term about latency constraint should be weighted significantly since theformula 4 is independent of the other slice embedding, and the satisfaction of this is toppriority. Thus, setting the constant µ large directs the evolution to satisfy the latency
constraint preferentially.
∑

P k (g | t) :=∑max {QN (c, X, k, g) − RN (c, X, k, t), 0}

X∈{C,M } c∈VC+∑

max {QL (e, k, g) − RL (e, k, t), 0}

e∈ES

+µ

∑

{
}
max LP (ψgk (er )) − L(er ), 0ker ∈ER

Here, when P k (g | t) = 0, the pair of mappings deck (g) is a feasible solution of the
NSE problem. Therefore, the mapping Ak is defined as1 if P k (g | t) = 0kA (g, t) :=0 otherwise
Additionally, there is a basic objective in NSE problem is to save the cost of hosting
network slices. This can be achieved by minimizing the total requested resource amount.
Since the amounts of requested resources of cloud nodes from applications are constant,
it is needed to consider only the requested throughput from virtual links.
Therefore, the fitness reflecting these objective can be composed as below.
−ρP k (g | t) −

∑
e∈ES

23

QL (e, k, g)where ρ is a constant, and setting this large leads to the evolution such that achieve theresource requirements preferentially.
However, this definition of fitness results in just local optimization about the network slice NRk since the other slices is not taken into account. To overcome this problem
and realize the cooperative behavior considering the other slices, we defined the unselfish
evaluation values additionally and incorporate them into the fitness.
4.2.1 Unselfish Behavior: Load Balancing
Generally, to accept more network slices, it is important to avoid the resource fragmentation, that is, to reduce the number of bottleneck, which is element with little residual
resource. This can be realized by using the elements with abundant residual resources preferentially. The total residual resources of the substrate elements assigned to the network
slice NRk is incorporated into the fitness, thus such solutions are selected preferentially.
Therefore, the mapping B k is defined as
B k (g | t) :=

∑

∑

∑

RN (ϕkg (a), X, k, t) +X∈{C,M } a∈VAk∑

RL (e, k, t)

k e∈E (ψ k (e ))er ∈ER
S
g r

where
in

ES (p) = {e ∈ ES | e ∼ p}
4.2.2 Unselfish Behavior: Giving Way
When MA (NRk | t) = 1, if the controller C k searches for such feasible solutions that do not
use the substrate element where the demand from not accepted slices concentrates, it isexpected more slices is accepted. The individual with the highest fitness in each controller
is the allocation best adapting to the current resource condition and highly possible to be
accepted later. Therefore, the total requested resource amounts by these individuals are
regarded as the demand of the substrate elements. At the time t, the values representing

24the demand of c ∈ VC and e ∈ ES denoted by DN , DL , respectively, are defined as
DN (c, k, t) :=

∑

∑

∑

Attr(a)X∈{C,M } n∈NkR a∈VAn (c|t)∑

DL (e, k, t) := T (e) −∑T (er )

n (e|t)n∈NkR er ∈ER

where
NR = {n | MA (NRn | t) = 0}, VAn (c | t) = {a ∈ VAn | ϕngtn (a) = c},
in

n
n
ER
(e | t) = {er ∈ ER
| e ∼ ψgntn (er )}, gtn = argmax F k (g, t)g∈Ptn

The controller C k should search for feasible solutions using the substrate elements wherethese values are lower, this means the population should evolve to minimize the total
demand of the substrate elements assigned to the slice. Thus, the mapping Dk is definedasDk (g | t) :=∑DN (ϕkg (a), k, t) +∑∑DL (e, k, t)

k e∈E (ψ k (e ))er ∈ER
S
g r

a∈VAk

where
in

ES (p) = {e ∈ ES | e ∼ p}
4.2.3 Definition of Fitness for Cooperative Control
Using the terms defined above, the mapping returning the fitness F k is defined as
F k (g, t) := −∑QL (e, k, g) − ρP k (g | t) + ϵB k (g | t) − MA (NRk | t) · δDk (g | t)e∈ESwhere ϵ, δ are constants, ϵ should be set small for the third term not to affect the otherrequirements. The fourth term is valid only when a feasible solution has been alreadyfound in the controller C k , and set large to give way to the other controllers.

