Hypergraph theory in wireless communication networks

First, in Chap.1, we introduce the basic preliminaries of hypergraph theory in general and develop two hypergraph-based polynomial algorithms, i.e., graph coloring and hypergraph cluster

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Yingjun Zhang

Hypergraph Theory in

Wireless Communication Networks

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With the explosive traffic demand and dense mobile devices, a new generation

of cellular networks is required to support resource sharing among multiplemobile devices and manage their collisions properly In this book, we focus oncommunication systems in which network devices are allowed to reuse the sameresource so as to improve the overall performance For example, mobile users withsparse code multiple access (SCMA) share the same codebook, or ultra-dense smallcells with massive multiple-input-multiple-output (MIMO) share the same limitedpilot sequences

Conventionally, considering the topology of network devices, graph theory is

a useful tool to seek the solutions of the resource sharing problem In the graphrepresentation of cellular network, each vertex represents a network device, and anedge exists between two vertices if they collide when sharing the same resource.Therefore, the resource allocation problem corresponds to finding the independentset in which the devices do not collide By modeling the pairwise relations, theoverall system performance increases

However, in many scenarios of future wireless systems, the network needs

to coordinate multiple devices in order to further improve the utilization of thescarce spectrum resources, and thus, the graph model is not accurate in modelingthe relation among multiple devices For example, in the subchannel allocationproblem in non-orthogonal multiple access (NOMA), the mobile users share thesame subchannel with multiple mobile users, and thus, these mobile users willbring cumulative interference, which cannot be captured by traditional graph In thisbook, we introduce a mathematical framework from hypergraph theory, in which ahyperedge can be a subset of the vertex set It provides a useful analytical tool for thereaders to analyze how the relations among multiple mobile users affect the systemperformance, and thus, can be applied to address the resource sharing scenarios infuture wireless networks

First, in Chap.1, we introduce the basic preliminaries of hypergraph theory

in general and develop two hypergraph-based polynomial algorithms, i.e., graph coloring and hypergraph clustering Then, in Chaps.2 and 3, we presenttwo emerging applications of hypergraph coloring and hypergraph clustering in

hyper-v

vi Preface

Device-to-Device (D2D) underlay communication networks, respectively, in order

to show the advantages of hypergraph theory compared with the traditional graphtheory Finally, in Chap.4, we discuss the limitations of using hypergraph theory infuture wireless networks and briefly present some other potential applications

