arXiv:0903.1684v5 [cs.NI] 9 Aug 2009
Connectivity of Heterogeneous Wireless Networks
Wei Ren, Qing Zhao
∗, Ananthram Swami
Abstract
We address the connectivity of large-scale ad hoc heterogeneous wireless networks, where secondary users exploit channels temporarily unused by primary users and the existence of a communication link between two secondary users depends on not only the distance between them but also the transmitting and receiving activities of nearby primary users. We introduce the concept of connectivity region defined as the set of density pairs — the density of secondary users and the density of primary transmitters
— under which the secondary network is connected. Using theories and techniques from continuum percolation, we analytically characterize the connectivity region of the secondary network and reveal the tradeoff between proximity (the number of neighbors) and the occurrence of spectrum opportunities.
Specifically, we establish three basic properties of the connectivity region – contiguity, monotonicity of the boundary, and uniqueness of the infinite connected component, where the uniqueness implies the occurrence of a phase transition phenomenon in terms of the almost sure existence of either zero or one infinite connected component; we identify and analyze two critical densities which jointly specify the profile as well as an outer bound on the connectivity region; we study the impacts of secondary users’ transmission power on the connectivity region and the conditional average degree of a secondary user, and demonstrate that matching the interference ranges of the primary and the secondary networks maximizes the tolerance of the secondary network to the primary traffic load. Furthermore, we establish a necessary condition and a sufficient condition for connectivity, which lead to an outer bound and an inner bound on the connectivity region.
Index Terms
Heterogeneous wireless network, cognitive radio, connectivity region, phase transition, critical densities, continuum percolation.
This work was supported in part by the Army Research Laboratory CTA on Communication and Networks under Grant DAAD19-01-2-0011, by the Army Research Office under Grant W911NF-08-1-0467, and by the National Science Foundation under Grants ECS-0622200 and CCF-0830685.
W. Ren and Q. Zhao are with the Department of Electrical and Computer Engineering, University of California, Davis, CA 95616. A. Swami is with the Army Research Laboratory, Adelphi, MD 20783.
∗Corresponding author. Phone: 1-530-752-7390. Fax: 1-530-752-8428. Email: qzhao@ece.ucdavis.edu.
I. INTRODUCTION
The communication infrastructure is becoming increasingly heterogeneous, with a dynamic composition of interdependent, interactive, and hierarchical network components with different priorities and service requirements. One example is the cognitive radio technology [1] for opportunistic spectrum access which adopts a hierarchical structure for resource sharing [2].
Specifically, a secondary network is overlaid with a primary network, where secondary users identify and exploit temporarily and locally unused channels without causing unacceptable interference to primary users [2].
A. Connectivity and Connectivity Region
While the connectivity of homogeneous ad hoc networks consisting of peer users has been well studied (see, for example, [3, 4, 5, 6, 7, 8, 9, 10]), little is known about the connectivity of heterogeneous networks. The problem is fundamentally different from its counterpart in homogeneous networks. In particular, the connectivity of the low-priority network component depends on the characteristics (traffic pattern/load, topology, interference tolerance, etc.) of the high-priority component, thus creating a much more diverse and complex design space.
Using theories and techniques from continuum percolation, we analytically characterize the connectivity of the secondary network in a large-scale ad hoc heterogeneous network. Specifi- cally, we consider a Poisson distributed secondary network overlaid with a Poisson distributed primary network in an infinite two-dimensional Euclidean space1. We define network connectivity as the existence of an infinite connected component almost surely (a.s.), i.e., the occurrence of percolation. We say that the secondary network is strongly connected when it contains a unique infinite connected component a.s.
Due to the hierarchical structure of spectrum sharing, a communication link exists between two secondary users if the following two conditions hold: (C1) they are within each other’s transmission range; (C2) they see a spectrum opportunity determined by the transmitting and receiving activities of nearby primary users (see Sec. II-B1). Thus, given the transmission power
1This infinite network model is equivalent in distribution to the limit of a sequence of finite networks with a fixed density as the area of the network increases to infinity, i.e., the so-called extended network [11]. It follows from the arguments similar to the ones used in [12, Chapter 3] for homogeneous ad hoc networks that this infinite ad hoc heterogeneous network model represents the limiting behavior of large-scale networks.
and the interference tolerance of both the primary and the secondary users, the connectivity of the secondary network depends on the density of secondary users (due to (C1)) and the traffic load of primary users (due to (C2)).
We thus introduce the concept of connectivity region C, defined as the set of density pairs (λS, λP T) under which the secondary network is connected, where λS denotes the density of the secondary users and λP T the density of primary transmitters (representing the traffic load of the primary users). As illustrated in Fig. 1, a secondary network with a density pair(λS, λP T) inside this region is connected: the secondary network has a giant connected component which includes infinite secondary users. The existence of the giant connected component enables bidirectional communications between distant secondary users via multihop relaying. On the other hand, a secondary network with a density pair(λS, λP T) outside this region is not connected: the network is separated into an infinite number of finite connected components. Consequently, any secondary user can only communicate with users within a limited range.
λP T
λS
λ∗P T
λ∗P T(λS)
λ∗S 0
Connectivity Region
Fig. 1. The connectivity regionC (the upper boundary λ∗P T(λS) is defined as the supremum density of the primary transmitters to ensure connectivity with a fixed density of the secondary users; the critical densityλ∗Sof the secondary users is defined as the infimum density of the secondary users to ensure connectivity under a positive density of the primary transmitters; the critical densityλ∗P T of the primary transmitters the supremum density of the primary transmitters to ensure connectivity with a finite density of the secondary users).