4.3 Design of Feature Space
Since MAP-Elites algorithm searches for solutions in feature space, it is important todesign such feature space that can capture the superior solutions spread in search space25

well. In addition, for the flexible adaptation to the environmental change occurring in
dynamic NSE problem, it is effective to use the features whose the optimal value varies by
the environment as variables. Based on above, the features below are used in this method.
1. The total resource amount of assigned nodes— This feature is the total substrate resource amount of the cloud nodes assigned
to the applications. In the layer substrate network model defined in section 3.1,
the resource amounts varies by the type of cloud node. Therefore, this featurereflects which type of nodes are mainly used. Since the optimal node usage
varies by the service requirements of the slice and the substrate resource condition, it is effective to search for solutions divergently about this feature.
However, the distribution of the solutions in the dimension whose variable isthis value is heterogeneous, which leads to the degradation of solution search
eﬀiciency. Thus, the value F1 is used as the substitution of this value. Notethat the numbers of the substrate nodes are assigned in order of Node B, edge
cloud, and central cloud.
2. The total latency— This feature is the total latency of the substrate paths assigned to the virtual
links. This value is correlated mainly with the number of hops of paths, but
the paths with the same number of hops can be distinguished by latency, whichleads to the diverse path selection. Since the diverse path selection results inavoiding usage of resources with high demand, it is effective in this problem.
For this feature, the value F2 is used as the substitution for the same reason
as above.
|VAk |

F1 (g, k) :=

∑

gi

i=1
k|
|ER

F2 (g, k) :=∑g|V k |+i
A

i=1

26

5

Evaluation

For the dynamic NSE problem of controlling multiple network slices in parallel, we compared our proposed method with the previous methods through computer simulations in
the two network slice request scenarios below.
Scenario 1
The scenario based on the network slice arrival model defined in 3.2. The parametersof the model are set as λ = 0.2, n = 40, and the duration of a single simulation is1000 time units. The time-out period is set to 100 time units. If a controller does
not find a feasible solution by the period, the network slice request is rejected. This
scenario is used as dynamic environment assuming realistic use case.
Scenario 2
The scenario where 100 network slice requests arrive at the beginning of simulation.
This is used for the evaluation of solution search eﬀiciency and total optimization
performance of each method under static environment.
In any simulation, 100 individuals (solution candidates) are evaluated per time unit. In
this paper, the real-time evaluation is not performed since the simulation program is not
optimized. It is assumed there is no difference about computation time among the methods
because the evaluation process is same in all methods.
The methods we evaluated are shown in Table 1. The proposed methods are notatedwith * in this table. The parameters in Table 1 are constants in the equation 4.2.3.
Elitism-based Genetic Algorithm (EGA) is a basic genetic algorithm that adopt elitism tomake its evolution steady and used as a conventional genetic algorithm in this comparison.
The method NR applies RW-BFS which is a heuristic in the VNE problem to this problem
in the same way as the heuristic proposed in [2].
The substrate network is generated based on the layer substrate network model definedin section 3.1 and common between the scenarios. The parameter settings of the substrate
network used in simulation are shown in Table 2. In Table 2, EA,B is the set of links
between elements of VA , VB , and |E| is the number of elements of VB connected to eachelement of VA . Note that elements of EU,N B are through VRAN actually.
27

Table 1: The methods used in this comparative evaluation
Parameter
Notation

Algorithm

µ

ρ

*ME-C
*ME-LB

MAP-Elites*ME-GW

104

102

ME-I
GA-C

EGA

GA-I
NR

104

RW-BFS [18]

-

102
-

ϵ

δ

10−5

102

10−5

0

0

102

0

0

10−5

102

0

0

-

-

Table 2: Parameter settings of the substrate network
Node Type

Link Type

VU

VN B

VEC

VCC

|V |

90

30

10

1

C, M

-

[100, 200]

[200, 700]

[5000, 10000]

28

EU,N B

EN B,EC

EEC,CC|E|[1, 3][2, 6][1, 1]

