Minimum number of neighbors for fully connected uniform ad hoc

Minimum Number of Neighbors for Fully Connected Uniform Ad Hoc Wireless Networks† Gianluigi Ferrari1,2 , and Ozan K. Tonguz1 1

Electrical and Computer Engineering Department, Carnegie Mellon University Pittsburgh, PA, 15213-3890, USA - Phone: 412-268-5991 Fax: 412-268-2860 2 Dipartimento di Ingegneria dell’Informazione, University of Parma I-43100, Parma, Italy E-mail: [email protected],[email protected]

Abstract— Determining the minimum number of neighboring nodes required to guarantee full connectivity, i.e., to ensure that a node can reach, through multiple hops, any other node in the network, is an important problem in ad hoc wireless networks. In this paper, we consider reservation-based wireless networks with stationary and uniform (on average) node spatial distribution. Assuming that any communication route is a sequence of minimum length hops, we show that, in an ideal case without inter-node interference (INI) and on the basis of a suitable definition of transmission range, the minimum number of neighbors required for full connectivity is, on average, π. Full connectivity is guaranteed if the transmitted power (in the case of fixed node spatial density) or, equivalently, the node spatial density (in the case of fixed transmitted power) are larger than critical minimum values. In a realistic case with INI, we prove that there are situations where full connectivity can not be guaranteed, regardless of the number of neighbors or the transmitted power.

I. I NTRODUCTION Connectivity is a fundamental feature of an ad hoc wireless network, since it is an indicator of the likelihood that a multihop communication route between any two nodes can be sustained. A definition of connectivity is not straightforward, but in the literature the number of neighbors of a node is usually associated with the connectivity level of the network. In [1]–[6], various approaches, based on continuum percolation, throughput maximization, and random graph theory, are proposed for the evaluation of the minimum number of neighbors needed for full connectivity in a wireless network. According to the considered criterion for connectivity and the particular network topology, various values of the minimum number of neighbors needed for full connectivity are obtained. In this paper, on the basis of insights gained with a recently developed communication-theoretic framework for the analysis of ad hoc wireless networks [7]–[9], we formulate the problem of finding the minimum number of nodes required for full connectivity assuming that a communication route is formed by a sequence of minimum length hops and considering a proper definition of transmission range. According to our definition, we show that in the case of a stationary † The authors were funded in part by Army Research Office (ARO) under Contract No. DAAD19-02-1-0389 and CyLab of Carnegie Mellon University (CMU). Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the ARO or CyLab of CMU.

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wireless network with (i) uniform—in an average spatial sense—node distribution, (ii) no inter-node interference (INI), and (iii) omnidirectional transmission from each node, the number of neighbors required to guarantee full connectivity is π, provided that the transmitted power is above a critical threshold. This can be reformulated by requiring that, for a fixed transmitted power, the node spatial density is above a critical value. Expressions for the critical values (in terms of both transmitted power and node spatial density), related to the major physical network parameters, are given in the case of a network communication scenario characterized by binary phase shift keying (BPSK) signaling over a channel with free space loss. Assuming a given node spatial density, while in the ideal (no INI) case a minimum required transmitted power guaranteeing full connectivity always exists, in a realistic (INI) communication scenario such a value might not exist. In other words, there might be situations where full connectivity can never be guaranteed, regardless of the transmitted power. II. C OMMUNICATION -T HEORETIC P RELIMINARIES We consider an ad hoc wireless network communication scenario characterized as follows: (i) peer-to-peer multi-hop reservation-based communication with disjoint routes is considered (a node can not serve as a relay in more than one route); (ii) the nodes are fixed at the vertices of a regular grid; (iii) the route creation phase is not considered (we assume that the routes are created and we focus on the analysis of the transmission phase following the creation); (iv) a node generates packetized information only after reserving a multi-hop route to its desired destination and the generation process has a Poisson distribution with parameter λ (dimension: [pck/s]). In the following, a characterization of an average uniform spatial node distribution is first presented. Afterwards, on the basis of [8], [9], we derive an expression for the BER over a multi-hop communication route and we consider a suitable definition of communication range. In the rest of the paper, we assume that (i) the transmitted power, (ii) the transmission data-rate Rb , and (iii) the modulation format are fixed. A. Uniform Node Distributions We assume that N nodes are placed inside a planar surface of area A and are uniformly distributed and define by ρS
N/A the node spatial density. We denote the average distance