The objective of this paper is to establish analytical characterizations of the connectivity region and to study the impact of system design parameters (in particular, the transmission power of the
secondary users) on the network connectivity. Main results are summarized in the subsequent two subsections.
B. Analytical Characterizations of the Connectivity Region
We first establish three basic properties of the connectivity region: contiguity, monotonicity of the boundary, and uniqueness of the infinite connected component. Specifically, based on a coupling argument, we show that the connectivity region is a contiguous area bounded below by the λS-axis and bounded above by a monotonically increasing functionλ∗P T(λS) (see Fig. 1), where the upper boundary λ∗P T(λS) is defined as
λ∗P T(λS)= sup{λ∆ P T : G(λS, λP T) is connected.},
withG(λS, λP T) denoting the secondary network of density λS overlaid with a primary network specified by the densityλP T of the primary transmitters. The uniqueness of the infinite connected component is established based on the ergodic theory and certain combinatorial results. It shows that once the secondary network is connected, it is strongly connected.
Second, we identify and analyze two critical parameters of the connectivity region: λ∗S and λ∗P T. They jointly specify the profile as well as an outer bound on the connectivity region.
Referred to as the critical density of the secondary users, λ∗S is the infimum density of the secondary users to ensure connectivity under a positive density of the primary transmitters:
λ∗S= inf{λ∆ S : ∃λP T > 0 s.t. G(λS, λP T) is connected}.
We show that λ∗S equals the critical density λc of a homogeneous ad hoc network (i.e., in the absence of primary users), which has been well studied [13]. This result shows that the “takeoff”
point in the connectivity region is completely determined by the effect of proximity—the number of neighbors (nodes within the transmission range of a secondary user).
Referred to as the critical density of the primary transmitters, λ∗P T is the supremum density of the primary transmitters to ensure the connectivity of the secondary network with a finite density of the secondary users:
λ∗P T= sup{λ∆ P T : ∃λS < ∞ s.t. G(λS, λP T) is connected}.
We obtain an upper bound on λ∗P T which is shown to be achievable in simulations. More importantly, this result shows that when the density of the primary transmitters is higher than
the (finite) value given by this upper bound, the secondary network cannot be connected no matter how dense it is. This parameter λ∗P T thus characterizes the impact of opportunity occurrence on the connectivity of the secondary network: when the density of the primary transmitters is beyond a certain level, there are simply not enough spectrum opportunities for any secondary network to be connected.
Since a precise characterization of the upper boundary λ∗P T(λS) of the connectivity region is intractable, we establish a necessary and a sufficient condition for connectivity to provide an outer and an inner bound on the connectivity region. The necessary condition is expressed in the form of the conditional average degree of a secondary user, and is derived by the construction of a branching process. The sufficient condition is obtained by the discretization of the continuum percolation model into a dependent site percolation model.
0 200 400 600 800 1000 1200 1400 1600 1800
0 2 4 6 8 10 12 14 16 18
λS (per km2) λ PT (per km2 )
small power large power
ptx
ptx
Fig. 2. Simulated connectivity regions for two different transmission powers (ptx denotes the transmission power of the secondary users, and the largeptxis3α times the smallptx, whereα is the path-loss exponent).
C. Impact of Transmission Power on Connectivity: Proximity vs. Opportunity
The study on the impact of the secondary users’ transmission power on the network con- nectivity reveals an interesting tradeoff between proximity and opportunity in the design of
heterogeneous networks. As illustrated in Fig. 2, we show that increasing ptx enlarges the connectivity region C in the λS-axis (i.e., better proximity leads to a smaller “takeoff” point), but at the price of reducing C in the λP T-axis. Specifically, with a large ptx, few secondary users experience spectrum opportunities due to their large interference range with respect to the primary users. This leads to a poor tolerance to the primary traffic load parameterized by λP T. The transmission power ptx of the secondary network should thus be chosen according to the operating point of the heterogeneous network given by the density of the secondary users and the traffic load of the co-existing primary users. Using the tolerance to the primary traffic load as the performance measure, we show that the interference range rI of the secondary users should be equal to the interference rangeRI of the primary users in order to maximize the upper bound on the critical density λ∗P T of the primary transmitters. Given the interference tolerance of the primary and secondary users, we can then design the optimal transmission power ptx of the secondary users based on that of the primary users.
D. Related Work
To our best knowledge, the connectivity of large-scale ad hoc heterogeneous networks has not been characterized analytically or experimentally in the literature. There are a number of classic results on the connectivity of homogeneous ad hoc networks. For example, it has been shown that to ensure either 1-connectivity (there exists a path between any pair of nodes) [5, 6]
or k-connectivity (there exist at least k node-disjoint paths between any pair of nodes) [8], the average number of neighbors of each node must increase with the network size. On the other hand, to maintain a weaker connectivity – p-connectivity (i.e., the probability that any pair of nodes is connected is at least p), the average number of neighbors is only required to be above a certain ‘magic number’ which does not depend on the network size [7].
The theory of continuum percolation has been used by Dousse et al. in analyzing the con- nectivity of a homogeneous ad hoc network under the worst case mutual interference [3, 4].
In [9, 10], the connectivity and the transmission delay in a homogeneous ad hoc network with statically or dynamically on-off links are investigated from a percolation-based perspective.
The optimal power control in heterogeneous networks has been studied in [14], which focuses on a single pair of secondary users in a Poisson network of primary users. The impacts of sec- ondary users’ transmission power on the occurrence of spectrum opportunities and the reliability
of opportunity detection are analytically characterized.
E. Organization and Notations
The rest of this paper is organized as follows. Sec. II presents the Poisson model of the heterogeneous network. In particular, the conditions for the existence a communication link in the secondary network is specified based on a rigorous definition of spectrum opportunity. In Sec. III, we introduce the concept of connectivity region and establish its three basic properties.
