Impact of Primary Users on the Connectivity of a

Cognitive Radio Network

Francesca Cuomo, Anna Abbagnale, Alessandro Gregorini

INFO-COM Dept, University of Rome “Sapienza”, Via Eudossiana 18, 00184, Rome, Italy

Tel. +39 06 44585640, e-mail: francesca.cuomo@uniroma1.it,abbagnale@infocom.uniroma1.it, gregalex@alice.it

Abstract—We analyze the impact of primary users on thesecondary network connectivity in a cognitive radio network. Theanalysis is based on the second smallest Laplacian eigenvalue, i.e.,the algebraic connectivity, re-elaborated in a cognitive scenario.The contribution of this paper is twofold: first we derive theform of the average Laplacian matrix of the network, averagedover the random activity of the primary users, and computethe second smallest Laplacian eigenvalue of this matrix. Wederive in this way the compact cognitive algebraic connectivityand we show that, in the scenarios we are interested in, itrepresents a valid estimation of the expected value of the cognitivealgebraic connectivity. The second contribution is the evaluationof the impact of different topological parameters on the compactcognitive algebraic connectivity.

I. INTRODUCTION

The advent of software defined radios is making possible

opportunistic access of temporarily unoccupied licensed spec-

trum bands by unlicensed Cognitive Radio (CR) nodes. These

nodes are assumed to be able to sense the spectrum and to

tune their transceiver parameters in order to establish commu-

nication links by occupying unused portions of spectrum.

In this way, a set of CR nodes can be arranged to form a self-

operating network called Cognitive Radio Ad-Hoc Network

(CRAHN) [1], [2]. In this scenario there are two kinds of

actors: Primary Users (PUs) and Secondary Users (SUs).

PUs are licensed users with high priority in the utilization

of the spectrum; SUs instead are cognitive devices able to

sense the spectrum and detect available Spectrum Opportunity

(SOP) to be used for transmission. SOPs are defined as a

set of frequency bands currently unoccupied by PUs and,

therefore, available for SUs. A SU can opportunistically access

a spectrum band not used by its licensed PU and it has to

immediately relinquish this channel when the PU becomes

active.

The key distinguish factors of the secondary networks is

that the topology is highly dynamic and nodes should coop-

erate in a distributed fashion to achieve network connectivity

and communication. The fact that the communication links

of the secondary networks vary as a function of the PUs

activities makes the design and the analysis of these networks

very challenging. In classical wireless Ad-Hoc network nodes

communicate by using the same frequency and distance among

nodes combined with the adopted transmission power are the

only parameters affecting network connectivity.

On the contrary, in a CRAHN also the physical location of

the PUs affects the SU network topology. In this framework,

the concept of connectivity has to be revised and worked up,

because not only the nodes’ position but also their commu-

nication frequency bands affect network connectivity. In our

initial work in [3] we laid the foundations of a mathematical

model for computing the k-connectivity (k = 1, 2, . . . ) of

a CRAHN. In the literature several works dealt with the k-

connectivity in wireless multi-hop Ad-Hoc networks ([4], [5],

[6]), seeking a power assignment to the nodes such that the

resulting communication network is k-connected and the total

power is minimum. We believe that also for CRAHNs the

analysis of the k-connectivity represents a fundamental step for

the network design and planning. To this aim in this work we

present the mathematical framework, based on the Laplacian

spectrum of graphs, to evaluate the CRAHN k-connectivity.

We then use this model to evaluate the impact of PUs behavior

on the secondary network connectivity. By discussing the

model we also present a new and key result of ours that

is the possibility to compute the CRAHN connectivity on a

model for the secondary network that is a cognitive graph

where edges are weighted with average activity factors of the

PUs. We derive in this way the Compact Cognitive Algebraic

Connectivity (CCAC) and we show that in all scenarios we

are interested in it represents a valid estimation of the expected

value of the cognitive algebraic connectivity.

The remainder of the paper is organized as follows. In

Section II we introduce the Laplacian matrix and the relation-

ships that can be established with the network k-connectivity.

We then present the cognitive radio scenario (Section III)

and formalize the model for evaluating the expected value

of the cognitive algebraic connectivity in CRAHNs (Section

III-A) and the unified approach for this evaluation by means

of the CCAC (Section III-B). In Section IV we present

some numerical results of our analysis. Finally, the paper is

concluded in Section V.