1 Basics of Hypergraph Theory 1

1.1 Basic Hypergraph Concepts 1

1.1.1 Preliminary Definitions 1

1.1.2 Incidence and Duality 3

1.1.3 Basic Hypergraph Operations 5

1.1.4 Subhypergraphs 8

1.2 Hypergraph Coloring 11

1.2.1 Basic Kinds of Hypergraph Coloring 11

1.2.2 Greedy Algorithm for Hypergraph Coloring 13

1.3 Hypergraph Clustering 14

1.3.1 Hypergraph Clustering Problem 15

1.3.2 Clustering Algorithm 16

References 19

2 Radio Resource Allocation for Device-to-Device Underlay Communications 21

2.1 Introduction 21

2.2 System Model and Problem Formulation 22

2.2.1 System Model 22

2.2.2 Problem Formulation 24

2.3 Traditional Graph Based Channel Allocation 25

2.3.1 Graph Construction 25

2.3.2 Channel Allocation Algorithm 26

2.4 Hypergraph Based Channel Allocation 26

2.4.1 Hypergraph Construction 27

2.4.2 Hypergraph Coloring Algorithm 29

2.5 Property Analysis 31

2.6 Simulation Results 33

2.7 Summary 37

References 38

vii

viii Contents

3 Resource Allocation for Cross-Cell Device-to-Device

Communications 41

3.1 Introduction 41

3.2 System Model and Problem Formulation 42

3.2.1 System Model 42

3.2.2 Cross-Cell D2D Communications 44

3.2.3 Problem Formulation 45

3.3 Hypergraph Based Algorithm 47

3.3.1 Hypergraph Construction 47

3.3.2 Hypergraph Clustering 49

3.3.3 Convergence 51

3.3.4 Alternating Optimization 52

3.4 Simulation Results 53

3.5 Summary 55

References 55

4 Conclusions and Future Works 57

4.1 Conclusions 57

4.2 Other Applications 58

4.2.1 Socially-Aware Content Delivery Networks 58

4.2.2 Codebook Assignment Using SCMA 59

4.2.3 Heterogeneous Cloud Radio Access Networks 60

4.2.4 Smartphone Sensing 61

References 61

3GPP Third Generation Partnership Project

5G Fifth Generation

AWGN Additive White Gaussian Noise

BS Base Station

CDF Cumulative Distribution Function

CDN Content Delivery Network

D2D Device-to-Device

eNB Evolved Node Base Station

H-Cran Heterogeneous Cloud Radio Access Networks

LTE Long-Term Evolution

MIMO Multiple Input Multiple Output

NOMA Non-orthogonal Multiple Access

OFDMA Orthogonal Frequency Division Multiple Access

QAM Quadrature Amplitude Modulation

Chapter 1

Basics of Hypergraph Theory

Graph theory is a useful tool to solve some problems in wireless communications,such as resource allocation [1], scheduling [2], and routing [3], etc However,the conception of edge in graph theory can only model the pairwise relation,which might not be sufficient to model the multiple users relation To model therelation among multiple users more accurately, such as cumulative interference, weintroduce the hypergraph theory [4] which allows any subsets of the vertices set to

be a hyperedge, instead of exactly two vertices defined in traditional graph As such,the hypergraph can achieve better approximation accuracy than the traditional graph

in wireless networks as it effectively captures the relation among multiple users

In this chapter, we first introduce the preliminaries of hypergraph theory In thefollowing, we present two problems: hypergraph coloring and clustering, which arewidely used to model the problems in wireless communications, and then provideefficient algorithms to solve these two problems, respectively

1.1 Basic Hypergraph Concepts

In this section, we first give the definitions of hypergraph, and then introduce theincidence and duality of hypergraph [5]

As shown in Fig.1.1, hypergraph is a generalization of a graph in which anysubset of a given set can be an edge rather than two-element subsets Specially,

© The Author(s) 2018

H Zhang et al., Hypergraph Theory in Wireless Communication Networks,

SpringerBriefs in Electrical and Computer Engineering,

DOI 10.1007/978-3-319-60469-5_1

1

23

the hyperedge which contains only one vertex is called singleton In what follows

we provide the definition of hypergraph which generalizes the respective graphconcept [6]

be a family of subsets of X such that

e i ¤ ; i D 1; 2; : : : ; m/;

m

S

The pair H D .X; E/ is called a hypergraph with vertex set X and hyperedge

set E The elements x1; x2; : : : ; x n of X are vertices of hypergraph H, and the sets

e1; e2; : : : ; e m are hyperedges of hypergraph H.

In a special case that a hyperedge is the subset of some other hyperedges, this

hyperedge is called included In special cases, some hyperedges may coincide, then they are called multiple A hypergraph is called simple if it does not contain

included hyperedges In this book, hypergraphs are referred to the simple ones In a

hypergraph, two vertices are said to be adjacent if there exists a hyperedge e i 2 E

that contains these two vertices And two hyperedges are said to be adjacent if their

intersection is not empty If a vertex x i 2 X belongs to a hyperedge e j 2 E, they are

called incident to each other.

In addition, E.x/; x 2 X denotes the set of all the hyperedges which contains vertex x The cardinality of E.x/, i.e., jE.x/j, is called the degree of vertex x The maximum degree of the hypergraph H is denoted by

.H/ D max

A hypergraph in which each vertex in the vertex set is with the same degree k> 0

is called k-regular, and a hypergraph in which each hyperedge in the hyperedge set

is with the same cardinality r > 0 is called r-uniform.

1.1 Basic Hypergraph Concepts 3

56

Fig 1.2 Example of a hypergraph H and its incidence matrix I .H/

with rows representing the vertices and columns representing the hyperedges of H

E D fe1; e2; : : : ; e m g The dual of the hypergraph H is a hypergraph H D Y; Z/ whose vertex set is Y D fy1; y2; : : : ; y mg, and the hyperedge set is given by

Z D fz1; z2; : : : ; z ng;

z i D fy j W x i 2 e j in Hg: (1.4)

Fig 1.3 Example of a hypergraph H and its dual H

Fig 1.4 Example of a graph G and its dual G

An example of the dual hypergraph is shown in Fig.1.3 Notice that edge e6is

singleton, and vertices 5 and 6 are incident to the same hyperedge Thus, in H,

vertex e6is of degree 1, and hyperedges 5 and 6 are multiple

It is important to mention that any graph as a special case of hypergraphs, has itsdual, which is not necessary to be a graph as well The duality of hypergraphs is apowerful tool, while we cannot use it if we are restricted by graphs only Figure1.4

shows an example of graph G such that its dual Gis a hypergraph The pendant

vertex 5 in G becomes a hyperedge with cardinality 1, and vertex 1 with degree 3

becomes a hyperedge with cardinality 3

The dual provides a convenience in the hypergraph construction: the vertices andhyperedges sets are equivalent, i.e., the vertices set of a hypergraph is the hyperedgesset of its dual hypergraph

