On the Connectivity Analysis over Large-Scale Hybrid Wireless Networks

Transcription

1 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM. On the Connectivity Analysis over Large-Scale Hybri Wireless Networks ChiYianWenyeWang Department of Electrical an Computer Engineering North Carolina State University Abstract Many real systems are hybri networks which inclue infrastructure noes in multi-hop wireless networks, such as sinks in sensor networks an mesh routers in mesh networks. However, we have very little unerstaning of network connectivity in such networks. Therefore, in this paper, we consier hybri networks enote by H(α, β) with a hoc noes an base stations an prove how base stations can improve the connectivity of a hoc noes in subcritical phase, that is, the a hoc noe ensity, is lower than the critical ensity λ c α. We fin that with the existence of a positive ensity of base stations, i.e., the ensity of base stations λ β > which have the same transmission range as a hoc noes, the number of connecte a hoc noes is Θ(n) with probability nearly, where n is the number of a hoc noes. However, the size of connecte a hoc component scales linearly with λ β when it is lower than c () with probability nearly, which emonstrates a tremenous benefit of using base stations to enhance the connectivity of a hoc noes. Further, we stuy a hybri network architecture that makes a significant connectivity improvement with transmission range r β larger than r α for a hoc noes. Therefore, our results provie a theoretical unerstaning of to what extent a hoc noes can benefit from base stations in multihop wireless networks. I. INTRODUCTION Connectivity is an essential issue in multi-hop wireless networks in that a wireless noe must establish a communication path to other noes in orer to successfully transmit or receive information. To this en, there have been extensive stuies on the connectivity of wireless a hoc networks an sensor networks, in which a large number of wireless noes, either mobile or static, communicate with each other in a multi-hop fashion [] [6]. For example, Gupta an Kumar in [] provie the critical transmission range for asymptotic connectivity in a ense network, where n noes are uniformly an inepenently place in a unit isc. Further, a funamental result in continuum percolation theory [7], [8] shows that in an extene network, where n noes are ranomly eploye in a square region by use of Poisson istribution with ensity, there exists a critical value, which is calle critical ensity λ c α. When noe ensity is above λ c α (supercritical), there exists a unique giant component containing Θ(n) noes asymptotically almost surely (a.a.s.); an if falls below λ c α (subcritical), the network will be partitione into a number of finite-size connecte components almost surely (a.s.). However, large-scale pure a hoc networks rarely exist in real wor, because the per-noe throughput can iminish to zero [9] []. The most wiely use systems are in fact Fig.. B B Subnet-A Subnet-A A hybri network with base stations an a hoc noes. hybri networks which inclue infrastructure noes in multihop wireless networks, such as sinks in sensor networks an mesh routers in mesh networks. In recent years, there have been consierable stuies of such networks [] [5], which have mainly focuse on the capacity of hybri networks. For example, Liu et al. in [] investigate the capacity of hybri networks, in which n a hoc noes are uniformly an inepenently locate in a unit isk, but m base stations are place on a regular gri within the area of the network. Their results show that the benefit of aing base stations on capacity is insignificant if m grows asymptotically slower than n. But if m grows asymptotically faster than n, the capacity increases linearly with the number of base stations, which provies an effective improvement over pure a hoc networks. One interesting problem, yet remains unsolve, is to what extent the base stations improve the network connectivity, even though it appears to be intuitive that base stations can surely improve network connectivity. As an example, Fig. shows that two connecte components of a hoc noes, Subnet-A an Subnet-A with size 6 an 3 respectively, are separate before two base stations, B an B, are ae in the network. Nevertheless, after B an B are ae, Subnet-A an Subnet-A are connecte together an thus form a larger connecte component. Therefore, in this paper, our objective is to stuy the connectivity in hybri networks, an we focus specifically on the case that a hoc noe ensity, is lower than the critical ensity λ c α, uner which the connectivity in pure a hoc network is ba. The challenge for our stuy is, in the existing literature, there are rarely analytical results on the connectivity in such inhomogeneous networks. Hence, our results can provie a funamental unerstaning an analytical support of the connectivity in hybri networks. First, we stuy a pure a hoc network which is in subcritical phase, an examine the upper boun of complementary //$6. IEEE

