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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 2, FEBRUARY 2012 263 Connectivity of Multiple Cooperative Cognitive Radio Ad Hoc Networks Weng Chon Ao, Shin-Ming Cheng, Member, IEEE, and Kwang-Cheng Chen, Fellow, IEEE Abstract—In cognitive radio networks, the signal reception activity of an SU is constrained by its interference to PUs.quality of a secondary user degrades due to the interference It is much harder to ensure the connectivity of the secondaryfrom multiple heterogeneous primary networks, and also the network. As a result, if multiple secondary networks can co-transmission activity of a secondary user is constrained by itsinterference to the primary networks. It is difﬁcult to ensure the operate with each other by having other secondary networks’connectivity of the secondary network. However, since there may nodes acting as relays, connectivity may be improved1.exist multiple heterogeneous secondary networks with different Connectivity of a homogeneous wireless ad hoc networkradio access technologies, such secondary networks may be has been widely studied in different aspects. In [5], n nodestreated as one secondary network via proper cooperation, to im- are uniformly distributed in a unit square where two nodesprove connectivity. In this paper, we investigate the connectivityof such a cooperative secondary network from a percolation- are connected if their distance is smaller than r(n). To ensurebased perspective, in which each secondary network’s user may 1-connectivity of the network, i.e., a path exists between anyhave other secondary networks’ users acting as relays. The con- pair of nodes, r(n) = log n+c(n) where c(n) → ∞ as n → πnnectivity of this cooperative secondary network is characterizedin terms of percolation threshold, from which the beneﬁt of ∞. In [6], k-connectivity of a network, i.e., at least k node-cooperation is justiﬁed. For example, while a noncooperative disjoint paths exist between any pair of nodes, was studied. Onsecondary network does not percolate, percolation may occur in the other hand, in many applications such as sensor networksthe cooperative secondary network; or when a noncooperative and mobile ad hoc networks, 1-connectivity is not required. Asecondary network percolates, less power would be required to weaker connectivity from a percolation-based perspective (i.e.,sustain the same level of connectivity in the cooperative secondarynetwork. a nonvanishing fraction of nodes in the network, say, 95%, are connected) is more suitable, for example, signiﬁcant savings Index Terms—cognitive radio networks, connectivity, cooper- in power consumption can be achieved by allowing a smallation, heterogeneous wireless networks, ad hoc networks fraction (say, 5%) of nodes to be disconnected [7], [8]. For an http://ieeexploreprojects.blogspot.com inﬁnite network with density λ and transmission range r (i.e., a I. I NTRODUCTION random geometric network), the network percolates in whichC OGNITIVE radio networking has emerged as a promis- an inﬁnite connected component appears when the average ing technology to enhance spectrum utilization by sens- degree of a network node is above a certain number (i.e.,ing the spectrum and opportunistically accessing the spectrum λπr2 ≥ 4.52 [9]). Otherwise, the network does not percolateof primary (licensed) systems [1], [2]. The spectrum sharing and consists of many isolated ﬁnite clusters. In our study, amodel can be practically divided into two main approaches: secondary user may be deactivated or disconnected from eachinterweave and underlay [3]. In the interweave approach, other due to the appearance of primary transmissions so thatsecondary (unlicensed) users (SUs) embedded with cognitive 1-connectivity in the secondary network may not be feasible,radios orthogonally access temporary spectrum holes of pri- justifying investigation of connectivity of secondary networkmary systems facilitated by the advanced spectrum sensing from a percolation-based approach.and signal processing techniques. In the underlay approach, In [10], [11], the effect of fading on the mean degreeprimary users (PUs) and SUs transmit at the same time while of a network node was discussed. In [7], [12], the impactSUs are considered of lower priority in spectrum access. of fading on the percolation of a wireless ad hoc networkTheir interference to PUs should be limited by an interference was investigated. The effect of self co-channel interferencetemperature [4] indicating the maximum tolerable interference was considered in [13] and the authors proposed a Signalpower. This paper focuses on the underlay approach. It is To Interference Ratio Graph (STIRG) to study percolation inimportant to understand the connectivity of the secondary ad hoc network. Connectivity of a network with unreliablenetwork so that proper network operations can be performed. links was studied in [14] in a percolation sense. PercolationHowever, different from a homogeneous network user, the of a cooperative wireless ad hoc network was addressed insignal reception quality of an SU further degrades due to the [15], in which nodes may form a group to perform distributedinter-system interference from PUs, and also the transmission beamforming so that signals are coherently combined at some destinations. Long range transmissions can be established Manuscript received 14 February 2011; revised 14 July 2011. This researchis sponsored by the National Science Council under the contract of NSC and thus connectivity is improved. Note that all the above98-2221-E-002-065-MY3, NSC 99-2911-I-002-001, and NSC 99-2911-I-002- works were focused on connectivity of a homogeneous ad hoc201. W. C. Ao, S.-M. Cheng, and K.-C. Chen are with the Graduate In- 1 Note that in this paper multiple secondary networks cooperate with eachstitute of Communication Engineering, National Taiwan University, Taipei, other by relaying (carrying) each other’s trafﬁc, which is different fromTaiwan (e-mail: r97942044@ntu.edu.tw, smcheng@cc.ee.ntu.edu.tw, and physical layer cooperative techniques such as distributed beamforming.chenkc@cc.ee.ntu.edu.tw). Digital Object Identiﬁer 10.1109/JSAC.2012.120204. 0733-8716/12/$25.00 c 2012 IEEE