1.1 Basic Hypergraph Concepts 5

to the sum of all hyperedge cardinalities, i.e.,

Proof Consider the incidence matrix I H/ If we sum the entries of incidence matrix

by columns, we obtain the left side of the equality And if we sum the entries ofincidence matrix by rows, we can obtain the right side of the equality Evidently,they coincide because both are equal to the sum of entries in the incidence matrix

uFor example in Fig.1.2, it gives

2 C 2 C 2 C 3 C 2 C 2 C 1 D 14 D 3 C 2 C 3 C 3 C 2 C 1: (1.6)

In hypergraph theory, there are a few basic operations which are used to obtain onehypergraph from another They are helpful in proofs of many theorems and useful

in many algorithms for solving optimization problems on hypergraph

delete all the hyperedges containing x from E, and delete x from X.

We provide an example of strong deletion of vertex x in Fig.1.5 If X1D X  fxg, and E1 D E  E.x/, then strong deletion of x from E results in obtaining a new hypergraph H1D X1; E1/ We denote this operation by H1D H  x By sequential

Fig 1.5 Strong deletion of x from hypergraph H

Fig 1.6 Weak deletion of x from hypergraph H

strong deletion of vertices, we can obtain a sequence of hypergraphs, and thisapproach is common and helpful in developing many algorithms and proving aseries of theorems by using mathematical induction

In some problems, there is a need to strongly delete an entire subset of vertices.This operation is equivalent to a sequential strong deletion of the respective vertices

in any order Such deletions is called strong because the vertices are also removedfrom a hypergraph along with the incident hyperedges

remove x from X and from each hyperedge in E.x/.

We provide an example of weak deletion of vertex x in Fig.1.6 If the same vertex

x is weakly deleted from hypergraph H, we will obtain a different hypergraph H1

We denote this operation by H1D Hnfxg As shown in Fig.1.6, the loop at vertex x becomes an empty hyperedge, the hyperedge of size 3 incident to vertex x becomes

a hyperedge of size 2 containing the remaining two vertices

remove hyperedge e i from E and then weak delete all the vertices contained by e i

from X.

An example of strong deletion of hyperedge e is shown in Fig.1.7 We strongly

delete hyperedge e and obtain a new hypergraph H1 In that case, we write H1 D

H  e to indicate that the strong deletion operation of hyperedge e.

remove hyperedge e i from E.

An example of weak deletion of hyperedge e is shown in Fig.1.8 It is the

simplest operation in a hypergraph We just remove hyperedge e and obtain a new hypergraph H1, and all the rest are remained We use H1D Hne to denote the weak deletion operation of hyperedge e.

Let us compare these four deletion operations In the incidence matrix, strongdeletion of a vertex corresponds to the deletion of the respective row and all the

1.1 Basic Hypergraph Concepts 7

Fig 1.7 Strong deletion of e from hypergraph H

Fig 1.8 Weak deletion of e from hypergraph H

columns which have intersections with that row In turn, the weak deletion of avertex corresponds to just remove the respective row As we have introduced inSect.1.1.2, transposition of the incidence matrix results in the incidence matrix ofthe dual hypergraph Rows in original hypergraph become the columns in the dualhypergraph, and columns become the rows Thus, removing of rows corresponds tothe removing of the columns in the transposed matrix, that is, strong (weak) deletion

of vertices is equivalent to the strong (weak) deletions of hyperedges in the dual Asfor weak and strong deletions, only in the special cases for isolated vertices andempty hyperedges, strong and weak deletions are the same

contrac-tion of hyperedge e is to weakly delete e from H, and then replace all the vertices

of e by one vertex belonging to each e02 E such that e0\ e ¤ ;.

Fig 1.9 Contraction of hyperedge e

Contraction of a hyperedge may significantly change the structure of a graph We provide an example of hyperedge contraction in Fig.1.9 We can easily

hyper-observe that hypergraph H is simple while H1is not As in graph theory, sequentialapplication of deletions to decompose a hypergraph and then to reconstruct it ininverse order is widely used in many algorithms

By strong and weak deletions of vertices and hyperedges from a hypergraph, we canobtain different types of subhypergraphs

such that X0 X, and E0 E can be called a subhypergraph of H.