T[50, 80][80, 150][200, 500]

L

[3, 7][3, 5][2, 4]

Evaluation Metrics

For the quantification of the degree of the total optimization, we

define the acceptance ratio of network slice requests as the metric.
At the time t, the acceptance ratio is given by|{NR ∈ R | ∃t (MA (NR | t) = 1)}||R|
Furthermore, the profitability is also an important metric to evaluate NSE methods, therefore we define the revenue and the revenue to cost ratio (R/C ratio) like the previous
work [18]. The revenue Rv(t) and the cost Cs(t) are defined as



∑  ∑
∑
∑
Rv(t) :=
X(a) +
T (l)


k
k∈NA
X∈{C,M } a∈VAk
l∈ER



∑  ∑
∑
∑
Cs(t) :=
X(a) +(|MLk (l)| − 1)T (l)


k
k
k∈NA

X∈{C,M } a∈VAl∈ER

Consequently, the R/C ratio is given by
Rv(t)
Cs(t)
In addition, the average revenue over the entire duration is called the long-term revenue, which is given by∑Tlim

t=0 Rv(t)T →∞T

Similarly, the long-term R/C ratio can also be defined as∑TRv(t)lim ∑t=0TT →∞t=0 Cs(t)5.1Comparison under the Dynamic Environment

For the performance comparison under the realistic dynamic environment, NSE using
each method is simulated. The parameter settings about network slice requests in thissimulation are shown in Table 3. The same network slice requests and arrival times areused in every simulation. For the methods except for NR, the simulations are performed
30 times in the same settings since genetic algorithms are probabilistic.
The simulation results for each method are shown in Table 4 and Figure 5–8. Table 4shows that our proposed method ME-C accepts the most slices and produces the most29

Table 3: The network slice requirements by service types in scenario 1
Service Type
URLLC

eMBB

mMTC

|VU′ |[1, 10][1, 10][15, 30]|VA |[1, 5][1, 10][1, 5]

C, M[3, 15][10, 40][1, 3]

T[5, 15][10, 20][1, 5]

L[10, 30][25, 50][50, 100]

long-term revenue, which reveals that the total optimization is realized by the cooperative
behavior in our methods. The metrics of all methods using MAP-Elites are larger thanthose of other methods. This means that MAP-Elites searches diverse feasible solutions
most eﬀiciently in larger search space. While the difference of the acceptance ratio among
the methods using MAP-Elites is small, the difference of the long-term revenue is large
relatively. Moreover, ME-LB accepts more slices than ME-GW, whereas ME-GW produces
more revenue than ME-LB. This result indicates that the requests with more resource
requirements tend to be accepted by the cooperative behavior “giving way”. On the other
hand, the R/C ratios of ME-C and ME-GW are less than the other ME, which is caused by
the selection of longer paths. It is found from these result that the selection of detour paths
for giving way results in the acceptance of the requests with more resource requirements.
Table 4: Evaluation metrics in scenario 1
ME-C

ME-LB

ME-GW

ME-I

GA-C

GA-I

NR

Acceptance ratio(%)

65.5

65.2

64.8

64.4

62.3

61.2

51.3

Long-term revenue

2861.02755.02849.42709.32589.82478.32102.7Long-term R/C ratio(%)

50.3

55.4

50.2

55.4

49.6

53.9

50.7

30

Figure 5: Acceptance ratio through a single simulation in scenario 1

Figure 6: Long-term revenue through a single simulation in scenario 1

31

Figure 7: The transition of the number of accepted network slices over time in scenario 1

Figure 8: The transition of R/C ratio over time in scenario 1

32

5.2

Comparison under the Static Environment

For the comparison the profits among the proposed method, we evaluate the methods
using MAP-Elites in scenario 2. The parameter settings about network slice requests inthis simulation are shown in Table 5.
Table 5: The network slice requirements by service types in scenario 2
Service Type
URLLC

eMBB

mMTC

|VU′ |[1, 5][1, 10][15, 30]|VA |[1, 3][1, 5][1, 5]

C, M[3, 15][10, 40][5, 10]T[2, 3][3, 5][1, 2]