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264 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 2, FEBRUARY 2012network. Connectivity of heterogeneous wireless networks, II. BACKGROUNDspeciﬁcally cognitive radio networks, has been recently inves- A. Degree distribution of a stand-alone wireless networktigated in [16]–[18]. In [16], the connectivity of a secondarynetwork is parameterized by the tuple (λP T , λSU ) under path We consider a stand-alone wireless ad hoc network, inloss channel model, where λP T is the density of primary which the spatial distribution of nodes is assumed to followtransmitters and λSU is the density of SUs. Percolation of a homogeneous Poisson point process (PPP) with density λ.the secondary network occurs only when λP T is below some Let Φ = {Xk } denote the set of locations of the nodesthreshold and λSU is above some threshold. Another parame- and let P denote the transmit power of a node. A pair ofterization of the secondary network (λSU , rSU ) is developed nodes are connected if the capacity of the channel betweenin [17] under Rayleigh fading channel model, where rSU is them can support an information rate R, which happens when G P r −αthe transmission range of an SU with an outage constraint. log 1 + XN0 ≥ R, where GX denotes the channelThe temporal activity of a spectrum opportunity is further power gain, r is the distance between the pair of nodes, αcaptured in [18]. In spite of the above efforts, the connectivity is the path loss exponent, and N0 is the background noiseof cooperative secondary network has not been investigated. power. In this paper, we therefore analyze the beneﬁt of cooperation Consequently, neighbors of a typical node (a reference nodeamong multiple secondary networks from a percolation-based located at the origin)3 are resulted from independently thinning Rperspective. Quantiﬁcation of connectivity improvement with of each node Xk ∈ Φ with probability P GX ≥ (2 −1)N0 . k P Xk −αnetwork-based cooperation facilitates network control and G denotes the channel power gain of the desired link from Xkdesign. To our knowledge, it is the ﬁrst framework to jointly the typical node to the node at X , which is independently kconsider the effects of cooperation (among multiple secondary drawn according to some distribution f , i.e., the distribution GXnetworks), interference (from multiple primary networks), function f GXk (x) = fGX (x), ∀k. Xk denotes the distanceheterogeneity (in different priorities of spectrum access), and between the typical node and the node at X . By the mapping kgeneral fading on the percolation of wireless networks. Our theorem [11], [22], the number of neighbors (or the degree)framework can be summarized into four steps. In the ﬁrst of the typical node is Poisson distributed with mean βstep, we derive the degree distribution of an SU in a sec- ∞ondary ad hoc network2 sustaining interference from multiple (2R − 1)N0 β= λP GX ≥ 2πrdrheterogeneous primary ad hoc networks under general fading 0 P r−αof desired signal and interfering signals. The relationship −δ (2R − 1)N0between the degree distribution of a http://ieeexploreprojects.blogspot.com wireless network node = λπ E[Gδ ], X (1) Pand the connectivity of the wireless network in percolation 2sense can be established by extending the methods of complex where δ = α . The probability that a node has k neighbors isnetworks [19], [20]. In the second step, we obtain the active denoted as pk , which satisﬁesprobability of an SU. An SU can transmit (or be active) exp(−β)β konly when its interference power to each primary receiver pk = , k = 0, 1, 2, . . . (2) k!in primary networks is below the interference temperature.In the third step, with both the degree distribution and the The probability that a node has no neighbors (or the nodeactive probability of an SU, percolation threshold of a single isolation probability) is exp(−β). Note that in the case withoutsecondary network is derived by using the site-percolation fading, the average number of neighbors of a node (or the −δ (2R −1)N0model [21]. The site-percolation model is widely used to study mean degree) is λπ P , implying that E[Gδ ] Xpercolation of a network (of some degree distribution) with corresponds to the connectivity fading gain [10], [11].random node failures. A random node failure is analogous tothe deactivation of an SU in our study. In the fourth step, weinvestigate the connectivity of cooperative secondary network B. Percolation analysisin which each secondary network’s node may have other The concept of percolation is originated from statisticalsecondary networks’ nodes acting as relays. The beneﬁt of co- physics and random networks [9], [20]. In our study, asoperation among multiple heterogeneous secondary networks the transmit power of a node P increases, the mean degreeis analytically characterized in terms of percolation threshold. β increases. If we draw links between each node and its The remainder of this paper is organized as follows. Section neighbors, the resulting network graph percolates (i.e., anII provides background on degree distribution of a stand-alone inﬁnite connected component appears in the network) whenwireless fading network and percolation analysis. In Section β is above a percolation threshold βth . On the other hand,III, we present the network model, the degree distribution when β is below the percolation threshold, the network doesof an SU, the active probability of an SU, the mapping not percolate and consists of many isolated ﬁnite components.to site-percolation model, and the optimal power allocation The percolation threshold βth can be computed by ﬁnding theamong SUs. In Section IV, we study percolation in cooperative critical point at which the mean component size blows up,secondary network. Numerical results are provided in Section indicating the formation of an inﬁnite connected component.V. Section VI gives the conclusion. 3 By the stationary characteristic of homogeneous PPP [22], the statistics 2 Inan ad hoc network, the nodes are distributed as a homogeneous Poisson measured by the typical node is representative for all other nodes. Therefore,point process, and two nodes are connected by a link if the channel between we focus our discussions on the typical node. A typical node may be athem can support a given information rate. transmitter or a receiver depending on the context.