2 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM. cumulative istribution function (ccf) of origin stretch, which is efine as the largest Eucliean istance between the origin an the noes in a connecte component containing the origin. Then we use a mapping approach calle box connection mapping to obtain the upper boun of ccf of origin stretch, an hereby fin the value c( ), by which origin stretch is boune with probability nearly. Then we procee to stuy a hybri network H(α, β) in which the transmission ranges of a hoc (α-)noes r α is normalize to unit an base stations (s) are multiples of this unit, enote by r β. We prove that when the Eucliean istance between ajacent base stations is larger than c ( )=c( )+ for r β = r α =, s connecte irectly by any two base stations, o not intersect with probability nearly. Here, we say s are connecte irectly by a base station irectly, iff these s can be connecte by the base station even if we remove all other base stations. For example, in Fig., the s in Subnet-A are connecte irectly by B,butthes in Subnet-A are not connecte irectly by B. Base on this result, we fin that with the existence of a positive ensity of base stations λ β >, the number of connecte a hoc noes is Θ(n) with probability nearly, where n is the number of a hoc noes. However, the size of connecte a hoc component scales linearly with λ β that is lower than c ( ) with probability nearly, which emonstrates a tremenous benefit of using base stations to enhance the connectivity of a hoc noes. Further, we stuy a hybri network architecture which can utilize large r β (> ) to obtain an aitional benefit on the connectivity of a hoc noes. Our results inicate that by increasing r β from unit to k c ( ) (k N), with proper placement base stations an aitional a hoc noes in the network, the connectivity of a hoc noes can be significantly improve, which provie a guieline on esign of a new hybri network. The rest of the paper is organize as follows. In Section II, we introuce network moels an formulate the problem. In Section III, we provie the value c( ) that origin stretch is lower than with probability nearly in pure a hoc networks. In Section IV, we present theoretical proof of the connectivity improvement ue to base stations, followe by the conclusions in Section V. II. NETWORK MODELS AND PROBLEM STATEMENT A. Hybri Network Moel There are two types of noes in the hybri network H(α, β). The first set consists of a hoc noes which are calle s. A set of n s A = {α,α,...,α n } are ranomly eploye at locations A = {a,a,...,a n } in a square region D with sie length, by use of Poisson istribution with ensity. This square region D is also calle a continuum box [8]. Every has the same transmission power an signal receiving capability, an thus has the same transmission range enote as r α. That is, for any two s α i,α j A, there is an unirecte link between them iff a i a j <r α, where enotes the Eucliean istance. Without loss of Fig.. m The layout of a hybri network. generality, we normalize r α to unit throughout the paper. Then we enote ranom geometric graph (RGG) G(H λα, ) as the ensemble of all s an the links between them. It is well known from continuum percolation theory [7], [8], that there exists a critical ensity <λ c α < in G(H λα, ) such that: ()for >λ c α, there exists a unique connecte component containing Θ(n) noes a.a.s. (supercritical phase); an () for <λ c α, the connecte components contain a finite number of noes a.s. (subcritical phase). Until recently, the exact value of λ c α is still unknown although some bouns have been given in the literatures. The analytical boun.696 <λ c α < 3.37 is given in [7], while the simulation boun.434 <λ c α <.438 is given in [6]. In this paper, we assume the ensity of α- noes,, is lower than λ c α. The secon set consists of infrastructure noes which are calle s. A set of ms B = {β,β,...,β m } (with ensity λ β = m ) are place regularly at the locations B = {b,b,...,b m } ([ + m + m i, + m + m j], i, j m ) in the region D as epicte in Fig.. The s are connecte by high banwith wirelines an act only as relay noes for routing an forwaring the packets from an to the s. We assume that s, like base stations, are much more powerful than s that are power-constraine [9] an s have a transmission range of r β r α =). That is, for any pair of α i Aan B, there is an unirecte link between them iff a i b j <r β. Figs. 3(a) an 3(b) emonstrate an example of ata flow from an α i to a in the case r β = an r β =. In both cases, there exists one, which is in the transmission range of, an can communicate with α i in a multi-hop fashion. Therefore, the ata sent from α i can reach through the relay of s. As a result, the hybri network H(α, β), is efine as the ensemble of all s, s, an links (wireline an wireless) between them. B. Definitions Since continuum percolation theory stuies the connectivity properties in homogeneous networks, there is no efinition in the theory for hybri networks. Therefore, we efine β- component an β-giant component in hybri network H(α, β) as follows: Definition (β-component): In hybri network H(α, β), the β-component enote by C (α,β) is efine as the connecte component containing all s, an a subset of s

3 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM. α i α i (a) Insufficient use of s (a) r β = (b) r β = Fig. 3. Data flow from an to a. α i OS() (b) Sufficient use of s Fig. 5. Insufficient use vs. sufficient use of s (r β =). Fig. 4. Origin stretch in pure a hoc network. that are connecte by the s. Note that, we assume all s are connecte via broaban wireline, that is, if a connecte component contains one, it will contain all s, of which the number are constant. Thus, we efine β-component size C (α,β) as the number of s in C (α,β). For example, as epicte in Fig., there are two s in hybri network, connecting two separate subsets together to form a larger connecte component. The connecte component incluing B, B, Subset-A an Subset-A is the β-component. Definition (β-giant Component): In hybri network H(α, β), the β-giant component, is the β-component with size C (α,β) =Θ(n). Next, we efine a new metric origin stretch in pure a hoc networks. Definition 3 (Origin Stretch): Let G(H λα,, ) = G(H λα, ) {}, an C be the connecte component in G(H λα,, ) containing the origin. Theorigin stretch of G(H λα,, ) is efine as OS( )=max αi C α i, where α i is the Eucliean istance between α i an the origin. For example, as epicte in Fig. 4, origin stretch OS( ) is the Eucliean istance between α i an the origin. C. Problems an Main Results ) Problem Formulation: Our objective in this paper, is to stuy how can improve the connectivity of α- noes, an investigate what are steps. Now, we formulate the problems aresse in this paper. First, we start from a special case, that is, s only an fin the upper boun of origin stretch for any given ensity, to unerstan the coverage of s. Then, we stuy how to use s efficiently. From previous iscussions, we have observe that s can improve network connectivity. An interesting question is that how many s nee to be ae to a network. Similar questions, like the placement of base stations in sensor networks have been stuie in [7], [8]. While this question is beyon the scope of this stuy, it is relevant to the evaluation of s utilization in network connectivity. For example, in Fig. 5(a), two β- noes, an, are connecte through the relay of a number of s. Let I be the number of s connecte irectly by. Here, as we explain in Section I, s are connecte irectly by a, iff these s can be connecte by the even if we remove all other s. Similarly, let J be the number of s connecte irectly by. For the particular case shown in the figure, we can easily fin that I =9an J =3. However, the total number of s, which are connecte irectly by an,is3 <I+ J. In Fig. 5(b), an cannot be connecte through the relay of s. Then the total number of s, which are connecte irectly by an,is4 = I + J. Apparently, in the later case, the two s are better use. Thus, in this case, we call the two s are sufficiently use. Further, we efine all s in hybri network H(α, β) are sufficiently use, iff any two β- noes cannot be connecte by s. Thir, we evaluate the impact of s on the connectivity in a hybri network. This problem is twofol: the effect of noe ensity λ β an the effect of transmission range r β on network connectivity, that is, the conitions for the existence of β-giant component in a hybri network H(α, β). ) Main Results: In orer to answer the first question, our approach is to fin the upper boun of origin stretch, that is, the value that origin stretch is lower than with probability nearly. Then, we use box connection mapping to investigate the upper boun of complementary cumulative istribution function (ccf) of origin stretch in the following theorem. Base on this result, we will be able to fin the answer regaring the first question.