II. LAPLACIAN MATRIX IN GRAPH THEORY

In graph theory the connectivity of a network is evaluated by

using the Laplacian matrix. In this Section we provide some

definitions and theorems related to the connectivity analysis

in graph theory.

Definition 1: A bidirectional graph G(N , E), where N is

the set of nodes, with cardinality N = |N |, and E is the set of

edges, is simple if a) there is at most one edge between any

978-1-4244-8435-5/10/$26.00 ©2010 IEEE

2

pair of nodes and b) there are no loops (no edges of the form

(i, i), with i ∈ E)).

Definition 2: A bidirectional graph G is connected if for

each couple of nodes in G there exists at least one path

between them. It is possible to quantify the graph connectivity

with two parameters: node connectivity and edge connectivity.

Definition 3: The node connectivity of a graph G, denoted

by κn(G), is equal to the minimum number of nodes whose

deletion from G causes the graph to be disconnected or reduces

it to a 1-node graph. A graph G is k-node connected if

κn(G) > k.

Definition 4: The edge connectivity of a graph G, denoted

by κe(G), is equal to the minimum number of edges whose

deletion from G causes the graph to be disconnected or reduces

it to a 1-node graph. A graph G is k-edge connected if

κe(G) > k.

Definition 5: The adjacency matrix of a graph G, denoted

with A, is a N ×N binary matrix and the generic element aij

is:

aij =

{

1 if (i, j) ∈ E0 otherwise

(1)

The adjacency matrix of simple graph is symmetric and has

all diagonal elements equal to 0.

Definition 6: The degree of any node i in a graph G, denoted

by degi, is equal to the number of edges incident on i; i.e.,

degi =∑

j aij =∑

j aji. The N × N matrix D contains the

degree of each node. The generic element dij is:

dij =

{

degi if i = j0 otherwise

(2)

Definition 7: The Laplacian matrix of a graph G, denoted

by L, is a N ×N matrix, computed as the difference between

D and A, where the generic element lij is:

lij =

{

dij if j = i−aij otherwise

(3)

For a simple bidirectional graphs, L is symmetric and all

its row and column sums are equal to 0 (that is L is singular).

Definition 8: The Laplacian eigenvalues of G are the roots

of the characteristic polynomial of L. Since L is symmetric, all

its eigenvalues are real. The eigenspectrum of L is the set of its

N eigenvalues that can be ordered according to the magnitude,

from the smallest to the greatest (λ1 6 λ2 6 ... 6 λN ). The

notation λk indicates the kth eigenvalue.

Theorem 1: The smallest eigenvalue of the Laplacian of

a bidirectional graph G is equal to 0 (i.e., λ1 = 0) and

the multiplicity of 0 as an eigenvalue of L is equal to the

number of connected components of G ([7]). Consequently,

λ2 = 0 iff G is disconnected: λ2 is generally called algebraic

connectivity.

The literature presents several properties of λ2 related to the

performance of a network modeled by a graph G:

• Jamakovic et al. [8] have shown that the algebraic con-

nectivity measures stability and robustness of complex

network models; a small perturbation in the network

Fig. 1. Coverage area associated to a PU

configuration will be attenuated back to the equilibrium

with a rate proportional to λ2;

• the work of Mohar [9] discussed the relationship between

the average of all distances between distinct vertices of

a graph G and λ2; this average distance is inversely

proportional to the algebraic connectivity.

A further property is that for any bidirectional graph G, the

second eigenvalue of its Laplacian is upper bounded by its

node connectivity.

Theorem 2: For any bidirectional graph G, the second

eigenvalue of its Laplacian is upper bounded by its node

connectivity, which in turn is upper bounded by its edge

connectivity:

λ2 6 κn(G) 6 κe(G) (4)

For all these reasons we believe that the concept of algebraic

connectivity, suitably extended taking into account PUs’ be-

havior, can be used to capture very fundamental aspects of

CRAHNs characterized by dynamic topologies.

III. COGNITIVE RADIO AD-HOC NETWORK MODEL

We consider Np PUs randomly scattered in a region where

Ns SUs exist. Both SUs and PUs are assumed motionless.