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AO et al.: CONNECTIVITY OF MULTIPLE COOPERATIVE COGNITIVE RADIO AD HOC NETWORKS 265 Desired link fGX Interfering link fGY1 Interfering link fGY2 = + + + ... Interfering link fGZ1 Interfering link fGZ2 Rx of SN0 Tx of SN0Fig. 1. P An illustration of the self-consistency relation H1 (x) = Rx of PN1x ∞ qk (H1 (x))k = xF1 (H1 (x)). A square represents the connected k=0 Tx of PN1component of nodes reached by following a randomly selected link, and a Rx of PN2circle represents the node ﬁrst reached. The component size can be expressedas the sum of the probabilities of having only a single node, having a single Tx of PN2node connected to one other component, etc. (a) Interfering link fGZA,1 Interfering link fGZA,2 Desired link fGXA,AWe describe the procedure of ﬁnding the percolation threshold Desired link fGXA,Bas follows. Interfering link fGY1,A Interfering link fGY1,B We ﬁrst deﬁne the generating function of the degree distri- ∞ Interfering link fGY2,Abution of a randomly selected node as F0 (x) = k=0 pk xk = Interfering link fGY2,Bexp(β(x − 1)). Also, we deﬁne the excess degree4 distri- Rx of PN1bution of a node at the end of a randomly selected link Tx of PN1 Rx of SNAas qk . Since it is more likely to arrive at a node that Rx of PN2 Rx of SNBhas a higher degree by following a randomly selected link, Tx of PN2 Tx of SNA (b)qk = (k + 1)pk+1 / k kpk = (k + 1)pk+1 /β is propor-tional to kpk [23]. The generating function of qk is de- Fig. 2. (a) A transmitter (Tx) in SN0 broadcasts a message while receivers ∞ k (Rx) in SN0 sustain interference from heterogeneous primary networks P N1ﬁned as F1 (x) = k=0 qk x , and we can easily verify and P N2 . The channel power gain of the desired link is drawn accordingthat F1 (x) = F0 (x)/β = exp(β(x − 1)), where F0 (x) to distribution fG , the channel power gain of the interfering link from Xdenotes the derivative of F0 (x). Moreover, the generating P N1 (P N2 ) to SN0 is drawn according to distribution fGY 1 (fGY 2 ),function of the total number of nodes reachable by following and the channel power gain of the interfering link from SN0 to P N1a randomly selected link (resp. node) is deﬁned as H1 (x) (P N2 ) is drawn according to of SN andfGZ 1 (fGZ 2 ). (b) A cooperative secondary network consisting distribution SNB sustains interference from A(resp. H0 (x)), which satisﬁes the self-consistency relation heterogeneous primary networks P N1 and P N2 .H1 (x) = x ∞ qk (H1 (x))k = xF1 (H1 (x)) (resp. H0 (x) = k=0 http://ieeexploreprojects.blogspot.comx ∞ pk (H1 (x))k = xF0 (H1 (x))). An illustration is pro- k=0vided in Fig. 1. Consequently, the average total number PPP with density λ. Let Φ = {Xk } denote the set of locationsof nodes (or the mean component size) reachable from a of the SUs and let P denote the transmit power of an SU. SN0 F0 (1)randomly selected node is H0 (1) = 1 + 1−F (1) , which blows coexists with N heterogeneous primary networks (denoted asup when 1 − F1(1) = 0 indicating the formation of an inﬁnite P Ni , 1 ≤ i ≤ N ), where the spatial distribution of PTs 1connected component. Here in the Poisson case, it corresponds (resp. PRs) in R Ni is assumed to follow a PPP with density Pto 1−β = 0, and the percolation threshold βth = 1 is obtained. μP T (resp. μP ).5 The set of locations of the PTs in P Ni is i iMore details can be found in [20], [23], [24]. denoted as ΨP T = {Yk } and the transmit power of a PT in i i Note that the above formulation ignores the spatial depen- P Ni P RPi . The set of locations of the PRs in P Ni is denoted is idency (i.e., two of a node’s neighbors are likely neighbors as Ψi = {Zk } and the interference temperature at a PR inthemselves). The actual percolation threshold is offset with a P Ni is τi . The interference from the PTs in P Ni to a typical i −αsmall constant depending on the fading distribution. In the case SU of SN0 is denoted as Ii = Yk ∈ΨP T GYk Pi Yk i i i .without fading, the network graph is the so-called random ge- GYki denotes the channel power gain of the interfering link iometric graph with βth ≈ 4.52, while channel fading produces from the PT at Yk to the typical SU, which is assumed to bespread-out connections and weakens the spatial dependency, independently drawn from fGY i , i.e., the distribution function iand thus reduces the percolation threshold [7], [12], [25]. In fGYk (x) = fGY i (x), ∀k. Yk is the distance between the PT iour analysis, the spatial dependency may be further reduced at Yki and the typical SU, and α is the path loss exponent. Alldue to the randomness from the interference, and we justify packet transmissions of the networks are assumed to be slottedthat the spatial dependency is small before the secondary and synchronized. Since SN0 operates in an interference-network percolates in Appendix A of [26]. In Section IV limited environment, we ignore the background noise.of the paper, we follow the approach in [20], [23], [24] to As a result, neighbors of a typical SU of SN0 are resultedprovide benchmark comparison of the percolation thresholds from independently6 thinning of each SU Xk ∈ Φ with −αof a noncooperative secondary network and the cooperative probability P log 1 + GXk PN Xk P ≥ R , where GXk de- Iione. i=1 5 For example, in P N , each PT has a dedicated PR at distance d away i III. N ETWORK M ODEL AND ANALYTIC METHODS with an arbitrary direction so that the PRs also form a PPP with density P R = μP T . μi i We consider a secondary network (denoted as SN0 ), in 6 We assume that the spatial correlation of interference observed at differentwhich the spatial distribution of SUs is assumed to follow a nodes in Φ can be ignored due to the i.i.d. channel gains in different pairs of nodes [27]. This assumption leads to the Poisson degree distribution of an 4 Excess degree of a node is one less than its total degree. e SU. Theorem III.1-III.3 (about the mean β) still hold without this assumption.