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00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111

00000 11111 11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111









N0 = 3 and rL ≈

0000000 1111111 1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111








 


•












•

between any pair of neighboring nodes as rL . The derivation of an average uniform model requires the introduction of some geometric regularity. The concept of Voronoi tessellation can be used to give a more precise definition of uniform node distribution. In [10], the authors prove a lemma according to which for every  > 0, there is a Voronoi tessellation of the plane with the property that every Voronoi cell1 contains a disk of radius  and is contained in a disk of radius 2. Based on this characterization of Voronoi tessellation, the following definition of uniform distribution will be used in the remainder of this paper. Definition 1 A node distribution is considered uniform if there exists a Voronoi tessellation such that every Voronoi cell is contained in a disk of radius rL . In other words, a uniform node distribution is such that the local structure is almost the same everywhere—for instance, this is not the case for a random distribution, where there could be significant variations between different regions of the network. Various uniform geometric distributions are possible. We indicate by N0 the average number of neighbors (i.e., at average distance) of any node—according to the uniformity of the node distribution, each node has approximately the same environment. If N0 is equal to 2, there must exist a single multi-hop path which connects all the nodes. For N0 = 1, the network is formed by disjoint pairs of connected nodes. We therefore restrict our attention to uniform node distributions where N0 ≥ 3. In particular, it is possible to show that the only perfectly uniform covering node distributions correspond to (i) square, (ii) triangular, and (iii) hexagonal tessellations [11]. Given the node spatial density ρS , it is of interest to determine its relationship with the average distance between neighboring nodes, i.e., rL . In Fig. 1, the covering polygons corresponding to perfectly uniform geometries are shown. Fig. 1 (a): a triangle is the basic covering polygon for

1 Given a distribution of nodes, we recall that the Voronoi cell of a node is defined as the set of all points, in the plane, which are closer to that node than to any of the other nodes.

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4 1 1 √ √ ≈ 0.88 √ . ρ ρS 3 3 S

(1)

Fig. 1 (b): a square of side rL is the basic covering polygon for N0 = 4 and 1 rL ≈ √ . ρS

Fig. 1. Covering polygons for possible perfectly uniform nodes distributions. Uniform nodes distributions where each node has N0 = 3, 4, and 6 neighbors at minimum distance (rL ) are considered in (a), (b) and (c), respectively.

•




(2)

Fig. 1 (c): a hexagon is the basic covering polygon for N0 = 6 and
2 1 1 rL ≈ √ √ ≈ 1.07 √ . (3) ρS 3 ρS

Relaxing the condition that the basic covering element is a regular polygon, many uniform tiling combinations can be found [11]. We do not elaborate further on this aspect, since it is beyond the scope of this paper. The existence of many possible tessellations motivates the assumption that, in an average uniform spatial node distribution, the neighbors2 of √ each node are at a distance rL ≈ 1/ ρS from it. The goal of this paper is to determine the minimum number of neighbors for full connectivity in a spatially uniform wireless network. B. BER at the End a Multi-Hop Communication Path We assume that a communication route in an ad hoc wireless network is formed by a sequence of hops of length rL , i.e., a sequence of minimum length hops. This is the most reasonable assumption, since we are considering a flat architecture, and it is desirable that each node uses the minimum possible amount of energy for transmitting to a neighboring node. We assume shortest path routing—details of routing will not be considered in the following.3 Indicating by BERL the BER at the end of a single link, assuming that (i) there is regeneration (i.e., detection and possibly error correction) at each intermediate node, and that (ii) the uncorrected errors made in successive links accumulate, it is possible to show that the BER at the end of the n-th link of a multi-hop route, indicated by BER(n) , can be written as BER(n) ≈ 1 − (1 − BERL )n .