Each PUp (p = 1, ..., Np), represents a primary network

composed by a primary transmitter and a set of primary re-

ceivers that have a licensed access to a given spectrum portion

denoted as channel cp. The number of PUs is then equal to the

number of channels. The coverage area CAp associated to the

PUp is the area where a SU cannot transmit/receive without

causing/receiving an interference in the licensed spectrum cp.

As illustrated in Figure 1, the transmitting SU A can transmit

to the SU B if the transmission from A does not interfere with

nearby primary receivers. Furthermore, the correct reception

of the SU B can happen only if B is not affected by nearby

primary transmitter. This PU model may represent a base

station of a cellular network which manages its own cell by

providing on the spectrum band cp services to its mobile

stations or a TV broadcaster operating on its own UHF

spectrum [10]. In both cases SUs may transmit on channel

cp when the primary users do not use it. For the sake of

convenience in Table I we report the frequently used notations.

3

TABLE INOTATIONS

Np Number of primary users

{cp} Set of available licensed channels

CAp Coverage area associated to PUp

bp Binary aleatory variable representing the activity state of PUp

ap Average activity factor of PUp

Ns Number of secondary users

L Laplacian matrix

λ2 Second smallest eigenvalue of the Laplacian matrix

We suppose that each PUp is characterized by an on-off

transmission and we associate to each PUp a binary aleatory

variable bp that represents the activity state of PUp, that is:

bp =

{

1 if the PUp is active0 otherwise

(5)

Consequently, we can characterize each PUp with its average

activity factor:

ap = E[bp] (6)

The average activity factor of each PU can be numerically

calculated by measuring the average duration of the PU

activity period and the average duration of the relevant silence

period. Besides, if we indicate with Ep the event ”PUp is not

active”(with p = 1, 2, ...Np), the probability Pr(Ep) that this

event happens is equal to 1 − ap. We assume that two events

Ep and Eq , with p, q = 1, 2, ..., Np and p 6= q are independent.

Each SU opportunistically exploits locally unused licensed

spectrum bands without interfering with PUs. Therefore, a SU

can transmit on cp either when it does not belong to CAp

or when it belongs to CAp and PUp is inactive (meaning

that PUp in not transmitting nor receiving). If one of these

two cases occurs, the channel cp is available for that SU. All

SUs transmit with the same maximum transmission power and

the correspondent transmission range is indicated with r. We

assume a spectrum sensing able to measure the average value

of the PUs activity factors on the different spectrum bands

[11]. We also suppose that SUs can potentially use all the cp

channels, with p = 1, ..., Np.

A. Expected value of cognitive algebraic connectivity

In Section II we listed some important properties of the

algebraic connectivity. In a cognitive scenario, this parameter

dynamically varies in accordance to PUs’ behavior. This

consideration depends on the fact that the graph that models

the secondary network is a dynamic one, since its edges vary

each time a PU activates or deactivates. For this reason, to

model the secondary network, we introduce the concept of

cognitive graph Gc(Ns, Ec), where Ns is the set of secondary

nodes with cardinality Ns and Ec is the dynamic set of

edges: an edge (i, j) ∈ Ec iff nodes i and j are in each

other’s transmission range and they have at least an available

common channel. In this way, the cognitive graph reflects the

dynamic changes of the secondary network topology due to

PUs behavior. By fixing positions and transmission ranges of

SUs, the presence of an edge (i, j) in the graph Gc is function

of the Np binary aleatory variable bp. The number of all the

possible combinations of these Np binary aleatory variables is

M = 2Np .

For the m− th combination (m = 1, 2, ..., M ), the graph Gcm,

characterized by the set Ecm, is obtained and consequently a

cognitive Adjacency matrix Acm and a cognitive Degree matrix

Dcm can be provided. These matrixes are different from a

combination to another. The generic element of the cognitive

Adjacency matrix related to the m− th combination (Acm) is:

acij,m =

0 if dist(i, j) > r

1 if (dist(i, j) ≤ r) ∧

(∃ PUz |i /∈ CAz ∧ j /∈ CAz)

¬∧Np

p=1bp,m otherwise

(7)

where m = 1, 2, ...,M , bp,m is the value of the aleatory

variable bp in the m − th combination and dist(.) is the

Euclidean distance. The generic element of Dcm is:

dcij,m =

{

∑Ns

j=1ac

ij,m if i = j

0 otherwise(8)

with m = 1, 2, ...,M . The cognitive Laplacian matrix related

to the m − th combination Lcm is obtained as difference

between Dcm and A

cm, therefore its generic element is:

lcij,m =

{

dcij,m if i = j

−acij,m otherwise

(9)