Evidently, H0 can be obtained from H by strong deletion of the vertices and

further weak deletion of hyperedges As shown in Fig.1.10, H0is obtained by strongdeletion of vertex 3, and then weak deletion of hyperedge f5; 6g

of a hypergraph H D X; E/ if X0  X, and the hyperedges of H.X; E/ completely contained in X0 form the hyperedge set E0 We can also say that H0

is a subhypergraph induced by X0

An induced subhypergraph H0 is a special case of subhypergraphs, which can

be obtained from H by strong deletion of vertices If at least one hyperedge of H being a subset of X0 is empty, the subhypergraph is not induced In a hypergraph

H D X; E/, it is convenient to denote a subhypergraph induced by a set Y  X by

H In Fig.1.11, the subhypergraph H0is induced by vertex set Y D f3; 5g

1.1 Basic Hypergraph Concepts 9

Fig 1.10 Hypergraph H and subhypergraph H0

Fig 1.11 Hypergraph H and induced subhypergraph H0

H is called a partial subhypergraph.

Partial subhypergraphs have the same vertex set as hypergraph itself and can beobtained by weak deletions of some hyperedges

contains no edge of H is called the stable set, or independent set.

Independent set of vertices includes an empty subhypergraph The largestcardinality of an independent set over all maximal by inclusion independent sets

is called stability (independent) number, denoted by˛.H/ For all hypergraphs

whose vertices are not all singleton, we have1  ˛.H/  jXj For a hypergraph, if

we weakly delete all vertices of an independent set, there does not exist an empty

hyperedge in the obtained subhypergraph As the hypergraph H in Fig.1.11, verticesf1; 3; 6g form a maximal by inclusion stable set, but it is not a maximum stable set.Vertices f1; 2; 4; 5g form a maximum independent set, thus, ˛.H/ D 4.

In the wireless communication system, independent set is an important concept

in the resource allocation If we construct an interference hypergraph, where thevertices represent the devices and hyperedges represent the interference relationamong these devices if they share the same radio channel, the objective of resourceallocation is to find the independent sets corresponding to the respective channels

jT \ Dj  1 for every edge e i 2 E.

The cardinality of a minimum transversal is denoted by .H/ According to

Definition1.12, a set which subsets to X  T is an independent set Thus, we can

have the following equality:

In Fig.1.11, vertices f3; 6g form the minimum transversal of H Thus, .H/ D 2.

Evidently,˛.H/ C .H/ D 4 C 2 D 6 D jXj If we strongly delete the vertices in a

transversal from a hypergraph, the subhypergraph will be a hypergraph with emptyhyperedge set

pairwise have no common vertices is called a Matching.

A perfect matching is a matching which contains every vertex of a hypergraph.

The maximum size of a matching over all matchings is denoted by.H/ Since any

matching is a set of pairwise non-intersecting hyperedges, any transversal must have

at least one vertex from one hyperedge of the matching This fact implies that for

any hypergraph H,

As the hypergraph H in Fig.1.11, it can be easily observed that this hypergraphdoes not have a perfect matching, hyperedges f2; 6g and f3; 4; 5g form a maximummatching, thus,.H/ D 2 Any transversal of H must have at least one vertex from

each of hyperedges f2; 6g and f3; 4; 5g, therefore, .H/  2 D .H/

Covering if the union of all the hyperedges in C coincides with X.

The minimum number of hyperedges in a covering is denoted by.H/ We can see that each covering of H is a transversal in its dual Hand vice versa, therefore,

For the hypergraph H in Fig.1.11, hyperedges f4; 6g, f5; 6g, and f1; 2; 3g form aminimum covering Thus,.H/ D 3.

1.2 Hypergraph Coloring 11

1.2 Hypergraph Coloring

In this section, we first introduce three basic kinds of hypergraph coloring lems [7], and then present a greedy algorithm to solve these hypergraph coloringproblems

of a hypergraph H D X; E/ is a labeling of the vertices set X with the colors set

C such that every hyperedge e 2 E with jej  2 has at least two vertices coloreddifferently

We do not need to use all the colors in C In the coloring process, we ignore the

hyperedges with cardinality less than 1, because one vertex is only allowed to becolored with one color This can be obtained by weak deletion of these hyperedges.The proper-coloring is called weak coloring as well.

The minimum when there exists a proper -coloring is called the chromatic

number of different colors needed in coloring is jXj, thus, .H/  jXj.