The two critical densities are analyzed, followed by a necessary and a sufficient condition for connectivity. In Sec. IV, we demonstrate the tradeoff between proximity and opportunity by studying the impacts of the secondary users’ transmission power on the connectivity region and on the conditional degree of a secondary user. The optimal transmission power of the secondary users is obtained under the performance measure of the secondary network’s tolerance to the primary traffic load. Sec V contains the detailed proofs of the main results, and Sec. VI concludes the paper.
Throughout the paper, we use capital letters for parameters of the primary users and lowercase letters for the secondary users.
II. NETWORK MODEL
We consider a Poisson distributed secondary network overlaid with a Poisson distributed primary network in an infinite two-dimensional Euclidean space. The models of the primary and secondary networks are specified in the following two subsections.
A. The Primary Network
The primary transmitters are distributed according to a two-dimensional Poisson point process with density λP T. To each primary transmitter, its receiver is uniformly distributed within its transmission range Rp. Here we have assumed that all primary transmitters use the same transmission power and the transmitted signals undergo an isotropic path loss. Based on the displacement theorem [15, Chapter 5], it is easy to see that the primary receivers form a two- dimensional Poisson point process with densityλP T. Note that the two Poisson processes formed by the primary transmitters and receivers are correlated.
B. The Secondary Network
The secondary users are distributed according to a two-dimensional Poisson point process with density λS, independent of the Poisson processes of the primary transmitters and receivers.
The transmission range of the secondary users is denoted by rp.
1) Communication Links: In contrast to the case in a homogeneous network, the existence of a communication link between two secondary users depends on not only the distance between them but also the availability of the communication channel (i.e., the presence of a spectrum opportunity). The latter is determined by the transmitting and receiving activities in the primary network as described below.
As illustrated in Fig. 3, there exists an opportunity fromA, the secondary transmitter, to B, the secondary receiver, if the transmission fromA does not interfere with nearby primary receivers in the solid circle, and the reception at B is not affected by nearby primary transmitters in the dashed circle [16]. Referred to as the interference range of the secondary users, the radius rI of the solid circle at A depends on the transmission power of A and the interference tolerance of the primary receivers, whereas the radius RI of the dashed circle (the interference range of the primary users) depends on the transmission power of the primary users and the interference tolerance of B.
00 11 00
11 00
11 0000
00 1111 11
A B
Interference
rI
RI
Primary Tx Primary Rx
Fig. 3. Definition of spectrum opportunity.
It is clear from the above discussion that spectrum opportunities depend on both transmitting
and receiving activities of the primary users. Furthermore, spectrum opportunities are asymmetric.
Specifically, a channel that is an opportunity when A is the transmitter and B the receiver may not be an opportunity when B is the transmitter and A the receiver. In other words, there exist unidirectional communication links in the secondary network. Since unidirectional links are difficult to utilize in wireless networks [17], we only consider bidirectional links in the secondary network when we define connectivity. As a consequence, when we determine whether there exists a communication link between two secondary users, we need to check the existence of spectrum opportunities in both directions.
To summarize, under the disk signal propagation and interference model, there is a (bidirec- tional) link between A and B if and only if (C1) the distance between A and B is at most rp; (C2) there exists a bidirectional spectrum opportunity between A and B, i .e., there are no primary transmitters within distance RI of either A or B and no primary receivers within distance rI of either A or B.
2) Connectivity: We interpret the connectivity of the secondary network in the percolation sense: the secondary network is connected if there exists an infinite connected component a.s.
Based on the above conditions (C1, C2) for the existence of a communication link, we can obtain an undirected random graphG(λS, λP T) corresponding to the secondary network, which is determined by three Poisson point processes: the secondary users with density λS, the primary transmitters with density λP T, and the primary receivers with density λP T (correlated to the process of the primary transmitters)2. See Fig. 4 for an illustration of G(λS, λP T).
The question we aim to answer in this paper is the connectivity of the secondary network, i.e., the percolation in G(λS, λP T).
III. ANALYTICAL CHARACTERIZATIONS OF THE CONNECTIVITY REGION
Given the transmission power and the interference tolerance of both the primary and the secondary users (i.e., Rp, RI, rp, andrI are fixed), the connectivity of the secondary network
2The two Poisson point processes of the primary transmitters and receivers are essentially a snap shot of the realizations of the primary transmitters and receivers. In different time slots, different sets of primary users become active transmitters/receivers.
Thus, even if a secondary user is isolated at one time due to the absence of spectrum opportunities, it may experience an opportunity at a different time and be connected to other secondary users.
00 11
00 0 11 1
0000 00 1111 11 00
0 11 1 00
0 11 1
00 11
00 0 11
1Primary Tx Primary Rx Secondary User rI
RI
Fig. 4. A realization of the heterogeneous network. The random graphG(λS, λP T) consists of all the secondary nodes and all the bidirectional links denoted by solid lines. The solid circles with radiiRI denote the interference regions of the primary transmitters within which secondary users can not successfully receive, and the dashed circles with radiirI denote the required protection regions for the primary receivers within which the secondary users should refrain from transmitting.
is determined by the densityλS of the secondary users and the densityλP T of the primary trans- mitters. We thus introduce the concept of connectivity region C of a secondary network, which is defined as the set of density pairs (λS, λP T) under which the secondary network G(λS, λP T) is connected (see Fig. 1).
C= {(λ∆ S, λP T) : G(λS, λP T) is connected.}.
A. Basic Properties of the Connectivity Region
We establish in Theorem 1 below three basic properties of the connectivity region.
Theorem 1: Basic Properties of the Connectivity Region.