4 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM. Upper boun of origin stretch ccf L (a) Upper boun of the ccf of OS() for =.8 Fig. 6. Upper boun of origin stretch (b) Upper boun of origin stretch for ifferent Upper boun of origin stretch. Theorem : For a ranomly given value L (, ), the probability that the origin stretch OS( ) is larger than L will be upper boune by the following inequality Prob(OS( ) >L ) < ( ( e λα(i+) l ) j ) 4 3(i +)j. in which j an i can be obtaine from Eq. (4) an (5), an the sie length of the iscrete lattice l satisfies L l =. () ((i +)j + ) +( ) Remark : The above theorem characterizes the upper boun of the ccf of origin stretch. We plot the numerical results in above theorem an show it in Fig. 6(a). In the numerical computation, we set =.8, which inicates <λ c α. From the figure, we know the probability that origin stretch OS( ) is larger than 5.84, is approximately equal to. That means, in the case =.8, with probability nearly, origin stretch OS( ) is less or equal to Then, from our efinition, 5.84 is the upper boun of origin stretch OS( ) for =.8. In Fig. 6(b), we show the upper boun of origin stretch for ifferent ensity. From the figure, we fin that when G(H λα,, ) is in subcritical phase, the upper boun of origin stretch almost scales linearly with the ensity. For the secon question, we first enote U OS ( ) as the upper boun of origin stretch OS( ), which can be foun by using Theorem. Then we have the following result. Theorem : For unit transmission range of s, i.e., r b =,ifthe ensity satisfies λ β <c ( ),allm s in hybri network H(α, β) are sufficiently use with probability nearly, where c ( )= c (λα) an c ( )= U OS ( )+. Remark : The above theorem characterizes a sufficient conition that all s in hybri network H(α, β) are sufficiently use, with probability nearly. From the results in Fig. 6(b), we know that for =.4the sufficient conition is λ β <. 3, while for =.8, the sufficient conition becomes λ β < As we iscuss before, if all s in hybri network H(α, β) are sufficiently use, any two s in the network cannot be connecte through the relay of s. Then, β-component size C (α,β) is the summation of the number () of s, which are connecte irectly by each. Therefore, from Theorem, for unit transmission range of s, if the ensity satisfies λ β < c ( ), we can compute β-component size C (α,β), by summing up the number of s connecte irectly by each. An then, we prove if <λ β <c ( ), β-component size scales linearly with λ β, as we show in Theorem 3. Our results provie a funamental unerstaning an analytical support of the connectivity in hybri networks. In aition, we investigate the effect of transmission range r β on the connectivity of s in hybri network H(α, β). Specifically, we propose a hybri network architecture, an stuy the connectivity improvement in this network. Our results show that through increasing r β from unit to k c ( ) (k N), aitional benefit on the connectivity of s can be obtaine. Hence, our results provie a guieline on esign of a new hybri network, which contains low ensity of a hoc noes. III. UPPER BOUND OF ORIGIN STRETCH OS( ) As we iscuss in Section II-C, if any two s in hybri network H(α, β) cannot be connecte through the relay of s, all s are sufficiently use. To fin out the sufficient conition that all s are sufficiently use, we will first explore the upper boun of origin stretch OS( ) in G(H λα,, ), to unerstan the coverage of s. Here, the upper boun is from the probabilistic perspective, i.e., the value that OS( ) is lower than with probability nearly. In orer to fin the upper boun of origin stretch OS( ), in this section, we will investigate the upper boun of the complementary cumulative istribution function of OS( ), that is, if the probability that OS( ) is larger than S, is nearly, we know that OS( ) is boune by S with probability nearly. Because of the ranomness of the noe positions in G(H λα,, ), the shape of the connecte component containing the origin is quite ifficult to escribe. Thus, it is har to erive the upper boun of the complementary cumulative istribution function of OS( ) irectly in the continuous two-imensional plane. Our strategy here, is to first map the pure a hoc network onto a iscrete lattice. A. Box-Connection Mapping In this section, we introuce our mapping approach calle box-connection mapping. First, we consier a iscrete lattice L with sie length l. The coorinates of the vertices of L are ( l i, l j) for (i, j) Z. Then, we construct its ual lattice L by moving L horizontally an vertically by l,as epicte in Fig. 7. Let S a be the square box centere at vertex a of L. Now,we consier a ranomly chosen path in L with length L l, starting from the origin through v,v,...,v L to v L. Then we efine the event E opath that this path is open iff the following two conitions hol. ) E A : There is at least one in S an one noe in S vl ;