(4)

The link BER depends of course on the signal-to-noise ratio (SNR) at the ending node of the link (indicated by SNRL ), the modulation, possible channel coding and channel characteristics (e.g., with frequency selective versus non-selective fading). In particular, the link BER is a decreasing function of SNRL . This motivates the following assumption. 2 Note that in a uniform topology, the average number of neighbors (at √ distance 1/ ρS ) of each node is, on average, π. To the best of our knowledge, there is no rigorous approach to prove that this number is also the minimum number of neighbors required for full connectivity. For example, in [12], the author notices that, on average, the number of nearest neighbors is π, but argues that in general a larger number of neighbors should be guaranteed to provide connectivity. 3 Although the absence of routing problems is an idealistic assumption, this allows to provide a physical layer criterion for the determination of the minimum number of neighbors.

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Assumption 1 Given a maximum tolerable value of the link BER, there exists a minimum value of the link SNR— depending on the expression of the link BER—such that if the link SNR is larger than this value, then the BER at the ending node of the link is lower than the maximum link BER: max , ∃SNRmin = SNRmin ∀BERmax L L L (BERL ) : max ∀ ≥ 0, SNRL = SNRmin L +  ⇒ BERL ≤ BERL .(5)

III. M INIMUM N UMBER OF N EIGHBORS FOR F ULL C ONNECTIVITY In order to evaluate the minimum number of neighbors needed to guarantee full connectivity (in an average sense), we distinguish two cases: (i) ideal (no INI) and (ii) realistic (INI).

A. No Inter-Node Interference (Ideal Case) In this case, the receiver is affected by thermal noise only. In particular, this noise is independent from the node Based on Assumption (1), the following lemma can be derived. spatial density and the transmitted power. We assume that the thermal noise power can be written as Pthermal = F kT0 B, Lemma 1 For any given maximum tolerable BER at the end where F is the noise figure, k ≈ 1.38 × 10−23 J/K is the of a multi-hop path with an arbitrary number of hops, there Boltzman’s constant, T0 ≈ 300 K is the room temperature, and exists a minimum value of the link SNR such that if the link B is the transmission bandwidth. We consider the following SNR is larger than this minimum value, then the BER at the example of a realistic communication scenario to motivate end of the multi-hop path is lower than the maximum tolerable future assumptions. BER: Example: (Uncoded BPSK Signaling over an AWGN Channel with Free-Space Loss and no INI) Assuming that the power max min min max , nmax ), ∃SNRL = SNRL (BER , nmax ) : ∀(BER transmitted by a node is Pt , the received power at distance d (nmax ) ∀ ≥ 0, SNRL = SNRmin ≤ BERmax . (6) from the transmitting node, indicated by P (d) , can be written, L +  ⇒ BER r according to Friis free space formula [13], as Lemma 1 refers to a generic number of hops nmax . Given 2 a particular geometry, it is desirable to identify the maximum αPt Gt Gr λc Pt Pr(d) = 2 = , (9) number of hops for any possible communication route. For d (4π)2 fl d2 example, assuming that the routing strategy is “intelligent,” where: Gt and Gr are the transmitter and receiver antenna the longest possible path approximately corresponds to the gains, respectively; λc = c/fc is the wavelength corresponding diameter of the network (i.e., the largest possible distance to the carrier frequency fc (c is the speed of light); fl ≥ 1 is between two nodes). Hence, for any finite area A and any a loss factor which takes into account the losses not related to given number of nodes N , it follows that nmax can be propagation. Note that Friis formula applies in the far-field or uniquely identified. For example, in the considered circular Fraunhofer region, i.e., it applies provided that the distance area, indicating by dA the diameter of the area and recalling between the two communicating nodes is larger than the 2 the assumption that each path is formed by a sequence of Fraunhofer distance [13], defined as df = 2Dc fc , where D is minimum length hops, one can write that nmax = dA /rL . the antenna diameter. It follows that, in order to apply the free Given that nmax can be determined, Lemma 1 implies that space loss propagation model, the condition rL ≥ df must be it is possible to identify a minimum value SNRmin L . We then satisfied—in most practical network scenarios, this condition is consider the following definition of transmission range for a (r ) satisfied. Since Pr L ≈ αρS Pt , in the case of BPSK signaling multi-hop network communication scenario. (B ≈ Rb ) over an additive white Gaussian noise (AWGN) channel the BER at the end of a communication route with n Definition 2 Consider a wireless network with N nodes unihops thus becomes formly placed over an area A. Assume that a prescribed max  n imum BER at the end of any multi-hop communication route 2αρS Pt (n) . (10) BER ≈ 1 − 1 − Q is assigned—the maximum possible number of hops, which F kT0 Rb depends on N and A, can be determined. The transmission range rT of a node is defined as the distance such that the SNR Assumption 2 In a spatially uniform wireless network withat the input of another node at this distance is the minimum out INI, for any given minimum value of the link SNR, value required to guarantee full (average) connectivity: there exists a minimum transmitted power such that if the transmitted power is larger than this value, then the SNR at ∀(N, A, BERmax ), ∃rT = rT (N, A, BERmax ) : the end of a link is larger or equal than the required SNRmin L : max SNR|d=rT = SNRmin = SNRmin ). (7) L L (N, A, BER min ∀SNRmin = Ptmin (SNRmin L , ∃Pt L ): min A neighbor of a node is then a node whose distance from Pt ≥ Pt ⇒ SNRL ≥ SNRmin (11) L . the given node is not larger than the transmission range. Assumption 2 can be reformulated in the following way. Assuming that the radio transmission pattern of each node If there is no INI, then it is always possible, by sufficiently is omnidirectional, it follows that the number of neighbors of increasing the transmitted power, to guarantee that the SNR a node can be written as at the end of a minimum length hop is above any consid(8) ered threshold SNRmin N0 = ρS πrT2 . L . This assumption can also be given