A-coloring of H D X; E/ which actually uses k   colors defines a feasible partition of X into k stable color sets S1; : : : ; S k Each color set S irepresents that the

vertices in this set are colored with color i According to Definition1.16, there is nohyperedges within each color set, i.e.,

number, transversal number, and chromatic number by ˛.H/, .H/, and .H/, respectively Then, the following inequalities holds:

˛.H/.H/  n;

Proof In a coloring scheme with .H/ colors, the cardinality of each color set must

be less than˛.H/, i.e., jS i j  ˛.H/ Thus, we have

Fig 1.12 A proper

23

45

6

Fig 1.13 A strong

23

45

6

Choose a minimum transversal of H, and color them differently, we need .H/

colors Then, we color the remaining vertices with a new color, and obtain a proper

coloring of H with .H/C1 colors Thus, .H/ cannot be greater than .H/C1 ut

An example of a hypergraph with a proper 3-coloring is shown inFig.1.12 According to the aforementioned definitions, we can learn that

.H/ D 2; ˛.H/ D 4, and .H/ D 2 It is easy to check that the inequalities in

Proposition 1.2hold The coloring procedure partitions the vertex set into threecolor classes: green, blue and red Vertices with red and green represent a minimumtransversal, and vertices with blue are a maximum independent set

colors such that all the vertices are colored differently

Define the smallest when there exists a strong coloring as the strong chromatic

.H/  .H/ Specially, the strong and weak colorings coincide when H is a graph.

An illustration of a hypergraph with a strong 3-coloring is provided in Fig.1.13

where.H/ D 3.

with color sets.S1; : : : ; S/ is uniform if the number of vertices of the same color

is always to within one, i.e.,

1.2 Hypergraph Coloring 13

Fig 1.14 A uniform

23

45

6

An example of a hypergraph with a uniform 3-coloring is shown in Fig.1.14.These six vertices are equally partitioned into three color classes The uniformcoloring problem arises in many scheduling problems, which will be introduced

in Chap.4

Let H D X; E/ be a hypergraph, and let E.x/ denote the set of hyperedges which contains vertex x 2 X A star like structure which only has one unique center is

called a monostar in hypergraph A vertex x can be a center for many monostars,

and in the following, we give the definition of monodegree to present the largest

number of hyperedges in a monostar with x as the center [5]

is the maximum cardinality of a hyperedge subfamily E1.x/  E.x/ such that two elements e i ; e j 2 E1.x/, e i \ e j D fxg.

Intuitively speaking, the hyperedge set looks like a star, where vertex x is in the

center of the star If a graph has no loops, which implies that the two vertices in anedge are not the same, the monodegree is equal to the degree in the graph

Consider the value

Next, we introduce a greedy hypergraph coloring algorithm which is related

to the value of M.H/ The idea is to find an ordering of the vertices by first decomposing H using the monodegrees of the vertices Then color H successively.

To color each vertex, we use the first available color in the set of colors If theredoes not exist any color available for coloring this vertex, we will use a new colorand add it into the set of colors The algorithm is presented in Table1.1

Table 1.1 Greedy hypergraph coloring

 i D n, H n D H Find a vertex of the minimum monodegree in H n and label it x n.

we will elaborate on how to select available colors in these three basic kinds ofhypergraph coloring, respectively

In weak coloring, the available color set is equal to the color set C except one

special case If the vertex is the last uncolored vertex in this hyperedge, and other

vertices are colored by the same color c, the available color set is the color set excluding color c, i.e., Cnfcg.

In strong coloring, all the vertices in the same hyperedge need to be different

Thus, the available color set is the color set C excluding the colors used in the same

1.3 Hypergraph Clustering

In this section, we first introduce the mathematical expressions of hypergraphclustering problem, and then provide a clustering algorithm to solve this problemefficiently

1.3 Hypergraph Clustering 15

Given the weighted hypergraph H D X; E; W/, where W D Œw i ;j represents

the weight of vertex x i in hyperedge e j To characterize the vertex-to-hyperedgemembership, we define an indicator function as:

As the result, we can obtain an incidence matrix H D h.x i ; e j// for capturing the

vertex-to-hyperedge relationships Thus, the pairwise weight S D Œs i ;j is derived as:

s i ;jDP

e l

ıl h x i ; e l /h.v j ; e l/D

Hypergraph clustering seeks an optimal hypergraph cut solution for effective

clustering K-way normalized cut [9] is a well-known hypergraph clustering

cri-terion, which aims to optimally partition the vertex set X into K disjoint subsets V k

where Y is an N  K clustering matrix such that YTY is a diagonal matrix,1ddenotes

a d  1 vector with each element being 1, D is an N  N diagonal matrix with the m-th diagonal element being the sum of the elements in the m-th row of S, and Y nis

the n-th column of Y.