T1.1 The connectivity regionC is contiguous, that is, for any two points (λS1, λP T 1), (λS2, λP T 2) ∈ C, there exists a continuous path in C connecting the two points.
T1.2 The lower boundary of the connectivity region C is the λS-axis. Let λ∗P T(λS) denote the upper boundary of the connectivity region C, i.e.,
λ∗P T(λS)= sup{λ∆ P T : G(λS, λP T) is connected.}, then we have that λ∗P T(λS) is monotonically increasing with λS.
T1.3 There exists either zero or one infinite connected component in G(λS, λP T) a.s.
Proof: The proofs of T1.1 and T1.2 are based on the coupling argument, a technique frequently used in continuum percolation [13, Section 2.2]. The proof of T1.3 is based on the ergodicity of the random model driven by the three Poisson point processes of the primary transmitters, the primary receivers, and the secondary users (the concept of ergodicity of a random model is reviewed in Sec. V-A5). The details of the proofs are given in Sec. V-B.
T1.1 and T1.2 specify the basic structure of the connectivity region, as illustrated in Fig. 1.
T1.3 implies the occurrence of a phase transition phenomenon, that is, there exists either a unique infinite connected component a.s. or no infinite connected component a.s. This uniqueness of the infinite connected component establishes the strong connectivity of the secondary network: once it is connected, it is strongly connected. It excludes the undesirable possibility of having more than one (maybe infinite) infinite connected component in the secondary network. We point out that such a property is not always present in wireless networks. Two examples where more than one infinite connected component exists in a homogeneous ad hoc network can be found in [18].
B. Critical Densities
In this subsection, we study the critical density λ∗S of the secondary users and the critical density λ∗P T of the primary transmitters. Recall that
λ∗S = inf{λ∆ S : ∃λP T > 0 s.t. G(λS, λP T) is connected}, λ∗P T = sup{λ∆ P T : ∃λS < ∞ s.t. G(λS, λP T) is connected}.
We have the following theorem.
Theorem 2: Critical Densities.
Given Rp, RI, rp, and rI, we have
T2.1 λ∗S = λc(rp), where λc(rp) is the critical density for a homogeneous ad hoc network with transmission range rp (i.e., in the absence of the primary network).
T2.2 λ∗P T ≤ 4 max{Rλc(1)2
I,r2I}−r2p, where the constant λc(1) is the critical density for a homogeneous ad hoc network with a unit transmission range.
Proof: The basic idea of the proof of T2.1 is to approximate the secondary network G(λS, λP T) by a discrete edge-percolation model on the grid. This discretization technique is
often used to convert a continuum percolation model to a discrete site/edge percolation model (see, for example, [13, Chapter 3], [4]). The details of the proof are given in Sec. V-C1.
The proof of T2.2 is based on the argument that if there is an infinite connected component in the secondary network, then an infinite vacant component must exist in the two Poisson Boolean models driven by the primary transmitters and the primary receivers, respectively. The key point is to carefully choose the radii of the two Poisson Boolean models in order to obtain a valid upper bound on λ∗P T. The details of the proof can be found in Sec. V-C2.
−500 0 500
−500
−400
−300
−200
−100 0 100 200 300 400 500
Fig. 5. A realization of the Poisson heterogeneous network when the percolation occurs (black stars denote primary transmitters, green plus signs denote primary receivers, red dots denote secondary users, and blue segments denote the bidirectional links between secondary users). We have removed secondary users who do not see opportunities for clarity. The simulation parameters are given byλP T = 10km−2, Rp= 50m, RI= 80m, λS= 650km−2,rp= 50m, rI = 80m, and the critical density in this case isλc(50) ≈ 576km−2.
Fig. 5 shows one realization of the Poisson heterogeneous network when λS is slightly larger than λc(rp) and λP T is small. At least one left-to-right (L-R) crossing and at least one top-to- bottom (T-B) crossing can be found in the square network. It is thus expected that these L-R and T-B crossings in finite square regions can form an infinite connected component in the whole
network on R2. If we slightly increase λP T, then we observe from Fig. 6 that the reduction in spectrum opportunities eliminates considerable communication links in the secondary network, creating several disjoint small components.
−500 0 500
−500
−400
−300
−200
−100 0 100 200 300 400 500
Fig. 6. A realization of the Poisson heterogeneous network when the percolation does not occur (black stars denote primary transmitters, green plus signs denote primary receivers, red dots denote secondary users, and blue segments denote the bidirectional links between secondary users). We have removed secondary users who do not see opportunities for clarity.
The simulation parameters are given byλP T = 20km−2,Rp= 50m, RI = 80m, λS= 650km−2,rp= 50m, rI= 80m, and the critical density in this case isλc(50) ≈ 576km−2.
Fig. 7 shows a simulation example of the connectivity region, where the upper bound on the critical density λ∗P T of the primary transmitters given in T2.2 appears to be achievable.
C. A Necessary Condition for Connectivity
In this subsection, we establish a necessary condition for connectivity which is given in terms of the average conditional degree of a secondary user. This condition agrees with our intuition:
the secondary network cannot be connected if the degree of every secondary user is small.
0 500 1000 1500 2000 0
1 2 3 4 5 6 7
λS (per km2) λ PT (per km2 )
Fig. 7. Simulated connectivity regions whenrp= 150m, rI= 240m, Rp= 100m, and RI = 120m. The blue dashed line is the upper bound 4 max{Rλc2(1)
I,r2I}−r2p on the critical densityλ∗P T of primary transmitters given in T2.2. The area of the simulated heterogeneous network is2000m×2000m. For a fixed density λSof the secondary users, the upper boundaryλ∗P T(λS) is equal to the minimum density of the primary transmitters such that over all the1000 realizations, the percentage of the ones in which there exists at least one L-R crossing is below50%. The intuitive reason for choosing the existence of an L-R crossing as the criterion for connectivity is illustrated in Fig. 5-6.