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266 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 2, FEBRUARY 2012notes the channel power gain of the desired link from the asymptotically tight), suggesting that outage is mainly causedtypical SU to the SU at Xk , which is assumed to be in- by dominating interferers [26] as the path loss exponent isdependently drawn from fGX , i.e., the distribution function large.fGXk (x) = fGX (x), ∀k. Xk denotes the distance between We conclude that the degree distribution of an SU sustainingthe typical SU and the SU at Xk , and R denotes the infor- interference from multiple heterogeneous primary networksmation rate. An illustration of above scenario is provided in (denoted as pk ) is a Poisson distribution with mean β, i.e., ˜Fig. 2(a). By the mapping theorem, the number of neighbors exp(−β)β kof a typical SU of SN0 is Poisson distributed with mean P pk = ˜ , k = 0, 1, 2, . . . (7) ∞ (2R −1)( N Ii ) k!β = 0 λP GX ≥ i=1 P r −α 2πrdr, which is furthercomputed in the next subsection. B. Active probability of an SU The interference power from a typical SU of SN0 to theA. Degree distribution of an SU PR at Zk ∈ ΨP R is GZk P Zk −α , where GZk denotes the i i i i iTheorem III.1. Under general fading of both desired signal channel power gain of the interfering link from the typicaland interfering signals, β is bounded above by i SU to the PR at Zk , which is assumed to be independently λP δ E[Gδ ] drawn from fGZ i , i.e., the distribution function fGZ i (x) = X β< β Upper , (3) i fGZ i (x), ∀k. Zk denotes the distance between the typical k N i=1 μP T Piδ E[Gδ i ] (2R − 1)δ i Y i SU and the PR at Zk . An illustration is provided in Fig. 2(a).where δ = 2 We deﬁne the set Ξi = {Zk ∈ ΨP R : GZk P Zk −α > τi }. PR i i i i α. PR Ξi consists of PRs in P Ni whose received interference Proof: The proof is presented in Appendix B of [26]. power from the typical SU exceeds the interference tempera-Theorem III.2. Under general fading of both desired signal ture τi .and interfering signals, β is computed in nearly closed form From mapping theorem, the number of PRs in ΞP R is i ∞ g(t) Poisson distributed with mean νi λP δ −∞ tδ dt ∞ β= . (4) νi = μP R P GZ i P r−α > τi 2πrdr N i i=1 μP T Piδ E[Gδ i ] i Y (2R − 1)δ Γ(1 − δ) 0 ∞ 1g(t) is the inverse Laplace transform of F GX (s) satisfying = EGZ i μP R 1 r < (GZ i P τi−1 ) α 2πrdr ∞ http://ieeexploreprojects.blogspot.comi 0F GX (s) = L{g(t)} = −∞ e−st g(t)dt, where F GX (s) =P(GX ≥ s) is the ccdf (complementary cumulative distribu- = EGZ i μP R π(GZ i P τi−1 )δ ition function) of the channel power gain of the desired link. = μP R πP δ E[Gδ i ]τi−δ , i Z (8) ∞ z−1 −tΓ(z) = 0 t e dt is the Gamma function. where 1(·) is the indicator function. An SU can be active only Proof: The proof is presented in Appendix C of [26]. when there are no PRs in ΞP R , ∀i, 1 ≤ i ≤ N . Thus, the i For example, when the desired link is Rayleigh faded, active probability of an SU (denoted as pa ) can be computedi.e., the channel power gain GX is exponentially dis- astributed (with unit mean, without loss of generality), we Nhave F GX (s) = e−s = L{δ(t − 1)} where δ(t) is the a p = P(ΞP R = φ) iDirac delta function. We can thus compute the expres- i=1 ∞ ∞sion −∞ g(t) dt = −∞ δ(t−1) dt = 1 and obtain β = tδ tδ N δ P λP . = exp(−νi )( N μP T Piδ E[Gδ i ])(2R −1)δ Γ(1−δ) i=1 i Y i=1 NTheorem III.3. Under general fading of both desired signal (a)and interfering signals, β is bounded below by = exp −μP R πP δ E[Gδ i ]τi−δ i Z i=1 2 λP δ E[Gδ ](1 − X α−2 ) + N β≥ β Lower , (5) = exp − μP R E[Gδ i ]τi−δ πP δ , (9) N i=1 μP T Piδ E[Gδ i ] (2R − 1)δ i Y i Z i=1where (x)+ max(x, 0). The lower bound is non-trivial when where (a) follows (8).α > 4. Proof: The proof is presented in Appendix D of [26]. C. Mapping to site-percolation model We compare β Upper with β Lower , it is observed that From Section III-A and Section III-B, we obtain the degree β Upper 1 distribution of an SU (i.e., pk ), which is Poisson with mean ˜ = 2 + . (6) β, and the active probability of an SU pa . Now, we are able to β Lower (1 − α−2 ) derive the percolation threshold of the secondary network SN0The ratio between the upper bound and the lower bound is by extending the analysis in Section II-B to the scenario witha constant depending on the path loss exponent. Note that random node failure (deactivation of an SU) with probabilitywhen α → ∞, β Upper /β Lower → 1 (i.e., the bounds are 1 − pa , using the site percolation model [21].