To simultaneously capture both intra-cluster compactness and the inter-clusterseparability among the vertices in a unified clustering framework, a discriminativehypergraph clustering criterion (DHCC) is adopted, which aims to solve thefollowing optimization problem:

max g.Y/ D 1

K K

As a result, an optimal hypergraph partitioning solution is obtained by

maxi-mizing g.Y/ in problem (1.20) For simplicity, let P n denote the vertex-to-cluster

membership vector associated with the n-th cluster such that P n D Y n YT

n Y n/1,

and P denote the vertex-to-cluster membership matrix such that P D P1P2:::P K/ D

Y YTY/1 It can be shown that P is an orthogonal matrix:

where YTY is a diagonal matrix.

According to the result in [9], we obtain

Y D Diag.diag 1

that is the corresponding inverse transform of P D Y.YTY/ 1

Here, Diag.:/ denotes a diagonal matrix formed from its vector argument, and diag.:/ represents

a column vector formed from the diagonal elements of its matrix argument

1.3 Hypergraph Clustering 17Consequently, the optimization problem in (1.20) can be rewritten as:

max g.Y/ D 1

K K

where tr.:/ denotes the trace of a matrix This is a trace-ratio-sum optimization

problem, which is nonconvex and difficult to solve Thus, we relax this optimizationproblem to the following sum-trace-ratio optimization problem:

Newton-QY is a real-valued hypergraph partitioning solution, and thus does not satisfythe discrete solution requirements for clustering As a result, an iterative refiningalgorithm proposed in [9] is utilized to find the optimal discrete hypergraph

Table 1.2 Hypergraph clustering algorithm

 Compute the graph Laplacian matrix Q D D  S.

 repeat

1 Compute the trace ratio tr tr .P .P T T QP SP//

2 Compute the K largest eigenvalues of S 

.P1P2: : : P K / as P.

 until The aforementioned steps are converged.

Given R, and let QY D QYR, problem (1.25) is transformed as

min e.Y; R/ D jjY  QYRjj2;

Given Y, the reduced problem, i.e.,

min e.Y; R/ D jjY QYRjj2;

Due to the orthonormal invariance of the continuous optima, our method is robust

to arbitrary initialization, from either Y or R A good initialization can nevertheless speed up convergence We initiate R in the following way:

are as orthogonal to each other as possible To derive Xon this non-orthogonal R

is exactly K-means clustering with the unit-length centers.

Given Y, we solve problem (1.28) to find a continuous optimum QYRclosest

to it Each step reduces the same objective e.Y; R/ through coordinate descent We

can only guarantee such iterations end in a local optimum, which may vary with theinitial estimation The iterative refining algorithm is shown in Table1.3

References 19

Table 1.3 Iterative refining algorithm

 Initiate R according to (1.30 ) and ( 1.30), and let e.Y; R/ D 0.

 repeat

1 Set e.Y; R/ D e.Y; R/.

2 Find the optimal discrete solution Yaccording to ( 1.27 ).

3 Find the optimal orthonormal matrix Raccording to ( 1.29 ).

4 Set e Y; R/ D tr.˝/ and RD VUT

 untilje.Y; R/  e.Y; R/j is less than predefined threshold.

References

1 A N Zaki and A O Fapojuwo, “Optimal and Efficient Graph-Based Resource Allocation

Algorithms for Multiservice Frame-Based OFDMA Networks”, IEEE Trans Mobile ing, vol 10, no 8, pp 1175–1186, Aug 2011.

Comput-2 J.-C Chen, Y.-C Wang, and J.-T Chen, “A Novel Broadcast Scheduling Strategy Using

Factor Graphs and the Sum-Product Algorithm”, IEEE Trans Wireless Commun., vol 5, no 6,

pp 1241–1249, Jun 2006.

3 H Zhu, H Zang, K Zhu, and B Mukherjee, “A Novel Generic Graph Model for Traffic

Grooming in Heterogeneous WDM Mesh Networks”, IEEE/ACM Trans Networking, vol 11,

no 2, pp 285–299, Apr 2003.

4 C Berge, Hypergraphs: combinatorics of finite sets North-Holland publishing company,

Amsterdam, Netherlands, 1989.