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a complementary interpretation. If the transmitted power is at the fixed, for any minimum required SNR value SNRmin L end of a communication link, there exists a critical node spatial such that if ρS ≥ ρmin then the SNR at the end density ρmin S S of a communication link is larger than SNRmin L . Assumption 2 refers to the transmitted power since it represents a network parameter under control, while the node spatial density might be uncontrollable. However, the alternative interpretation in terms of minimum required node spatial density has interesting connections with the theory of percolation [14]. Based on Assumption 2, the following proposition can be derived. Proposition 1 For a given maximum tolerable BER at the end of any possible multi-hop communication path in a uniform ad hoc wireless network with finite area, the minimum number of neighbors required for full connectivity is, on average, π. In other words, indicating by N0 the number of neighbors of a node, ∀(BERmax , nmax ), ∃Ptmin = Ptmin (BERmax , nmax ) : Pt = Ptmin ⇒ N0 = π, BER(nmax ) = BERmax .(12) Proof: According to Lemma 1, max ∀(BERmax , nmax ), ∃SNRmin = SNRmin , nmax ) : L L (BER min (max) max SNRL ≥ SNRL ⇒ BER ≤ BER . (13)

Based on Assumption 2, ∃Ptmin

=

Ptmin (SNRmin L )

: Pt ≥

Ptmin

⇒ SNRL ≥

SNRmin L . (14)

In the limiting case for Pt = Ptmin , it follows that SNRL =SNRmin L . Hence, the transmission range rT corresponds to the distance rL between two neighboring nodes. In this case (where full connectivity is guaranteed), the number of neighbors of a node, in an average spatially uniform network, is 2 N0 = ρS πrT2 ≈ ρS πrL = π.

(15)

Since the result in (15) is derived for any possible maximum number of hops nmax of a communication route and for any maximum tolerable BER, we can conclude that it is independent from these values. Note that if Pt > Ptmin , then rT > rL and N0 > π. Proposition 1 can be given a simple and intuitive interpretation. In the case with a uniform network topology with fixed node spatial density, in order to support multiple hops it is necessary that each node reaches its nearest neighbors with an amount of power which guarantees sufficient regeneration of the transmitted signal along a multi-hop route in order to have a minimum prescribed BER at the final node. This situation can happen if the transmitted power is larger than a minimum critical value (for fixed node spatial density) or if the node spatial density is larger than a minimum critical value (for fixed transmitted power). At this point, full connectivity, through multi-hop routing, is guaranteed and the minimum number of neighbors required is π. We