We name the second smallest eigenvalue of the matrix Lcm

cognitive algebraic connectivity related to m−th combination,

λc2,m. This parameter measures the algebraic connectivity of

the secondary network when the sets of active PUs and inactive

PUs are fixed. We are interested in the expected value of

cognitive algebraic connectivity E[λc2] (named ECAC), since

in this way we obtain a connectivity measure averaged over

the random activity of the PUs. ECAC is given by:

E[λc2] = ECAC =

M∑

m=1

Pr(m) · λc2,m (10)

where Pr(m) is the probability of occurrence of the m − thcombination. This probability is a function of PUs’ activity

factors and, by assuming that the activations of PUi and PUj

are independent (for i, j = 1, ...., Np, i 6= j) it is equal to:

Pr(m) =

Np∏

p=1

(1 − ap − bp,m) · (1 − 2 · bp,m) (11)

where m = 1, 2, ...,M .

With this approach it is possible to calculate the ECAC of a

secondary network with a given number of SUs and a given

number of PUs characterized by their activity factors. Since the

computation of the ECAC requires the computation of 2Np

cognitive Laplacian matrices and for each of them the relevant

second smallest eigenvalue, in the next section we introduce a

methodology that reduces the computational complexity of this

4

average measure and obtains a good estimation of ECAC. We

named this parameter compact cognitive algebraic connectivity

(CCAC) to underline that it is obtained from only one

averaged Laplacian matrix.

B. From ECAC to the compact cognitive algebraic connec-

tivity

As stated before the secondary network topology changes

when a PU becomes active or inactive: as a consequence it

is not possible to define the set Ec in a deterministic fashion.

However, we can characterize this set in a probabilistic way,

indicating the probability that a generic link (i, j) belongs to

Ec, as function of PUs activity factors. To this aim we calculate

the expected value of the Ns × Ns cognitive Adjacency

matrix E[Ac] and the expected value of the Ns×Ns cognitive

Degree matrix E[Dc], whose elements are probabilistic ones

depending on PUs’ activity. In order to obtain these two

matrices, we define the Np×Ns Influence matrix I. The latter

matrix indicates for each PUp which are the SUs within its

coverage area CAp: these SUs can not use the channel cp

when p is active. The generic element µpj of matrix I is:

µpj =

{

1 if SUj ∈ CAp

0 otherwise(12)

The generic element E[acij ] of the matrix E[Ac] represents the

probability that SUs i and j are connected, that is equal to:

• 0 if SUs i and j are not in each other’s transmission

range;

• 1 if SUs i and j are in each other’s transmission range

and they have at least one available common channel

regardless of PUs’ activity (this is true when there is at

least a PU that does not influence nodes i and node j);

• 0 < 1 −∏Np

p=1ap < 1 if SUs i and j are in each other’s

transmission range but the possibility to have at least one

available common channel depends on PUs’ activity.

The term 1−∏Np

p=1ap is the probability that at least one PU is

inactive, that is equal to the probability that at least one event

Ep happens (probability of the union event):

π = Pr(∪Np

p=1Ep) (13)

This means that it is sufficient that at least one PU is inactive

to guarantee the connectivity between two SUs adjacent in

terms of distance. In fact, two SUs i and j, that are in each

other transmission range but that do not have any available

channel in common regardless of PUs, can communicate by

using a given channel in one of these three cases:

• a PUp influencing the SUi and not the SUj (i ∈CAp, j /∈ CAp) becomes inactive; in this case i and

j can communicate by using the channel cp, since it

is always available for SUj and in that moment it is

experienced free also by SUi;

• a PUp influencing the SUj and not the SUi (j ∈CAp, i /∈ CAp) becomes inactive; in this case i and j can

communicate by using the channel cp, since it is always

available for SUi and in that moment it is experienced

free also by SUj ;

• a PUp influencing both i and j (i ∈ CAp, j ∈ CAp)

becomes inactive; in this case i and j can communicate by

using the channel cp because this channel is temporarily

available for both these SUs.