L[16, 20][25, 50][50, 100]

Simulations are conducted for each of three the sequence of slice requests generated
based on these settings. The transition of number of accepted requests is depicted in
Figure 9. According to this figure, the methods are classified into two groups; the group
including ME-C and ME-GW, and the others. It is shown that the number of accepted
requests of ME-C and ME-GW increases rapidly, which means that the search of feasible
solutions using resources with less demands promotes the acceptance of the other slices.
The more early acceptance of more slices is desirable, and the increase of acceptance ratio
in the evaluation under scenario 1 should be caused by this effect. On the other hand, it
depends on the case which method accept the most slices finally. This suggests that theproperties of the network slice requests affect the effectiveness of the cooperative behaviors.
Based on these figures, it seems the difference between ME-C and ME-LB is negatively
correlated with the final number of accepted requests, which implies that the advantage
of giving way becomes large if the substrate resources are abundant, and the advantage
of load balancing becomes large if not. However, since no statistic analysis is conducted
about this relation, the research about the relation between the properties of requests andthe effectiveness of methods is feature work.

33

(a)

(b)

(c)

Figure 9: The transition of the number of accepted requests over time in scenario 234

6Conclusion

For the dynamic NSE problem in which it is needed to deal with multiple network slices
with diverse requirements, we proposed the total optimization method that optimizes
each slice with individual controller behaving cooperatively in parallel. Taking advantageof the design flexibility of the objective function in QD algorithm, we incorporated theunselfish evaluation values into the fitness to realize cooperative behavior, which enables
the total optimization by separate controllers. Through computer simulations under therealistic dynamic scenario and the static scenario, the comparative evaluation revealed theeffectiveness of our proposed method and the properties of each variant.
In the evaluation, the weighting parameters in the fitness are fixed to certain values,
although we did not conduct the search for the appropriate setting of the parameters. For
example, setting the parameter δ large caused of excess usage of substrate resources in the
conducted simulations. As the finer parameter settings may lead to the improvement of
the performance of proposed methods, this is future work.
Although we made the dedicated simulation program for the evaluation, a suﬀicientnumber of simulations cannot be conducted due to the execution time. For further detailed
research, the simulation program must be redesigned to reduce of the execution time. The
evaluation of the solution candidate can be formulated as matrix operation and accelerated
by using GPU. Subsequently to the acceleration, the large scale simulation should be
conducted to analyze the properties of the methods statistically. These are also future
works.

35

Acknowledgments
For the completion of this thesis, I have been given a lot of great advice and insightful
suggestion by Professor Masayuki Murata of the Graduate School of Information Scienceand Technology at Osaka University, and I appreciate it. In regular meetings, he sometimes
gave me compliments, and that encouraged me. I also thank for his tolerance for my fault
and his continuous direction.
I was also daily instructed and helped with the progress of my research by Associate
Professor Daichi Kominami of the Institute for Open and Transdisciplinary Research Initiative, Osaka University. Thanks to his considerable assistance, I proceeded with myresearch appropriately and completed this thesis finally. I always depended on him foreven the trivial daily things not related in the contents of research. I would like to express
my greatest gratitude to him.
The subject of this thesis is affected directly by the previous work of Assistant Professor Tatsuya Otoshi of the Graduate School of Economics, Osaka University. In the earlyphase of my research, I asked him for many knowledge about the problem and the implementation. Without his cooperation, I would have much trouble advancing my research.
I am most grateful to him.
I also would like to express my gratitude for giving me invaluable advice to Associate
Professor Shin’ichi Arakawa and Specially Appointed Assistant Professor Masaaki Yamauchi of the Graduate School of Information Science and Technology, Osaka University.
Finally, I would like to thank all the members of the Advanced Network Architecture
Research Laboratory.