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emphasize that the minimum number of neighbors required for connectivity corresponds to π under the proposed physical layer-based criterion related to a maximum acceptable BER at the end of a multi-hop communication route and under the assumption of a uniform node topology. In [2], the authors consider a connected network where the nodes are distributed according to a two-dimensional Poisson distribution, and, upon a criterion based on the maximization of the network throughput, they show that the ideal number of neighbors is around 5.89. Hence, different criteria (minimization of the BER at the end of a multi-hop route or maximization of the throughput defined as the ratio between the average sourcedestination distance and the progress per hop) and topologies (regular or random) lead to different conclusions on the “best” number of neighbors [15]. Example: (Uncoded BPSK Signaling over an AWGN Channel with Free-Space Loss and no INI) Let us assume that nmax represents the maximum possible number of hops in the considered ad hoc wireless network, and that BERmax is the minimum required BER to be guaranteed at the end of an nmax -hop communication path. Considering the single hop SNR threshold implied by Lemma 1, the condition can be equivalently rewritten as [9] SNRL ≥SNRmin L ρS Pt SNRmin L F kT0 . ≥ ρmin
E Rb α

(16)

The quantity ρmin (dimension: [J/m2 ]) can be interpreted as E the minimum spatial energy density required to support full connectivity in the network [8]. For a given node spatial density ρS , it follows that the minimum transmitted power can be written as Ptmin = ρmin E Rb /ρS , whereas, for a given transmitted power Pt , the critical minimum node spatial den= ρmin sity required for full connectivity is ρmin S E Rb /Pt . B. Inter-Node Interference (Realistic Case) In the case of communications affected by INI, Assumption 2 is no longer valid. In order to motivate this statement, we extend the previously considered example to the INI case. Example: (Uncoded BPSK Signaling over an AWGN Channel with Free-Space Loss and INI) In the case of INI, the link SNR can be written as follows: Psignal T SNRIN = (17) L Pthermal + PIN T where PIN T is the interfering power. Expression (17) underlies the assumption, usually considered in multiple access schemes, that the sum of the interfering signals can be considered as an additive white noise process. The expression for PIN T depends on the specific MAC protocol used. In the following, we consider the simple case of the first reservationbased MAC protocol4 proposed in [8], [9]. This MAC protocol is characterized by the fact that a node, after reserving a multihop route to the destination, starts transmitting regardless of 4 In [8], [9], this MAC protocol is defined as Aloha, for its resemblance with the classic Aloha MAC protocol. However, compared to regular Aloha protocol, there are significant differences in terms of (i) route reservation and (ii) absence of retransmission in intermediate links.

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the activity of the other nodes. For this reason, we refer to this MAC protocol as reserve-and-go (RESGO). In this case, on the basis of a bit-level interference analysis it is possible to show that RESGO PIN T

−λL/Rb

≈ αPt ρS (1 − e

)∆A (N )

Since lim

T SNRIN = L

1 (1 − e−λL/Rb )∆A (N )

(20)

it is immediate to conclude that the maximum achievable link SNR is intrinsically limited by the multiple access interference. Motivated by the considered example (and by other examples not reported here for lack of space), a reasonable assumption to characterize an ad hoc wireless network communication environment with INI is the following. Assumption 3 In a wireless network with INI, the interference power at the end of a communication link between two neighboring nodes is proportional to the transmitted power Pt and to the node spatial density ρS . In other words, PIN T ∝ ρS Pt .

(21)

It is intuitive to expect that the interference noise power, given by the contributions of all nodes, is proportional to Pt . Moreover, it is intuitive to assume that the denser the nodes, the larger the interference noise power. An important consequence of Assumption 3 is the fact that, in the case with INI, there exists a maximum value of the link SNR which can not be exceeded, regardless of the transmitted power. In (r ) fact, since the received power Pr L is proportional to the transmitted power, based on Assumption 3 it follows that the T , is a link SNR in the case with INI, indicated by SNRIN L monotonically increasing function of ρS Pt and is always lower than the following maximum value: T,max SNRIN
L

lim

Pt or ρS →∞

T SNRIN . L

(22)

The following proposition can then be derived. Proposition 2 In an ad hoc wireless network communication scenario with INI where Assumption 3 holds, there might be two mutually exclusive situations. • There is not full connectivity, regardless of the transmitted power. In other words, indicating by N0 the number of neighbors of a node, ∃(BERmax , nmax ) : max ∀Pt , SNRL < SNRmin = SNRmin , nmax ). (23) L L (BER