Therefore the generic element E[acij ] is:

E[acij ] =

0 if dist(i, j) > r

1 if (dist(i, j) ≤ r) ∧

(∃ PUz | µzi ∨ µzj = 0)

π otherwise

(14)

The generic element E[dcii] of the matrix E[Dc] represents

the probabilistic number of edges incident to i, that is the

number of edges incident to i weighted with the probability

that each edge incident to i is present. The probability that an

edge between nodes i and j is present is equal to the prob-

ability that these two nodes are in each other’s transmission

range and have at least one available common channel.

The generic element E[dcij ] can be expressed as:

E[dcij ] =

∑Ns

j=1E[ac

ij ] =

= n · 1 + h · π if i = j

0 otherwise

(15)

where:

• n is the number of i’s neighbors, in terms of radio

visibility, which satisfies E[acij ] = 1; in this case the

probability that an edge between nodes i and j is present

is equal to 1 because nodes i and j have always at

least one available common channel regardless of PUs’

activity;

• h is the number of i’s neighbors, in terms of radio

visibility, which satisfies E[acij ] = π; in this case the

probability that an edge between nodes i and j is present

is equal to π.

By computing the difference between E[Dc] and E[Ac] we

derive the expected value of the Ns × Ns Laplacian matrix,

named average Laplacian matrix, E[Lc] where the generic

element is defined as:

E[lcij ] =

{

E[dcii] if i = j

−E[acij ] otherwise

(16)

In accordance with the E[acij ] and E[dc

ij ] definition we can

notice that E[Lc] is symmetric and all its row and column

sums are equal to 0, as the matrix L: this means that the

matrix E[Lc] has the same properties of the Laplacian L. The

main difference between the generic element of L and E[lcij ] is

that the former is a deterministic value, whereas the latter is a

probabilistic one and it is a function of PUs activity factors. In

fact, E[lcij ] depends on the behavior of PUs characterized by

their activity factors: therefore each value E[lcij ] is function

of the probability π = Pr(∪Np

p=1Ep) = 1 −

∏Np

p=1ap. Con-

sequently, also the value of the second smallest eigenvalue

(λ2{E[Lc]}) is a function of the same probability:

λ2{E[Lc]} = λ2(π) (17)

5

(a) (b) (c)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.15

0.2

0.25

0.3

0.35

0.4

Activity factor of PU2

Co

gn

itiv

e a

lge

bra

ic c

on

ne

ctivity

CCAC

ECAC

(d)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.45

0.5

0.55

0.6

0.65

Activity factor of PU2

Co

gn

itiv

e a

lge

bra

ic c

on

ne

ctivity

CCAC

ECAC

(e)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.2

1.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

1.4

Co

gn

itiv

e a

lge

bra

ic c

on

ne

ctivity

Activity factor of PU2

CCAC

ECAC

(f)

Fig. 2. Network topology affected by PUs activity: (a) potential 1-node connected; (b) potential 2-node connected; (c) potential 3-node connected; (d)ECAC and CCAC as a function of activity factor of PU2 for the network topology (a) with a1 = 0.3 and a3 = 0.2 ; (e) ECAC and CCAC as functionof activity factor of PU2 for the network topology (b) with a1 = 0.3 and a3 = 0.2; (f) ECAC and CCAC as function of activity factor of PU2 for thenetwork topology (c) with a1 = 0.3 and a3 = 0.2.

We refer to this second smallest eigenvalue of the average

Laplacian matrix as CCAC. It is interesting to study the trend

of the CCAC when the probability π varies, that is when the

activity factors of PUs vary. In fact, by analyzing this trend

it is possible to determine the probability π (i.e., the activity

factor of PUs) that allows to obtain the desired value of the

CCAC and consequently the k-connectivity.

Let κcn be the cognitive node connectivity . The Theorem 2

continues to be valid also in cognitive networks, because the

PUs’ activity impacts both on CCAC and on κcn. If we want

to obtain a cognitive network with the constraint κcn = k, it is

possible to replace this condition with an heuristic one [12]:

CCAC > k − 1 (18)

In fact if CCAC > k − 1 ⇒ κcn ≥ λc

2> k − 1 ⇒ κc

n is

at least equal to k, considering that κcn is an integer. In case

of 1-connectivity the condition that needs to be satisfied is

CCAC = 0, as stated in Theorem 1.