5 V I Voloshin, Introduction to Graph and Hypergraph Theory, Nova Science Publishers, New

York City, NY, 2008.

6 C Berge, Graphs and Hypergraphs North-Holland publishing company, Amsterdam,

Nether-lands, 1973.

7 V Voloshin, Coloring Mixed Hypergraphs: North-Holland publishing company, Amsterdam,

Netherlands, 2002.

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Radio Resource Allocation for Device-to-Device Underlay Communications

2.1 Introduction

With the increasing demand for local traffic, device-to-device (D2D) tions under the control of evolved Node B (eNB) have recently received a greatdeal of attention [1 4] Reusing the same spectrum as in, not as for the cellularcommunications, user equipments (UEs) in a cellular network in proximity can set

communica-up direct transmissions, which potentially increases the overall spectral efficiency[5] In the Third Generation Partnership Project (3GPP), UEs are provided with

a resource pool (time and frequency) in which they attempt to receive schedulingassignments, and eNB controls whether UEs may apply scheduled or autonomousD2D transmissions [6] However, D2D communications generate interference to thecellular network if the radio resources are not properly allocated [7 9] In addition,multiple D2D pairs in the same channel also create mutual interference [10] Thus,interference management becomes one critical issue for D2D communicationsunderlaying cellular networks

In the literature, much attention has been paid to manage the interference inD2D networks The studies in [11] propose a radio resource allocation algorithmusing fractional frequency reuse to alleviate the interference between D2D pairs andcellular UEs The work in [12,13] tackles the interference management issues fromthe economy perspectives In [12], the authors formulate the resource allocationproblem as a reverse iterative combinatorial auction game, and propose a jointradio resource and power allocation method to increase energy efficiency In [13],

a sequential second price auction mechanism is designed to allocate the spectrumresources for D2D communications with multiple user pairs

As shown in the literature, though D2D communication may generate additionalinterference to cellular systems, it improves the system throughput with properinterference management [14] Therefore, the allocation of radio resources for D2Dunderlay communications needs further studies for efficient solutions with low

© The Author(s) 2018

H Zhang et al., Hypergraph Theory in Wireless Communication Networks,

SpringerBriefs in Electrical and Computer Engineering,

DOI 10.1007/978-3-319-60469-5_2

21

22 2 Radio Resource Allocation for Device-to-Device Underlay Communications

complexity Graph theory is a useful tool to solve this kind of resource allocationproblems in wireless communications [15,16] With graph theory, cellular UEs andD2D pairs are modeled as vertices in a graph, and the interference links betweenthe UEs are constructed as edges [17,18] In [17], the weight of edges is used

to represent the interference between two vertices, and the channel allocation

is to iteratively gather vertices from the corresponding channel, taking both theinterference value and the cluster value into account In [18], the system model

is constructed as a weighted bipartite graph, and the channel allocation problem isformulated as a matching problem to maximize the capacity

However, it is worth mentioning that the conception of edge in graph theorymight not be sufficient in modeling the interference relation due to the cumulativeeffect of the interference Specifically, the interference from several vertices mayconstitute a strong interferer, even though the interference from each individualvertex is weak [19,20] When the cumulative interference from neighboring D2Dpairs or cellular UEs exceeds a threshold, it may reduce the communication quality

of all the users Hence, it is necessary to take into account the cumulative impact ofmultiple interference sources to the cellular UEs and D2D pairs as victims

To this end, in this paper, we use the hypergraph to solve the interferencemanagement problem for D2D communication underlaying cellular networks

A hypergraph is a generalization of an undirected graph, in which the hyperedgesare any subsets of the given set of vertices, instead of exactly two vertices defined

in the traditional graph [21] In wireless networks, the hypergraph achieves betterapproximation accuracy than the traditional graph as it effectively captures thecumulative interference As such, the system capacity can be further improved bythe hypergraph based method, compared to the traditional graph approach [22].The goal of this chapter is to develop a resource allocation model for D2Dcommunications underlaying cellular networks, in which the data sum-rate of D2Dpairs and cellular UEs are maximized We first formulate a resource allocationproblem for multiple D2D pairs sharing channel resources with one cellular UE tomaximize the cell capacity Subsequently, we study the resource allocation problemusing hypergraph theory A hypergraph coloring method with low complexity isproposed to address the channel allocation for both D2D pairs and cellular UEs,which effectively increases the cell capacity Simulation results show that theproposed hypergraph based method can achieve a performance very close to theoptimal result, and perform much better than the traditional graph based method