Let I(A, d, rx/tx) denote the event that there exists primary receivers/transmitters within dis- tanced of a secondary user A. Let I(A, d, rx/tx) denote the complement of I(A, d, rx/tx). Since a secondary user is isolated if it does not see a spectrum opportunity, we focus on secondary users who experience spectrum opportunities and define the conditional average degree µ of such a secondary user A as
µ = E[deg(A)| I(A, rI, rx) ∩ I(A, RI, tx)], (1) where deg(A) denotes the degree of A, rI the interference range of the secondary users, and RI the interference range of the primary users. Notice that the degree of A is the number of secondary users within the transmission range of A and experiencing opportunities. We arrive at the following necessary condition for connectivity.
Theorem 3: A necessary condition for the connectivity of G(λS, λP T) is µ > 1, where µ is the conditional average degree of a secondary user defined in (1).
Proof: The basic idea is to construct a branching process, where the conditional average degreeµ is the average number of offspring. This branching process provides an upper bound on the number of secondary users in a connected component. If µ ≤ 1, then the branching process is finite a.s. It thus follows that there is no infinite connected component a.s. in G(λS, λP T).
Details can be found in Sec. V-D.
To apply the necessary condition given in Theorem 3, the conditional average degree µ of a secondary user A needs to be evaluated based on the network parameters. Let B be a secondary user randomly and uniformly distributed within the transmission range rp of A. Let g(λP T, rp, rI, Rp, RI) denote the probability of a bidirectional opportunity between A and B conditioned on the event that A sees an opportunity. Based on the statistical equivalence and independence of different points in a Poisson point process, the conditional average degree µ of a secondary user A is given by this conditional probability g(·) of a bidirectional opportunity between A and a randomly chosen neighbor multiplied by the average number of neighbors of A, i.e.,
µ = λSπr2p · g(λP T, rp, rI, Rp, RI). (2) The detailed derivation for (2) and the expression for g(·) are given in Appendix A. It is also shown in Appendix A that g(·) is a strictly decreasing function of λP T. Thus g−1(·), the inverse of g(·) with respect to λP T, is well-defined.
Combining (2) with Theorem 3, we obtain an outer bound on the connectivity region. Specif- ically, let µ(λS, λP T) denote the conditional average degree of a secondary user in G(λS, λP T).
Then those density pairs(λS, λP T) satisfying µ(λS, λP T) ≤ 1 are outside the connectivity region.
Corollary 1: Given Rp, RI, rp, andrI, an outer bound on the connectivity region C is given by
λP T = g−1
1
λSπr2p
,
where g−1(·) is the inverse of the conditional probability g(·) with respect to λP T.
D. A Sufficient Condition for Connectivity
In this subsection, we establish a sufficient condition for connectivity, which provides an inner bound on the connectivity region and a criterion for checking whether a secondary network is connected.
d d
O
O1 O2 O3
O4 O5
O6 O7 O8
Fig. 8. An illustration of the dependent site-percolation modelL with side length d (solid dots denote sites, solid lines denote edges connecting every two sites, and dashed lines denote the squared partition).
The sufficient condition for connectivity is established by using the discretization technique.
The continuum percolation model is mapped onto a dependent site-percolation model L in the following way. As illustrated in Fig. 8, we partition R2 into (dashed) squares with side length d and locate a site at the center of each square. Sites whose associated dashed squares share at least one common point are considered connected (as illustrated by solid lines in Fig. 8). Thus each site is connected to eight neighbors3 (see the eight neighbors O1,...,O8 of siteO in Fig. 8).
Let BO be the associated dashed square of O, then O is occupied if there exists in BO at least one secondary user who sees an opportunity.
Since the largest distance between two points in two neighboring dashed squares is 2√ 2d, it follows that if we set d = 2r√p2, then for every pair of secondary users in two neighboring
3For the commonly used square site-percolation model, each site has four neighbors. The site-percolation model constructed here can provide a better inner bound.
dashed squares, they are within the transmission range rp of each other. Based on the definitions of occupied site in L and communication link in the secondary network, we conclude that the existence of an infinite occupied component (a connected component consisting of only occupied sites) in L implies the existence of an infinite connected component in the secondary network.
Due to the fact that spectrum opportunities are spatially dependent, the state of one site is correlated with the states of its adjacent sites. Thus, the above site-percolation model L is a dependent model. Define the dependence range k as the minimum distance such that the state of any two sites at distance d > k are independent, where the distance between two sites is the minimum number of neighboring sites that must be traversed from one site to the other. Then the dependence range of L is given by
k =
&
8 maxRI+r4p, rI+ r4p rp
'
− 1. (3)
Let pc denote the upper critical probability of L which is defined as the minimum occupied probability p∗ such that if the occupied probability p > p∗, an infinite occupied component containing the origin exists in L with a positive probability (wpp.). Since the dependence range k of L is finite, it follows from Theorem 2.3.1 [12] that pc < 1. Now we present the sufficient condition for connectivity in the following theorem.
Theorem 4: Letpc denote the upper critical probability of the dependent site-percolation model L specified above. Define
I(r, Rp, rI) = 2 Z r
0
tSI(t, Rp, rI)
πR2p dt, (4)
whereSI(t, Rp, rI) is the common area of two circles with radii Rp andrI and centered t apart.
Then the secondary network is connected if
1 − exp
−λSr2p 8
exp−λP TπR2I+ r2I − I (RI, Rp, rI) > pc.
Proof: The proof is based on the ergodicity of the heterogeneous network model and its relation with the constructed dependent site-percolation model L. Details can be found in Sec. V-E.