IV. PERFORMANCE ANALYSIS

A. On the comparison between ECAC and CCAC

In Section III we defined ECAC and the CCAC. It is to

be noticed that in general these two parameters have not the

same value. We implemented the computation of both these

parameters, by using matlab, in order to have a numerical

comparison between the two. This analysis is provided to

estimate if it is possible to use CCAC instead of ECAC. We

computed both values in a large number of topologies with

different number of SUs and PUs and for each topology we

calculated the percentage difference between the two values

defined as:

δ[%] =CCAC − ECAC

ECAC∗ 100 (19)

This numerical study pointed out that:

1) ECAC and CCAC have the same behavior as a func-

tion of PUs’ activity;

2) δ[%] decreases when the activity factors of PUs decrease

and when the node connectivity calculated without con-

sidering PUs increases;

3) given two different network topologies TA and TB , if

ECACTA< ECACTB

then CCACTA< CCACTB

.

As for the points 1) and 2), we considered different topologies

and analyzed the trend of ECAC and CCAC as a function

of activity factor of one PU, whereas all the others PUs

activities are fixed. In all these cases we obtained that both

ECAC and CCAC are monotonically decreasing function of

the PU activity: this means that both are able to capture the

impact of PUs’ behavior on network connectivity. Besides, it

is possible to notice that, when the activity factors of one

PU increases, CCAC decreases slower than ECAC, i.e.,

the difference between ECAC and CCAC increases when

the activity factor of PU increases. However, since the FCC

estimates that the average utilization of licensed bands varies

6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11%

2%

3%

4%

5%

6%

7%

8%

9%

10%δ [

%]

Activity factor of PU2

1−connectivity

2−connectivity

3−connectivity

Fig. 3. Percentage difference between ECAC and CCAC for the networktopologies shown in Figures 2(a), 2(b) and 2(c).

between 15-85% [13], we are more interested in scenarios

where the PUs activity factors are low. In this case the

percentage difference between these two measures is very low.

Moreover, these results highlighted that δ[%] decreases when

the node connectivity, calculated without considering PUs (in

the following named potential node connectivity), increases.

In order to clarify this concept, we show the results obtained

in three network scenarios with the same number of SUs

but a different potential node connectivity, that are shown in

Figures 2(a), 2(b) and 2(c), where links between SUs represent

radio visibility relationships and circles indicate PUs’ CA. In

Figures 2(d), 2(e) and 2(f) we plot the trend of ECAC and

CCAC as a function of activity factor of PU2 for network

topologies potential 1-node connected, 2-node connected and

3-node connected. In all the three cases we can observe

that both ECAC and CCAC are monotonically decreasing

function of the PU activity. Figure 3 shows that the percentage

difference between ECAC and CCAC decreases when the

activity factors of PU2 decreases and when the potential node

connectivity increases. However, it is possible to notice that

this percentage difference does not exceed the 10%. This figure

confirms that the difference between ECAC and CCAC is

very small when the average utilization of primary bands is

low, that is the case we are interested in.

In order to generalize the last concept, we considered

network topologies with 3 PUs and five different sets of SUs

(Ns=8, 10, 12, 14, 16). For each number of SUs we randomly

generated 100 topologies, varying each time the position of

the SUs, and calculated the percentage differences δ[%], by

using Equ. (19); then we averaged all these values obtaining

the mean percentage difference δmean[%]. These results are

shown in Figure 4, where for each value of Ns it is represented

the mean percentage difference between CCAC and ECACaveraged on 100 randomly generated network topologies. It is

possible to notice that δmean[%] increases when the activity

factors of PUs increase, but it remains under 10% also when

the average activity factor of PUs is equal to 0.6. In Table II

we report the mean value of CCAC and ECAC respectively,

averaged on the 100 random topologies in case of Ns=12.

Moreover, when the aim is comparing the stability of two

different secondary networks, it is possible to indifferently use

8 9 10 11 12 13 14 15 160

2%

4%

6%

8%

10%

12%

14%

16%

Number of SUs

δm

ea

n [%

]

a1=0.2 − a

2=0.5 − a

3=0.2

a1=0.2 − a

2=0.7 − a

3=0.3

a1=0.4 − a

2=0.5 − a

3=0.4

a1=0.4 − a

2=0.8 − a

3=0.3

a1=0.6 − a

2=0.9 − a

3=0.5

Fig. 4. Mean percentage difference between ECAC and CCAC for randomnetwork topologies with 3 PUs and 5 different values of SUs.