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AO et al.: CONNECTIVITY OF MULTIPLE COOPERATIVE COGNITIVE RADIO AD HOC NETWORKS 267 A B We deﬁne the generating function of the degree distribution with set of locations of nodes ΦA = {Xk } and ΦB = {Xk }, ∞of a randomly selected SU as F0 (x) =˜ ˜ k = k=0 pk x with transmit power PA and PB , and with information rateexp(β(x − 1)). The excess degree distribution of an SU at RA and RB . The channel power gain of the link betweenthe end of a randomly selected link is deﬁned as qk , which ˜ a transmitting node in SNu , u ∈ {A, B} and a receivingsatisﬁes qk = (k + 1)˜k+1 / k k pk = (k + 1)˜k+1 /β. The ˜ p ˜ p node in SNv , v ∈ {A, B} is assumed to be drawn according ∞generating function of qk is deﬁned as F1 (x) = k=0 qk xk , ˜ ˜ ˜ to some distribution fGX u,v . Furthermore, the channel power ˜and we can easily verify that F1 (x) = F0˜ (x)/β = exp(β(x − gain of the interfering link from a PT in P Ni , 1 ≤ i ≤ N1)). Moreover, the generating function of the total number to a receiving node in SNv , v ∈ {A, B} is assumed to beof active SUs reachable by following a randomly selected drawn according to some distribution fGY i,v . An illustration ˜ ˜link (resp. SU) is deﬁned as H1 (x) (resp. H0 (x)), which is provided in Fig. 2(b). The average number of neighbors insatisﬁes the self-consistency relation H1 (x) = 1 − pa + ˜ SN v , v ∈ {A, B} of a node of SN u , u ∈ {A, B} is denoted ∞ k as βu,v .pa x k=0 qk (H1 (x)) = 1 − pa + pa xF1 (H1 (x)) (resp. ˜ ˜ ˜ ˜ ∞ k We take βA,B as an example. With cooperation, a node inH0 (x) = 1 − p + p x k=0 pk (H1 (x)) = 1 − pa + ˜ a a ˜ ˜ a ˜ ˜ SNB listens the message from a node of SNA , and decodingp xF0 (H1 (x))). Consequently, the average total number of succeeds with probabilityactive SUs reachable from a randomly selected SU (or themean size of a cluster of connected and active SUs) is GX A,B PA r−αH0 (1) = pa 1 + pa F0 (1)/(1 − pa F1 (1)) , which blows up ˜ ˜ ˜ P log 1 + N ≥ RA , (11) i=1 Iiwhen 1 − pa F1 (1) = 0 indicating the formation of an inﬁnite ˜ where r is the distance between the pair of nodes. By usingconnected component. Here in the Poisson case, it corresponds Theorem III.1, the number of neighbors in SNB of a node ofto 1 − pa β = 0 and we obtain the percolation threshold SNA is Poisson distributed with mean βA,B , whereβth = 1/pa . ∞ N (2RA − 1)( i=1 Ii ) βA,B = λB P GX A,B ≥ 2πrdrD. Optimal power allocation 0 PA r−α We observe that when an SU’s transmit power P decreases, λB PA E[Gδ A,B ] δ Xits active probability increases; however, its mean degree β < . (12) N P T P δ E[Gδdecreases. We maximize the effective mean degree of an SU i=1 μi i Y i,B ] (2RA − 1)δwith respect to P , which is deﬁned as the product of its active Similarly, the average number of neighbors in SN of a node Aprobability and its mean degree. That http://ieeexploreprojects.blogspot.com is, of SN , β , satisﬁes B B,A N pa β = exp − μP R E[Gδ i ]τi−δ πP δ λA PB E[Gδ B,A ] δ X i Z βB,A < . (13) N i=1 i=1 μP T Piδ E[Gδ i,A ] (2RB − 1)δ i Y λP δ E[Gδ ] X × . (10) Note that βA,B and βB,A are different. βA,A and βB,B can be N i=1 μP T Piδ E[Gδ i ] (2R − 1)δ i Y computed similarly. In the mean time, exact value and lowerNote that we use the upper bound on β in Theorem III.1. bound of the mean degree can be respectively derived by usingAfter some calculations, the optimal transmit power is P ∗ = Theorem III.2 and Theorem III.3. −δ −1 In addition, the channel power gain of the interfering link N π i=1 μP R E[Gδ i ]τi−δ i Z , which maximizes the effec- from a transmitting node in SNu , u ∈ {A, B} to a PR intive mean degree of an SU. P Ni , 1 ≤ i ≤ N is assumed to be drawn according to some distribution fGZ u,i . An illustration is provided in Fig. 2(b).IV. C OOPERATION AMONG HETEROGENEOUS SECONDARY The active probability of an SU in SNA (resp. SNB ) is NETWORKS denoted as pa (resp. pa ). By using (9), pa and pa can be A B A B As discussed in Section III-C, when the mean degree computed asof a secondary network node is lower than the percolation Nthreshold βth , the secondary network does not percolate. We pa = exp − A μP R E[Gδ A,i ]τi−δ i Z δ πPA ;here propose and prove that different secondary networks i=1may cooperate with each other by having other secondary Nnetworks’ nodes acting as relays to improve network con- pa = exp − B μP R E[Gδ B,i ]τi−δ i Z δ πPB . (14)nectivity. The beneﬁt of cooperation among heterogeneous i=1secondary networks can be quantiﬁed in terms of percolation Now, we have the following theorem.threshold. We ﬁrst focus on the case with cooperation betweentwo heterogeneous secondary networks coexisting with N Theorem IV.1. The percolation condition of the cooperativeheterogeneous primary networks. Then, the result is extended secondary network consisting of SNA and SNB isto the case with cooperation among multiple heterogeneous 1 − pa βA,A −pa βA,B A A = 0, (15)secondary networks. −pa βB,A 1 − pa βB,B B B Suppose that there are two secondary networks (denoted asSNA and SNB ), respectively with node density λA and λB , or, (1 − pa βA,A )(1 − pa βB,B ) − pa pa βA,B βB,A = 0. A B A B