36

References[1] F. Debbabi, R. Jmal, L. C. Fourati, and A. Ksentini, “Algorithmics and modelingaspects of network slicing in 5G and beyonds network: Survey,” IEEE Access, vol. 8,
pp. 162 748–162 762, 2020.
[2] K. Ludwig, A. Fendt, and B. Bauer, “An eﬀicient online heuristic for mobile network
slice embedding,” in Proceedings of Conference on Innovation in Clouds, Internet andNetworks and Workshops (ICIN), 2020, pp. 139–143.
[3] A. Fendt, C. Mannweiler, L. C. Schmelz, and B. Bauer, “A formal optimization modelfor 5G mobile network slice resource allocation,” in Proceedings of IEEE Annual Information Technology, Electronics and Mobile Communication Conference (IEMCON),
2018, pp. 101–106.
[4] C. Jiang and P. Zhang, Virtual Network Embedding Based on Modified Genetic Algorithm. Springer Singapore, 2021, pp. 109–127.
[5] N. M. M. K. Chowdhury, M. R. Rahman, and R. Boutaba, “Virtual network embedding with coordinated node and link mapping,” in Proceedings of IEEE Conference
on Computer Communications (INFOCOM), 2009, pp. 783–791.
[6] H. Cao, H. Hu, Z. Qu, and L. Yang, “Heuristic solutions of virtual network embedding:A survey,” China Communications, vol. 15, no. 3, pp. 186–219, 2018.
[7] J.-B. Mouret and J. Clune, “Illuminating search spaces by mapping elites,” 2015.
[8] A. Otsuki, D. Kominami, T. Otoshi, H. Shimonishi, and M. Murata, “Adaptive
network slicing control method for unpredictable network variations using qualitydiversity algorithms,” in Proceedings of IEEE Consumer Communications and Networking Conference (CCNC), 2024.
[9] D. Tarapore, J. Clune, A. Cully, and J.-B. Mouret, “How do different encodings
influence the performance of the map-elites algorithm?” in Proceedings of the Genetic
and Evolutionary Computation Conference.
2016, p. 173–180.
37

Association for Computing Machinery,

[10] NGMN, NGMN 5G white paper 2. NGMN Alliance, Jul. 2020.
[11] S. Vassilaras, L. Gkatzikis, N. Liakopoulos, I. N. Stiakogiannakis, M. Qi, L. Shi,
L. Liu, M. Debbah, and G. S. Paschos, “The algorithmic aspects of network slicing,”
IEEE Communications Magazine, vol. 55, no. 8, pp. 112–119, 2017.
[12] H.-T. Chien, Y.-D. Lin, C.-L. Lai, and C.-T. Wang, “End-to-end slicing as a service
with computing and communication resource allocation for multi-tenant 5G systems,”
IEEE Wireless Communications, vol. 26, no. 5, pp. 104–112, 2019.
[13] K. Chatzilygeroudis, A. Cully, V. Vassiliades, and J.-B. Mouret, “Quality-diversity
optimization: a novel branch of stochastic optimization,” 2020.
[14] J. Lehman and K. O. Stanley, “Abandoning objectives: Evolution through the search
for novelty alone,” Evolutionary Computation, vol. 19, no. 2, pp. 189–223, 2011.
[15] ——, “Evolving a diversity of virtual creatures through novelty search and local
competition,” in Proceedings of annual conference on Genetic and evolutionary computation, Jul. 2011, pp. 211–218.
[16] J. K. Pugh, L. B. Soros, and K. O. Stanley, “Quality diversity: A new frontier for
evolutionary computation,” Frontiers in Robotics and AI, vol. 3, pp. 1–17, Jul. 2016.
[17] I. Labrador Pavon, J. (Atos, A. Colazzo, R. Ferrari, P. Arnold, M. Breitbach,
H. Droste, D. Telekom, V. Friderikos, S. London, V. Sciancalepore, Z. (NEC, M.L. Alberi-Morel, M. Doll, B. Gajic, S. Kerboeuf, C. Mannweiler, D. Michalopoulos,
P. Rost, and A. Madrid, “5G NORMA network architecture – intermediate report,”
Jan. 2017.
[18] X. Cheng, S. Su, Z. Zhang, H. Wang, F. Yang, Y. Luo, and J. Wang, “Virtual network embedding through topology-aware node ranking,” ACM SIGCOMM Computer
Communication Review, vol. 41, no. 2, p. 38–47, Apr. 2011.

38