By applying a general upper bound on the upper critical probability pc for a site-percolation model with finite dependence range [12, Theorem 2.3.1], we arrive at the following corollary.
Corollary 2: A sufficient condition for the connectivity of G(λS, λP T) is
λP T < 1
π [R2I+ r2I− I(RI, Rp, rI)]ln1 − exp
−λS8r2p 1 − 13(2k+1)2 ,
whereI(RI, Rp, rI) is defined in (4) and k is the dependence range of the site-percolation model defined in (3).
IV. IMPACT OF TRANSMISSION POWER: PROXIMITY VS. OPPORTUNITY
In this section, we study the impact of the secondary users’ transmission power on the connectivity and the conditional average degree of the secondary network. As has been illustrated in Fig. 2, there exists a tradeoff between proximity and opportunity in designing the secondary users’ transmission power for connectivity. Specifically, increasing the transmission power of the secondary users leads to a smaller critical density λ∗S of the secondary users, but at the same time, a lower tolerance to the primary traffic load manifested by a smaller critical density λ∗P T of the primary transmitters.
A. Impact on the Conditional Average Degree
As discussed in Sec. III-C, the expression for the conditional average degree µ can be decomposed into the product of two terms:λSπrp2 andg(λP T, rp, rI, Rp, RI). The first term is the average number of neighbors of a secondary user, which increases with the transmission power ptxof the secondary users (i.e., enhanced proximity). The other term g(λP T, rp, rI, Rp, RI) is the conditional probability of a bidirectional opportunity, which decreases with ptx due to reduced spectrum opportunities. This tension between proximity and opportunity is illustrated in Fig. 9, where we observe that the impact ofptx on proximity dominates when ptx is small (µ increases with ptx) while its impact on the occurrence of opportunities dominates when ptx is large (µ decreases with ptx).
Corollary 3: Let ptx be the transmission power of secondary users and µ the conditional average degree defined in (1), then under the disk signal propagation and interference model we have4
µ = O (ptx)−2/α
as ptx → ∞,
4Here we use the Big O notation:f (x) = O(g(x)) as x → ∞ if and only if ∃ M > 0, x0> 0 such that |f(x)| ≤ M|g(x)|
for allx > x0.
where α is the path-loss exponent.
Proof: We show this corollary by deriving an upper bound on the conditional average degree µ. Details can be found in Appendix B.
For a homogeneous network, the average degree of a user is λπr2p, which increases with ptx
at rate (ptx)2/α. In sharp contrast, this corollary tells us that for a heterogeneous network, when ptx is large enough, the conditional average degree µ of a secondary user actually decreases with ptx at least as fast as (ptx)−2/α.
0 100 200 300 400 500 600 700 800 900
0 1 2 3 4 5 6 7 8
transmission range rp
conditional average degree µ
Fig. 9. Conditional average degreeµ of secondary users vs transmission range rp of secondary users (rp∝ (ptx)α1, where ptx is the transmission power of secondary users and α is the path-loss exponent, and simulation parameters are given by λP T = 2.5km−2, Rp= 200m, RI= 250m, λS= 25km−2,rI= rp/0.8).
B. Impact on the Connectivity Region
From the scaling relation of the critical density [13, Proposition 2.11], we know that in a homogeneous two-dimensional network,
λc(rp) = λc(1) (rp)−2 ∝ (ptx)−α2 ,
where the constant λc(1) is the critical density for a homogeneous ad hoc network with a unit transmission range. Thus, if each secondary user adopts a high transmission power, then
λc(rp) reduces. It follows from T2.1 that the critical density λ∗S of secondary users to achieve connectivity reduces due to the enhanced proximity.
On the other hand, from the upper bound on the critical densityλ∗P T of the primary transmitters given in T2.2, we have that
λ∗P T = O (ptx)−2/α
as ptx→ ∞,
where we have assumed that rp = βrI for some β ∈ (0, 1) under the disk signal propagation and interference model5. Thus, when the transmission power ptx of the secondary network is large enough, the critical density λ∗P T of the primary transmitters decreases with ptx at least as fast as (ptx)−2/α due to reduced spectrum opportunities.
C. Optimal Design of Transmission Power
Due to the tension between proximity and opportunity, there does not exist a transmission power of the secondary users that leads to the “largest” connectivity region (largest in the sense that its connectivity region contains all regions achievable with any finite transmission power ptx of the secondary users). Thus, the optimal design of ptx depends on the operating point of the heterogeneous network. For instance, when a sparse secondary network is overlaid with a primary network with low traffic load, a large ptx may be desirable to achieve connectivity. The opposite holds when a dense secondary network is overlaid with a primary network with high traffic load.
Focusing on a sufficiently dense secondary network, we address the design of its transmission power for the maximum tolerance to the primary traffic. Due to its tractability and achievability indicated by simulation examples (see Fig. 7), the upper bound on the critical density λ∗P T of the primary transmitters given in T2.2 is used as the performance measure.
Theorem 5: Let rI and RI denote the interference range of the secondary and the primary users, respectively. For a fixed RI, the upper bound on λ∗P T given in T2.2 is maximized when the primary and secondary networks have matching interference ranges: rI = RI.
5Since the minimum received signal power required for successful reception is, in general, higher than the maximum allowable received interference power , the transmission rangerpis smaller than the interference rangerI, i.e.,β < 1.
Proof: Since under the disk signal propagation and interference model, rp = βrI for some β ∈ (0, 1), the upper bound on λ∗P T can be written as
λ∗P T ≤
λc(1)
4R2I−β2rI2 for rI ≤ RI,
λc(1)
(4−β2)rI2 for rI > RI.