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268 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 2, FEBRUARY 2012In addition, the mean component size of a cluster of connected 60and active SUs reachable from an SU of SNA (resp. SNB ) Mean component size ˜ H0 (1)is denoted as H0A (1) (resp. H0B (1)), which satisﬁes ˜ ˜ 40 1/(1 − pa β) H0A (1) ˜ pa pa β pa β = a + a A,A a A,B A A A H0B (1) ˜ pB pB βB,A pB βB,B 20 −1 1 − pa βA,A −pa βA,B A A pa A 0 . (16) −pa βB,A 1 − pa βB,B B B pa B 0.1 0.2 0.3 0.4 0.5 0.6 Transmit power P 0.7 0.8 0.9 1 (a) Proof: The proof is presented in Appendix E of [26]. Effective Mean degree pa β With cooperation, βA,B and βB,A are greater than zero. 1.1 1 − pa βA,A −pa βA,B A ALet C , the mean component −pa βB,A 1 − pa βB,B B B 1sizes H0A (1) and H0B (1) are proportional to 1/ det(C). When ˜ ˜ 0.9the ﬁrst zero of det(C) is reached (i.e., equation (15)), the 0.8mean component size blows up indicating the formation of 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Transmit power Pan inﬁnite connected component in the cooperative secondary (b)network. In the case without cooperation, βA,B and βB,Aequal zero, recovering the percolation condition of a single Fig. 3. Percolation of the secondary network SN0 occurs (or the mean component size blows up) when the effective mean degree is greater thansecondary network SN A (or SN B ). That is, (15) reduces to e one, i.e., pa β ≥ 1. The system parameters are set as λ = 10−3 , R = 4,the percolation threshold βA,A = 1/pa (or βB,B = 1/pa ) as A B α = 4, N = 1, μP T = μP R = 10−4 , P1 = 0.15, and τ1 = 3.5 × 10−8 . 1 1described in Section III-C. All links are Rayleigh faded with unit average power.Corollary IV.1. By extending Theorem IV.1 to the case withcooperation among M heterogeneous secondary networks, the βA,A + βA,B decreases. We maximize the effective totalpercolation condition becomes mean degree of an SU of SNA with respect to PA . After ∗ 1 − pa β1,1 −pa β1,2 · · · −pa β1,M 1 1 1 some calculations, the optimal transmit power is PA = −δ −1 −pa β2,1 1 − pa β2,2 · · · −pa β2,M 2 2 2 π N μP R E[Gδ A,i ]τi−δ . = 0, (17) i=1 i Z . . . . .. . . . . . http://ieeexploreprojects.blogspot.com . −pa βM,1 −pa βM,2 · · · 1 − pa βM,M M M M V. N UMERICAL RESULTSwhere βi,j is the average number of neighbors in secondary Fig. 3 illustrates the site-percolation model in Section III-C.network j of an SU of secondary network i, and pa is the i In Fig. 3(b), the relationship between the effective meanactive probability of an SU in secondary network i. degree of an SU pa β and its transmit power P is shown. When pa β ≥ 1, the secondary network percolates and the When all the off-diagonal entries are zero, (17) reduces to ˜ mean component size H0 (1) blows up as shown in Fig. 3(a),the percolation threshold of a single noncooperative secondary indicating the formation of an inﬁnite connected componentnetwork. in the secondary network. Note that when P is between 0.24 and 0.82, the secondary network percolates; while P is tooA. Optimal power allocation large (i.e., the active probability pa is extremely small) or too small (i.e., the mean degree β is extremely small) such that In the cooperative secondary network consisting of SNA pa β < 1, the secondary network does not percolate.and SNB , the effective total mean degree of an SU of SNA Fig. 4 and Fig. 5 demonstrate the beneﬁt of cooperation.(or SNB ) can be deﬁned as the product of its active probability In Fig. 4, the behavior of the cooperative secondary networkand its total mean degree in both SNA and SNB . For example, consisting of SNA and SNB is studied. The transmit powerthe effective total mean degree of an SU of SNA is of SNA is varied from PA = 0.005 to PA = 0.5. When PA N increases, the changes of pa βA,A and pa βA,B are shown in A Apa (βA,A + βA,B ) = exp − A μP R E[Gδ A,i ]τi−δ i Z δ πPA Fig. 4(b). pa βB,A and pa βB,B do not change since they do B B i=1 not depend on PA (note that PB is ﬁxed). With cooperation ⎛ λA PA E[Gδ A,A ] δ among SNA and SNB , the mean component size H0A (1) ˜ ×⎝ X ˜ B (1)) reachable from an SU of SNA (resp. SNB ), (resp. H0 N i=1 μP T Piδ E[Gδ i,A ] (2RA − 1)δ i Y which is proportional to 1/det(C), blows up when the ﬁrst ⎞ zero of det(C) is reached, indicating the formation of an in- λB PA E[Gδ A,B ] δ + X ⎠. (18) ﬁnite connected component in the cooperative secondary net- N μP T Piδ E[Gδ i,B ] (2RA − 1)δ i=1 i Y work. In other words, percolation of the cooperative secondary network occurs when det(C) = (1 − pa βA,A )(1 − pa βB,B ) − A BNote that we use the upper bound on βA,A and βA,B . When pa pa βA,B βB,A ≤ 0. In Fig. 5, without cooperation, the mean A Bthe transmit power of an SU of SNA , PA , decreases, its component size of SNA (resp. SNB ) never blows up sinceactive probability increases; however, its total mean degree pa βA,A < 1 (resp. pa βB,B < 1). Note that βA,B = βB,A = 0 A B