CCAC and ECAC if these two parameters define the same

order relationship on the set of all possible topologies. In other

words, we are not interested in the absolute value of ECAC,

but it is important that if ECAC calculated in a given network

topology TA is lower than ECAC calculated in a different

network topology TB , then CCAC calculated in the network

topology TA continues to be lower than CCAC calculated in

the network topology TB :

if ECACTA< ECACTB

then CCACTA< CCACTB

(20)

In order to numerically verify this property we performed

1000 different tests: during each test we generated two differ-

ent topologies with 3 PUs, choosing the number of SUs and

their position in a random fashion. Then we calculated ECACand CCAC for both of them, verifying the condition (20). The

result of this analysis is that this condition is respected in all

the 1000 cases. A mathematical proof of this result will be

addressed in a future work.

To sum up, in the scenarios we are looking in, CCACgives a valid estimation of ECAC with low computational

complexity. In fact, the advantage of using of CCAC instead

of ECAC is that it is possible to compute only one average

Laplacian matrix and its second smallest eigenvalue to achieve

a measure of the network connectivity instead of 2Np Lapla-

cian matrices and their second smallest eigenvalues.

B. On the behavior of the compact cognitive algebraic con-

nectivity

In the previous section we showed that in the scenarios we

are interested in, it is possible to use the parameter CCACwith the aim of analyzing the connectivity of secondary net-

works. In this Section we tested the mathematical framework

TABLE IIPERCENTAGE DIFFERENCE BETWEEN CCAC AND ECAC IN DIFFERENT

NETWORK TOPOLOGIES.

CCAC ECAC

a1 = 0.2 a2 = 0.5 a3 = 0.2 0.3339 0.3306a1 = 0.2 a2 = 0.7 a3 = 0.3 0.3949 0.3872a1 = 0.4 a2 = 0.5 a3 = 0.4 0.3649 0.3516a1 = 0.4 a2 = 0.8 a3 = 0.3 0.3573 0.3420a1 = 0.4 a2 = 0.9 a3 = 0.5 0.3312 0.3020

7

20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

P1

P2P3

X [m]

Y [

m]

1

2

3

45

6

7 8

(a)

20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

P1

P2P3

1

2

3

45

6

7 8

X [m]

Y [

m]

(b)

20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

P1

P2P3

1

2

3

45

6

7 8

X [m]

Y [

m]

(c)

Fig. 5. Three different network scenarios with 3 PUs having coverage area of radius equal to 35 meters, 8 SUs and three values of secondary userstransmission range (a): r=28 meters, (b): r=36.5 meters, (c): r=40 meters

proposed in Section III-B, by presenting the behavior of

CCAC in some network topologies. The aim is to derive the

trend of CCAC as function of the probability π when some

characteristics of the network scenario vary. In particular, we

considered how the variation of i) transmission range of SUs,

ii) coverage area radius of PUs, and iii) number of SUs, impact

on compact cognitive algebraic connectivity.

We first performed the analysis of the impact of transmis-

sion range of the SUs on the compact cognitive algebraic

connectivity, by considering the three scenarios represented

in Figure 5. Three PUs are placed in an area of 100 × 100meters and eight SUs are scattered in this area. The radius

of PUs’ CAs is 35 meters, generating CAs represented by

circles in Figure 5. The reason behind these reduced PU CAs

is that we would test scenarios where the transmission of a

SU may affect near PU receivers and where SUs are affected

by different PUs. Besides, in this figure, links between SUs

represent radio visibility relationships. In Figure 6 it is shown

the behavior of CCAC as a function of the π probability in

the three scenarios of Figure 5. As expected, in all the three

cases, the value of CCAC increases when the probability πincreases. From this result we can infer that the PUs’ activity

impacts on degree connectivity and consequently on routing

performance in terms of reliability. Moreover, for a given value

of the probability π, CCAC improves when the transmission

range of SUs increases. It is possible to notice that for r equal

to 28 or 36.5 meters the secondary network is at most 1-

connected, in accordance with the heuristic condition (18),

also in the best case (π = 1). Instead, with r = 40 meters, if

π > 0.35 the secondary network becomes 2-connected. This

analysis can be used to find the minimum probability π (that

is the maximum value of PUs’ activities) that allows to obtain

the desired value of compact cognitive algebraic connectivity.