Then the above theorem can be readily shown by finding the maximal point for the two cases:
rI ≤ RI and rI > RI.
An example of the upper bound on λ∗P T is plotted as a function of rI in Fig. 10. Notice that there is a distinct difference in the slope on the two sides of the optimal point. As a consequence, the operating region ofrI < RI is preferred over that ofrI > RIwhen the optimal pointrI = RI
cannot be achieved. We point out that the desired operating region of rI < RI is the typical case of a secondary network coexisting with a privileged primary network.
50 100 150 200 250 300 350
0 5 10 15 20 25 30
rI UpperBoundforλ∗ PT(perkm2)
rI= RI
Fig. 10. An example of the upper bound onλ∗P T as a function ofrI (Parameters are given byRI= 120m, rp= 0.625rI).
V. PROOFS
In this section, we present proofs of the main results presented in Sec. III-IV. We start with a brief overview of several basic results in percolation and ergodic theory that will be used in the proofs.
A. Percolation and Ergodic Theory
1) Poisson Boolean Model: Poisson Boolean model is a common model in continuum perco- lation [13]. Often referred to as B(X, ρ, λ), the model is specified by two elements: a Poisson point process X on Rd with densityλ and a radius random variable ρ with a given distribution.
Under this model, each point inX is the center of a circle in Rd with a random radius distributed according to the distribution of ρ. Radii associated with different points are independent, and they are also independent of points in X. Under a Poisson Boolean model, the whole space is partitioned into two regions: the occupied region, which is the region covered by at least one ball, and the vacant region, which is the complement of the occupied region. We define occupied (vacant) components as those connected components in the occupied (vacant) region.
Assume that nodes in a homogeneous ad hoc network form a Poisson point process with densityλ and their transmission range is r. It is easy to see that the connectivity of this network can be studied through examining the occupied connected components in the corresponding Poisson Boolean model B(X, r/2, λ).
2) Sharp Transition in Two Dimensions: Phase transition is a well-known phenomenon in percolation. For the Poisson Boolean model in two dimensions, this phenomenon appears more remarkable in the sense that the critical density for the a.s. existence of infinite occupied components is equal to that for the a.s. existence of infinite vacant components. Let λc(2ρ) denote the critical density for the Poisson Boolean model B(X, ρ, λ), then we have that
when λ < λc(2ρ), there is no infinite occupied component a.s. and there is a unique infinite vacant component a.s.;
when λ > λc(2ρ), there is a unique infinite occupied component a.s. and there is no infinite vacant component a.s.
The exact value ofλc is not known. For a deterministic radius ρ, simulation results [19] indicate that λc(2ρ) ≈ 0.36ρ−2, while rigorous bounds 0.192ρ−2 < λc(2ρ) < 0.843ρ−2 are provided in [13, 20].
3) Crossing Probabilities: A continuous curve in the occupied region is called an occupied path. An occupied pathγ is an occupied L-R crossing of the rectangle {0 ≤ x ≤ l1} × {0 ≤ y ≤ l2} if γ intersects with both the left and the right boundaries of the rectangle, i.e., γ ∩ ({x = 0} × {0 ≤ y ≤ l2}) 6= φ, γ ∩ ({x = l1} × {0 ≤ y ≤ l2}) 6= φ, and the segment between the two
intersecting points is fully contained in the rectangle (see Fig. 11(a)). Similarly, we define an occupied T-B crossing by requiring that γ intersects with the top and bottom boundaries of the rectangle (see Fig. 11(b)). Let
σ((l1, l2), λ, L-R) = Pr{∃ an occupied L-R crossing of [0, l1] × [0, l2]}, σ((l1, l2), λ, T-B) = Pr{∃ an occupied T-B crossing of [0, l1] × [0, l2]},
denote the two crossing probabilities in the rectangle [0, l1] × [0, l2]. Then for a Poisson Boolean model B(X, ρ, λ) in two dimensions with a.s. bounded ρ, we have [13, Corollary 4.1] that for any k ≥ 1,
n→∞limσ((kn, n), λ, L-R) =
1, if λ > λc(2ρ);
0, if λ < λc(2ρ). (5)
Due to the symmetry of the Poisson Boolean model, similar results hold for the T-B crossing probability σ((n, kn), λ, T-B).
0 0
l1
l1
l2
l2
γ
γ
x x y
y
(a)
(b)
Fig. 11. An illustration of the L-R crossing (a) and the T-B crossing (b) in a rectangle{0 ≤ x ≤ l1} × {0 ≤ y ≤ l2}.
4) Dependent Edge-Percolation Model: Let L be a square lattice on R2 with side length d (see Fig. 12). In an edge-percolation model, every site in L is occupied but every edge in L exists with some probability p. An existing edge is often referred to as an open edge, and an
edge that is not open is called closed. When the states (open/closed) of edges are correlated, we have a dependent edge percolation model.
O
L L+
d
d d
2 d 2 d
2
Fig. 12. Part of the latticeL together with its dual L+ (solid dots and solid segments are sites and edges inL, and hollow dots and dashed segments are sites and edges inL+). The dual latticeL+is the`d
2, d2´-shifted version of L, which is used in the proof of T2.1. Since distinct edges inL are crossed by distinct edges in L+ and vice versa, there is a one-to-one mapping from the edges ofL to the edges of L+. In this case, we claim an edge inL+ being open if and only if its corresponding edge (i.e., the edge that it crosses) in L is open.
Consider a special case of dependent edge-percolation model L where the state of an edge e is only correlated with its six adjacent edges (edges that share a common point with e). We have the following known result.
Fact 1: [4, Proposition 1]
For any collection {ei}ni=1 of n distinct edges in L, we have
Pr{(C1 = 0) ∩ (C2 = 0) ∩ · · · ∩ (Cn= 0)} ≤ qn4,
where Ci is the indicator of ei being open,, and q = 1 − p is the probability of an edge being closed.