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AO et al.: CONNECTIVITY OF MULTIPLE COOPERATIVE COGNITIVE RADIO AD HOC NETWORKS 269 ˜ Cooperation among SNA and SNB : H0A (1) 200 100 ˜ Cooperation among SNA and SNB : H0B (1)Mean component size ˜ Cooperation among SNA and SNB : H0A (1) Mean component size 150 80 Cooperation among SNA and SNB : 1/det(C) ˜ Cooperation among SNA and SNB : H0B (1) 60 100 Cooperation among SNA and SNB : 1/det(C) 40 50 20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Transmit power PA 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 (a) ˜ pa βA,A Transmit power PA A 0.6 ˜ ˜ pa βA,A pa βA,B A Effective mean degree A ˜ ˜ pa βA,B pa βB,A A Effective mean degree 0.4 B 0.8 ˜ ˜ pa βB,A pa βB,B B 0.6 B 0.2 det(C) ˜ pa βB,B B 0.4 det(C) 0 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 Transmit power PA (b) −0.2 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Transmit power PA Fig. 4. With cooperation among SNA and SNB , percolation of the cooperative secondary network occurs (or the mean component size reach- able from an SU of SNA , H0A (1), or SNB , H0B (1), blows up) when ˜ ˜ Fig. 6. With cooperation among SNA and SNB , percolation of the e e e e det(C) = (1 − pa βA,A )(1 − pa βB,B ) − pa pa βA,B βB,A ≤ 0. The cooperative secondary network occurs when det(C) ≤ 0. The system A B A B system parameters are set as λA = 1.1 × 10−3 , RA = 4, λB = 1 × 10−3 , parameters are set as λA = 1.8 × 10−3 , RA = 4, λB = 1 × 10−3 , RB = 3, PB = 0.01, α = 4, N = 1, μP T = μP R = 10−4 , P1 = 0.15, RB = 3, PB = 0.01, α = 4, N = 1, μP T = μP R = 10−4 , P1 = 0.15, 1 1 1 1 and τ1 = 1 × 10−8 . PA is varied from 0.005 to 0.5. All links are Rayleigh and τ1 = 1 × 10−8 . PA is varied from 0.005 to 0.05. All links are Rayleigh faded with unit average power. faded with unit average power. 1.5 of SNA (resp. SNB ) blows up when PA = 0.0075 as shown Mean component size in Fig. 6. In Fig. 7, without cooperation, the mean component http://ieeexploreprojects.blogspot.com only when P = 0.1 (corresponding Without cooperation SNA size of SNA blows up 1 A Without cooperation SNB to pa βA,A = 1); however, the mean component size of SNB A 0.5 never blows up since PB = 0.01 is ﬁxed (corresponding to pa βB,B = 0.47 < 1). This example demonstrates that B 0 percolation occurs much earlier (at lower PA , 0.0075 < 0.1) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 with cooperation among SNA and SNB , suggesting that SNA Transmit power PA (a) can operate at lower transmit power to sustain the same level 0.7 of connectivity. Power efﬁciency is therefore improved. Effective mean degree 0.6 0.5 0.4 ˜ pa βA,A VI. C ONCLUSION A 0.3 ˜ pa βB,B B We investigate the connectivity of cooperative secondary 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 network from a percolation-based perspective by the following Transmit power PA (b) steps. First, the degree distribution of an SU is derived considering interference from multiple heterogeneous primary Fig. 5. Without cooperation, SNA (resp. SNB ) does not percolate (or networks under general fading of desired signal and inter- e the mean component size does not blow up) since pa βA,A < 1 (resp. e A fering signals. Second, the active probability of an SU is pa βB,B < 1). The system parameters are the same as in Fig. 4. B obtained considering interference temperature constraints at PRs. Third, the relationship between the degree distribution of an SU and the connectivity of the secondary network (with when there is no cooperation between SNA and SNB . This random node deactivation) in percolation sense is established example demonstrates that percolation occurs only when both by using the site-percolation model. Fourth, the above steps SNA and SNB cooperate with each other by having other are extended to the analysis of connectivity of cooperative network’s nodes acting as relays. secondary network, in which each secondary network’s node In Fig. 6 and Fig. 7, we provide another example to illustrate may have other secondary networks’ nodes acting as relays. the beneﬁt of cooperation among SNA and SNB . The system The beneﬁt of cooperation among multiple secondary net- parameters are the same as in Fig. 4 except that λA = works is analytically characterized in terms of the percolation 1.8×10−3. With cooperation among SNA and SNB , the mean threshold. Connectivity and energy efﬁciency of the CRN can component size H0A (1) (resp. H0B (1)) reachable from an SU ˜ ˜ be improved by the proposed cooperation framework.
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270 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 2, FEBRUARY 2012 [13] O. Dousse, F. Baccelli, and P. Thiran, “Impact of interferences on 100 connectivity in ad hoc networks,” IEEE/ACM Trans. Netw., vol. 13, no. 2, pp. 425–436, Apr. 2005. Without cooperation SNAMean component size 80 [14] Z. Kong and E. Yeh, “Connectivity and latency in large-scale wireless Without cooperation SNB networks with unreliable links,” in Proc. IEEE INFOCOM ’08, Apr. 60 2008. [15] D. Goeckel, B. Liu, D. Towsley, L. Wang, and C. Westphal, “Asymptotic 40 connectivity properties of cooperative wireless ad hoc networks,” IEEE J. Sel. Areas Commun., vol. 27, no. 7, pp. 1226–1237, Sept. 2009. 20 [16] W. Ren, Q. Zhao, and A. Swami, “Connectivity of heterogeneous wireless networks,” IEEE Trans. Inf. Theory, vol. 57, no. 7, pp. 4315– 4332, July 2011. 0.05 0.1 0.15 0.2 0.25 0.3 [17] W. C. Ao, S.-M. Cheng, and K.-C. Chen, “Phase transition diagram Transmit power PA for underlay heterogeneous cognitive radio networks,” in Proc. IEEE GLOBECOM ’10, Dec. 2010. [18] P. Wang and I. Akyildiz, “Dynamic connectivity of cognitive radio 1 ad-hoc networks with time-varying spectral activity,” in Proc. IEEE GLOBECOM ’10, Dec. 2010. Effective mean degree 0.9 [19] R. Albert and A.-L. Barab´ si, “Statistical mechanics of complex net- a 0.8 works,” Rev. Mod. Phys., vol. 74, no. 1, pp. 47–97, Jan. 2002. ˜ pa βA,A [20] M. E. J. Newman, “The structure and function of complex networks,” A 0.7 SIAM Rev., pp. 167–256, 2003. 