In this work we did not consider mutual interference among

SUs and consequently we allowed that two interfering SUs use

the same channel. As a consequence, when the transmission

range increases the degree connectivity increases too, but also

the mutual interference among SUs increases. We will address

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

π

CC

AC

r= 28 mt

r = 36.5 mt

CA radius = 40 mt

Fig. 6. Compact cognitive algebraic connectivity for different SUs transmis-sion ranges.

this problem in future works, taking into account also the

mutual interference in the secondary network with the aim

of finding the right trade-off between the degree connectivity

and the mutual interference among SUs.

In Figure 7 it is shown the behavior of the CCAC as a

function of the π probability in Scenario of Figure 5(b), for

three different values of CA radius of PUs. In this situation,

for a given value of π, CCAC increases when the CA radius

of PUs decreases, since the lower is the CA radius, the smaller

is the number of SUs influenced by PUs. We observe that the

potential algebraic connectivity, calculated without considering

PUs, corresponds to the value of CCAC when π = 1, that

is it is equal to 0.713. The value of π such that CCAC is

equal to the potential algebraic connectivity is influenced by

the CA radius of PUs. In fact, the lower is the CA radius,

the lower is the probability π necessary to obtain the potential

algebraic connectivity, that is values of PUs’activity factors

may be higher without compromising the compact cognitive

algebraic connectivity. From the Figure 7 it is possible to

notice that when the CA radius is equal to 25 meters the

potential algebraic connectivity is reached for π = 0.37,

instead when the CA radius is equal to 40 meters the potential

algebraic connectivity is reached only for π = 1.

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

π

CC

AC

CA radius=40

CA radius=30

CA radius=25

Fig. 7. CCAC for different radius of PUs’ coverage areas (CA radius= 25, 30 and 40 meters).

Fig. 8. Three-dimensional behavior of the CCAC as function of theprobability π and of the number of SUs

In Figure 8 it is represented the tree-dimensional behavior of

the CCAC as function of the probability π and of the number

of SUs in the network. We considered a network scenario with

3 PUs and Ns SUs (Ns = 5, 10, 15, 20) scattered in an area

of 100 × 100 meters. The CA radius of PUs is 40 meters

while the transmission range of the SUs is 45 meters. For each

value of Ns we randomly generated 10 topologies analyzed by

means of CCAC and averaged these results. From this figure

we can see that CCAC increases when π and the number

of SUs increase. Besides, we can argue that the increase

of CCAC related to an increment of π is bigger than the

increase of CCAC related to the same increment of Ns. The

result is that the CCAC is mainly limited by the impact of

PUs rather than the number of SUs composing the secondary

network. Consequently, it is harder to play with the number of

SUs to achieve a given value of compact cognitive algebraic

connectivity rather than operating on the activity factors of

PUs.

V. CONCLUSIONS

In this paper, we studied the impact of a primary user

network on the connectivity of a secondary cognitive radio

network. To this aim we leveraged the concept of algebraic

connectivity computed on a Laplacian graph enhanced to

model the impact of the primary users on the links of the

secondary network. The contribution of the work is twofold.

On one side we proposed a compact cognitive algebraic

connectivity that is a unified parameter able to reproduce the

behavior of the expected value of cognitive algebraic connec-

tivity averaged on all possible scenarios of PUs activities. On

the other side, we used this metric to evaluate the impact

of the primary activities on the secondary network and we

showed that both the extension of the coverage area of the

PUs and the transmission range of the SUs have an impact

on the connectivity. Moreover, we derived that to achieve a

given connectivity target in the secondary network, it is more

effective to play, when possible, with the activity factors of

the PUs rather than with the number of secondary nodes.

Our results may be used to plan a secondary cognitive radio

network on the basis of both the topology of the primary

one and the mean activity factors of the licensed users.

Furthermore, the proposed methodology can find applications

in the field of the cognitive radio routing to determine the

number of primary-free paths in a networks, as well as to

characterize the stability of a path as a function of the primary

users activities. As for this latter aspect we already started a

work that has been presented, in its embryonal stage, in [14].

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