2.2 System Model and Problem Formulation

As shown in Fig.2.1[23], we consider an uplink transmission scenario in a cellular

network that consists of N cellular UEs and M D2D pairs We denote a cellular UE

by U n,1  n  N, and a D2D pair by D m,1  m  M Here, we use D t to represent

InterferenceD2D links

multiple access for both the cellular and D2D communications, where a set of K

channels are available for resource allocation In this system, the eNB coordinatesthe resource allocation between cellular UEs and D2D pairs We assume that D2D

pairs transmit with the power denoted by P d, and cellular UEs use the transmission

power P c

The channel is modeled as a Rayleigh fading channel, and the channel gains can

be calculated by

8ˆˆˆˆ

24 2 Radio Resource Allocation for Device-to-Device Underlay Communications

where L n c , L t m ;r , L t m , L c n ;r ;m , and L t i ;r ;mdenote the corresponding distance-dependent path

loss, and h c , h t ;r

m , h t

m , h c ;r

n ;m , and h t i ;r ;mdenote the fading channel, respectively,1  n 

N, 1  m  M, 1  i  M, and i ¤ m The thermal noise satisfies independent

Gaussian distribution with zero mean and variance2.

The instantaneous SINR of the received signal at the eNB from cellular UE U n

over channel k can be written as

and the instantaneous SINR at the D2D receiver D r

m over channel k is given by

where A NK D Œ˛n ;k represents the channel allocation matrix for the cellular UEs,

and B MK D Œˇm ;k stands for the channel allocation matrix for the D2D pairs,

1  n  N, 1  m  M, 1  k  K The value of ˛ n ;kandˇm ;kare defined as

Our objective is to maximize the cell capacity by optimizing the channelallocation variables f˛n ;kI ˇm ;kg for the cellular UEs and D2D pairs, which can beformulated as

Note that the aforementioned resource allocation problem in (2.7) is a NP-hardcombinatorial optimization problem with nonlinear constraints, graph coloring is

an approximate and efficient method for such a resource allocation problem [24]

Thus, we formulate the channel resources as K different colors, the cellular UEs as N (cellular) vertices, and the D2D pairs as M (D2D) vertices in the

plane Consequently, the channel allocation problem is transformed into a coloringproblem of the vertices with fixed colors [25] In the following two subsections, wewill demonstrate the graph and the hypergraph based methods, respectively

2.3 Traditional Graph Based Channel Allocation

Before introducing the hypergraph based channel allocation method, we describethe conventional graph based method In a graph, vertices represent the cellularUEs and the D2D pairs, and edges indicate that the interference between connectedvertices does not allow them to use the same channel simultaneously [26] The graphbased method contains the graph construction and the channel allocation algorithm

as follows

We transform the interference information into a graph A cellular UE U nand a D2D

pair D mare connected by an edge which satisfies that the wanted signal ratio to theinterference is below a threshold:

26 2 Radio Resource Allocation for Device-to-Device Underlay Communications

P c g c

P d g t m

< ıcI at the eNB receiver; (2.9)or

P d g t ;r

m

P c g c n ;r ;m < ıd I at the D2D receiver D m; (2.10)where ıc and ıd are the thresholds selected to determine the severity of theinterference at the eNB and the receiver of a D2D pair, respectively Two D2D pairs

D i and D mare connected by an edge if

g t m ;r

g t i ;r ;m < ıd I at the D2D receiver D m; (2.11)which indicates that if the interference from another D2D pair is strong, these two

D2D pairs cannot share the same channel Besides, two cellular UEs U i and U j

always form an edge for the assumption that two cellular UEs cannot share thesame channel In this way, an interference graph is constructed

After the graph construction, we use the greedy coloring algorithm in [24] to colorthe constructed graph We define the available color set by all the colors exceptthe colors used in the connected vertices The algorithm successively colors thevertices in a color randomly chosen in the corresponding available color set, in thedescending order of degree If the available color set becomes empty, the vertexremains uncolored In this way, the cellular UEs and the D2D pairs are classified intoclusters with different colors, where the colors represent the channels Finally, thechannels are allocated to the D2D pairs and cellular UEs with mutual interferencebelow the given threshold These detailed algorithms are shown in Table2.1

2.4 Hypergraph Based Channel Allocation

In the traditional graph based method of Sect.2.3, the edge connecting two vertices

is not sufficient to model the interference in a wireless network, because some weakinterferers together may constitute a strong cumulative interferer to affect the linkquality In this subsection, the hypergraph method, in which a hyperedge containsseveral vertices, is used for interference modeling