5) Ergodic Theory: The study object of ergodic theory is the so-called measure-preserving (m.p.) dynamical system (Ω, F, µ, T ), which consists of a set Ω, a σ-algebra F of measurable subsets ofΩ, a nonnegative measure µ on (Ω, F), and an invertible m.p. transformation T : Ω →
Ω such that µ(T−1F ) = µ(F ) for all F ∈ F. A set F ∈ F is said to be T-invariant if T−1F = F . Obviously, all T-invariant sets in F form a σ-algebra.
An m.p. dynamical system (Ω, F, µ, T ) is said to be ergodic if the σ-algebra of T-invariant sets is trivial, i.e., for any invariant set, either it has measure zero or its complement has measure zero. Another property of the m.p. dynamical system that implies ergodicity is called mixing:
an m.p. dynamical system (Ω, F, µ, T ) is said to be mixing if for all E, F ∈ F, µ(TnE ∩ F ) − µ(E)µ(F ) → 0 as n → ∞. For a m.p. dynamical system which is a product of two m.p.
dynamical systems, we have the following classical result in ergodic theory.
Fact 2: [22, Theorem 2.6.1]
The product system of a mixing m.p. dynamical system and an ergodic m.p. dynamic system is ergodic, that is, for a mixing (Ω, F, µ, T ) and an ergodic (Ψ, L, ν, S), the product system (Ω × Ψ, F × L, µ × ν, T × S) is ergodic, where F × L is the σ-algebra on Ω × Ψ generated by subsets of the form F × L (F ∈ F, L ∈ L) and µ × ν is the corresponding product measure.
The concepts of ergodicity and mixing can also be defined for a random model under a probability space (Ω, F, µ), where the m.p. transformation T is replaced by a transformation group{Sx : x ∈ Rd or Zd} indexed by Rd or Zd. For a point process model, the transformation Sx is usually to shift the realization ω ∈ Ω by x. A random model under a probability space (Ω, F, µ) is said to be ergodic if there exists a transformation group {Sx : x ∈ Rd or Zd} that acts ergodically on (Ω, F, µ). A transformation group {Sx : x ∈ Rd or Zd} is said to act ergodically if the σ-algebra of events invariant under the whole group is trivial, i.e., any invariant event has measure either zero or one. Moreover, a random model under a probability space (Ω, F, µ) is said to be mixing if there exists a transformation group {Sx : x ∈ Rd or Zd} such that for all E, F ∈ F, we have µ(SxE ∩ F ) − µ(E)µ(F ) → 0 as |x| → ∞. One direct consequence of an ergodic random model is presented as below.
Fact 3: For an ergodic random model (Ω, F, µ), if an event E ∈ F invariant under the whole transformation group {Sx: x ∈ Rd or Zd} occurs wpp., i.e., µ(E) > 0, then it occurs a.s., i.e., µ(E) = 1.
B. Proof of Theorem 1
1) Proof of T1.1: To prove T1.1, it suffices to show that for any two given points(λS1, λP T 1) and (λS2, λP T 2) in C, we can find a path in C that connects these two points. In particular, the
path we constructed is given by a horizontal segment and a vertical segment as shown in Fig. 13, where we assume, without loss of generality, that λS1≤ λS2.
(λS1, λP T 1)
(λS1, λP T 1)
(λS2, λP T 2)
(λS2, λP T 2) (λS2, λP T 1)
(λS2, λP T 1) λP T
λP T
λS λS
(a) λP T 1 ≤ λP T 2 (b) λP T 1 > λP T 2
Fig. 13. The continuous path connecting the two points(λS1, λP T1) and (λS2, λP T2) in the connectivity region C.
Consider case (a) in Fig. 13 where λP T 1 ≤ λP T 2. Case (b) can be proven similarly. First we show every point (λS, λP T 1) (λS1 ≤ λS ≤ λS2) on the horizontal segment belongs to C.
Let λ′ = λS − λS1. A Poisson point process X with density λS is statistically equivalent to the superposition of a Poisson point process X1 with density λS1 and an independent Poisson point process X′ with density λ′. It follows that any realization of the heterogeneous network with densities λS and λP T 1 can be generated by adding more secondary nodes to a realization of the heterogeneous network with densities λS1 and λP T 1. Thus, the existence of an infinite connected component in G(λS1, λP T 1) implies the existence of an infinite connected component in G(λS, λP T 1). We thus have that (λS, λP T 1) ∈ C for (λS1≤ λS ≤ λS2).
Now we know that the two end points (λS2, λP T 1) and (λS2, λP T 2) of the vertical segment belong to C. For a point (λS2, λP T) (λP T 1 ≤ λP T ≤ λP T 2) on the vertical segment, let λ′ = λP T 2− λP T, then any realization of the heterogeneous network with densitiesλS2 and λP T can be obtained by independently removing each primary transmitter-receiver pair with probability λ′/λP T 2 from a realization of the heterogeneous network with densitiesλS2 andλP T 2. It follows from the definition of communication link in the secondary network (see Sec. II-B1) that the existence of an infinite connected component inG(λS2, λP T 2) implies the existence of an infinite connected component in G(λS, λP T). Thus, we have (λS2, λP T) ∈ C (λP T 1 ≤ λP T ≤ λP T 2).
2) Proof of Theorem 1.2: Suppose that (λS, λP T) ∈ C (λP T > 0), then by using the coupling argument for showing that the vertical segment belongs to C in the above proof of T1.1, we conclude that (λS, 0) ∈ C, i.e., the λS-axis is the lower boundary of C.