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There is full connectivity, on average, and the minimum required number of neighbors is π. In other words, ∀(BERmax , nmax ), ∃Ptmin = Ptmin (BERmax , nmax ) : Pt = Ptmin ⇒ N0 = π, BER(nmax ) = BERmax . (24)

(18)

where λ is the average packet transmission rate, L (dimension: [b/pck]) is the packet dimension, and, in the case of square grid topology,    √
N /2 i−1 1 6  +8  − 1 . (19) ∆A (N )
 i2 i2 + j 2 i=1 j=1

Pt or ρS →∞

•

IV. C ONCLUSIONS In this paper, the theoretical problem of determining the minimum number of neighbors for full connectivity in spatially uniform ad hoc wireless networks has been considered. Based on a suitable definition of spatially uniform node distribution and transmission range, it has been proved that in an ideal network communication scenario without INI, the minimum number of neighbors needed for full connectivity in uniform networks is, on average, π. Based on a reasonable assumption characterizing the case with INI, it has been shown that in uniform ad hoc wireless networks networks with INI two mutually exclusive situations are possible: (i) there is connectivity and the minimum number of neighbors is, on average, π or (ii) there is no connectivity, regardless of the number of neighbors. R EFERENCES [1] E. N. Gilbert, “Random plane networks,” SIAM J., vol. 9, no. 4, pp. 533–543, December 1961. [2] L. Kleinrock and J. A. Silvester, “Optimum transmission radii for packet radio networks or why six is a magic number,” in Nat. Telecommun. Conf., dec 1978, pp. 4.3.1–4.3.5. [3] T. K. Philips, S. S. Panwar, and A. N. Tantawi, “Connectivity properties of a packet radio network model,” IEEE Trans. Inform. Theory, vol. 35, no. 5, pp. 1044–2047, September 1989. [4] Y.-C. Cheng and T. G. Robertazzi, “Critical connectivity phenomena in multihop radio models,” IEEE Trans. Commun., vol. 37, no. 7, pp. 770–777, July 1989. [5] C. Bettstetter, “On the minimum node degree and connectivity of a wireless multihop network,” in Proc. ACM Int. Symp. on Mobile Ad Hoc Network. and Comput. (MOBIHOC), Lausanne, Switzerland, June 2002, pp. 80–91. [6] O. Dousse, P. Thiran, and M. Hasler, “Connectivity in ad-hoc and hybrid networks,” in Proc. IEEE Conf. on Computer Commun. (INFOCOM), New York, U.S.A., June 2002, pp. 1079–1088. [7] O. K. Tonguz and G. Ferrari, “A Communication-Theoretic Framework for Ad Hoc Wireless Networks,” Carnegie Mellon University, ECE Dept., Tech. Rep., TR-043-2003, February 2003. [8] G. Ferrari and O. K. Tonguz, “Performance of circuit-switched ad hoc wireless networks with Aloha and PR-CSMA MAC protocols,” in Proc. IEEE Global Telecommun. Conf. (GLOBECOM), San Francisco, USA, December 2003, pp. 2824 – 2829. [9] ——, “MAC protocols and transport capacity in ad hoc wireless networks: Aloha versus PR-CSMA,” in Proc. IEEE Military Comm. Conf. (MILCOM), Boston, USA, October 2003. [10] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Trans. Inform. Theory, vol. 46, pp. 388–404, March 2000. [11] B. Grunbaum and G. C. Shephard, Tilings and Patterns. New York: W. H. Freeman, 1987. [12] T. J. Shepard, “A channel access scheme for large dense packet radio networks,” in Proc. ACM Conference of the Special Interest Group on Data Communication (SIGCOMM), Palo Alto, CA, 1996, pp. 219–230. [13] T. S. Rappaport, Wireless Communications. Principles & Practice. Upper Saddle River, NJ, U.S.A.: Prentice-Hall, 2002, second edition. [14] R. Meester and R. Roy, Continuum Percolation. Cambridge, U.K.: Cambridge University Press, 1996. [15] O. K. Tonguz and G. Ferrari, “Is the number of neighbors in ad hoc wireless networks a good indicator of connectivity?” in Proc. International Zurich Seminar on Communications (IZS): Access-TransmissionNetworking, Zurich, Switzerland, February 2004, pp. 40 – 43.

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