0.6 ˜ pa βB,B B [21] D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, “Network robustness and fragility: Percolation on random graphs,” Phys. 0.5 Rev. Lett., vol. 85, no. 25, pp. 5468–5471, Dec. 2000. [22] J. Kingman, Poisson Processes. Oxford University Press, 1993. 0.4 [23] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, “Random graphs 0.05 0.1 0.15 0.2 0.25 0.3 with arbitrary degree distributions and their applications,” Phys. Rev. E, Transmit power PA vol. 64, no. 2, p. 026118, July 2001. [24] M. E. J. Newman, “Mixing patterns in networks,” Phys. Rev. E, vol. 67, no. 2, p. 026126, Feb. 2003. Fig. 7. Without cooperation, percolation of SNA occurs when PA = 0.1, [25] M. Franceschetti, L. Booth, M. Cook, R. Meester, and J. Bruck, and SNB does not percolate. The system parameters are the same as in Fig. 6. “Continuum percolation with unreliable and spread out connections,” J. of Statistical Physics, vol. 118(3/4), pp. 721–734, Feb. 2005. [26] W. C. Ao, S.-M. Cheng, and K.-C. Chen, “Connectivity of multiple cooperative cognitive radio ad hoc networks (extended version),” 2011. R EFERENCES [Online]. Available: http://santos.ee.ntu.edu.tw/11jsac-long.pdf http://ieeexploreprojects.blogspot.com “Spatial and temporal correlation of the [27] R. Ganti and M. Haenggi, [1] I. Akyildiz, W. Lee, M. Vuran, and S. Mohanty, “NeXt genera- interference in aloha ad hoc networks,” IEEE Commun. Lett., vol. 13, tion/dynamic spectrum access/cognitive radio wireless networks: A no. 9, pp. 631–633, Sept. 2009. survey,” Comput. Netw., vol. 50, no. 13, pp. 2127–2159, Sept. 2006. [2] S. Haykin, “Cognitive radio: brain-empowered wireless communica- tions,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. Weng Chon Ao was born in Macau. He received the B.S. degrees in both [3] A. Goldsmith, S. A. Jafar, I. Mari´ , and S. Srinivasa, “Breaking spectrum c computer science and physics from National Taiwan University, Taiwan, in gridlock with cognitive radios: An information theoretic perspective,” 2008, and the M.S. degree in communications engineering from National Proc. IEEE, vol. 97, no. 5, pp. 894–914, May 2009. Taiwan University, Taiwan, in 2010. From 2010, he has been a research [4] Y. Xing, C. N. Mathur, M. Haleem, R. Chandramouli, and K. Subbalak- assistant with Intel-NTU lab, Taiwan. His research interests include stochastic shmi, “Dynamic spectrum access with qos and interference temperature modeling and control of communication networks. constraints,” IEEE Trans. Mobile Comput., vol. 6, no. 4, pp. 423–433, Apr. 2007. [5] P. Gupta and P. Kumar, “Critical power for asymptotic connectivity,” in Proc. IEEE CDC ’98, 1998. Shin-Ming Cheng (S’05-M’07) received the B.S. and Ph.D. degrees in com- [6] H. Zhang and J. Hou, “Asymptotic critical total power for k-connectivity puter science and information engineering from National Taiwan University, of wireless networks,” IEEE/ACM Trans. Netw., vol. 16, no. 2, pp. 347– Taipei, Taiwan, in 2000 and 2007, respectively. He joined the Graduate 358, Apr. 2008. Institute of Communication Engineering, National Taiwan University, Taipei, [7] X. Ta, G. Mao, and B. Anderson, “On the giant component of wireless Taiwan, as a postdoctoral research fellow in 2007. His research interests multihop networks in the presence of shadowing,” IEEE Trans. Veh. include wireless communications, network security, cognitive radio networks, Technol., vol. 58, no. 9, pp. 5152–5163, Nov. 2009. and network science. [8] P. Santi and D. Blough, “The critical transmitting range for connectivity in sparse wireless ad hoc networks,” IEEE Trans. Mobile Comput., vol. 2, no. 1, pp. 25–39, Jan. 2003. [9] M. Franceschetti and R. Meester, Random Networks for Communication: Kwang-Cheng Chen (M’89-SM’93-F’07) received his B.S. degree from From Statistical Physics to Information Systems. Cambridge University National Taiwan University in 1983, and M.S. and Ph.D. degrees from the Press, 2008. University of Maryland, College Park, in 1987 and 1989, all in electrical [10] D. Miorandi, E. Altman, and G. Alfano, “The impact of channel ran- engineering. From 1987 to 1998 he was with SSE, COMSAT, the IBM domness on coverage and connectivity of ad hoc and sensor networks,” Thomas J. Watson Research Center, and National Tsing Hua University, IEEE Trans. Wireless Commun., vol. 7, no. 3, pp. 1062–1072, Mar. 2008. Hsinchu, Taiwan, working on mobile communications and networks. He is [11] M. Haenggi, “A geometric interpretation of fading in wireless networks: a Distinguished Professor and Director of the Graduate Institute of Com- Theory and applications,” IEEE Trans. Inf. Theory, vol. 54, no. 12, pp. munication Engineering and the Communication Research Center, National 5500–5510, Dec. 2008. Taiwan University. He has received numerous awards and honors, including [12] R. Hekmat and P. Mieghem, “Connectivity in wireless ad hoc networks co-authoring 3 IEEE papers to receive ISI Classic Citation Award, the IEEE with a log-normal radio model,” Mob. Netw. Appl., vol. 11, no. 3, pp. ICC 2010 Best Paper Award, and IEEE GLOBECOM 2010 GOLD Best Paper. 351–360, June 2006. His research interests include wireless communications and network science.
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