5 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM. Fig. 7. α v v v 3 v 4 v 5 L α v5 L The lattice L (real), an its ual L (ashe). α v v v 3 α v4 v 4 L l = Fig. 8. A single link can go across 5 square boxes at most for l =. ) E B : Among all s in S an S vl, there exists at least one pair of s, one in S an the other in S vl, that can be connecte by several links. An, these links are all containe in S,S v,...,s vl an go through all these square boxes. Otherwise, this path is efine to be close. Fig. 7 gives an example for an open path in L with length 5 l. The path begins from to v 5. There is one in S, α, an another in S v5, α v5. The two links connecting α an α v5 are all containe in S,S v,...,s v5 an go through all these 6 square boxes. Thus, from the above efinition, the path is open. B. Upper Boun of Origin Stretch OS( ) In this section, we stuy how to use box-connection mapping to erive the upper boun of origin stretch OS( ). First, we will stuy the relation between the largest number of square boxes that a single link can go across an the box sie length l. Here, we only consier l <, to make sure a single link can go across multiple square boxes. As an example for illustration, we take l = as epicte in Fig. 8. From the figure, it is not ifficult to fin out a single link can go across at most 5 square boxes. Then, we may ask, for any given l, what is the largest number of square boxes that a single link can go across? Base on this question, we give the following lemma. Lemma : If l satisfies (i +) + i < l <, i N. (3) i + i a single link can go across at most (i +3) square boxes. Proof: We first efine the coorinates of square boxes in L. The square box containing the origin is assigne coorinate (, ). Then the box at the right sie of box (, ) is assigne l coorinate (, ), an the box at the right sie of box (, ) is assigne coorinate (, ), an so on. Similarly, the square box below box (, ) is assigne coorinate (, ), an the box below box (, ) is assigne coorinate (, ), an so on. From this efinition, we know if a single link reaches box (m, n) (without loss of generality, we assume m n), it can go across (m + n +) square boxes. Thus, the problem of fining the largest number of square boxes a single link can go across, is switche to the problem of fining the maximum of (m + n +) uner the constraint that box (m, n) has nonempty intersection with the isk centere at with raius. That is, this problem can be escribe as the following constraine optimization problem. max i,j N subject to m + n + (4) ((m ) l ) +((n ) l ) < (5) If the conition (3) hols, the above optimization problem can be solve as: when m = i+,n= i+ or m = i+,n= i, then m + n + reaches its maximum value i +3. Lemma provies the largest number of square boxes that a single link can go across, if the sie length l satisfies Eq. (3). Base on this result, we have the following lemma. Lemma : If the sie length of each square box l satisfies Eq. (3), the probability P (i+)j that there exists an open path in L with length (i +)j l (i, j N) will satisfy P (i+)j < ( ( e λα(i+) l ) j ) 4 3(i+)j. (6) Proof: For a ranomly chosen open path in L with length (i +)j l, from Lemma, there exists at least one in every (i +) consecutive square boxes. Let P be the probability that there exists at least one in (i +) consecutive square boxes, an P be the probability that a ranomly chosen path in L with length (i +)j l is open. Then, we have P < (P ) j. (7) We enote θ(k) (k N) as the number of paths starting from with length k l. Further, let P (i+)j be the probability that there exists an open path in L with length (i +)j l, an P (i+)j be the probability that any open path in L with length (i +)j l is close. Using FKG s inequality [7], [9], we know P (i+)j will satisfy the following conition. P (i+)j ( P ) θ((i+)j). (8) Base on the relation P (i+)j = P (i+)j,wehave P (i+)j ( P ) θ((i+)j). (9) We know θ(k) < 4 3 k from many literatures (e.g., [7]). An, since the s obey Poisson point process, we can easily know P = e λα(i+) l. Hence, Eq. (9) further satisfy P (i+)j < ( ( e λα(i+) l ) j ) 4 3(i+)j. ()

6 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM. Lemma provies the upper boun of the probability of existence of open path with a given length. From this result, we can erive the upper boun of the ccf of graph stretch OS( ) by the following theorem. Lemma 3: If the sie length of lattice L l satisfies Eq. (3), the probability that OS( ) is larger than ((i +)j + ( ) + l ) (i, j N) will satisfy ( ) Prob OS( ) > ((i +)j + ) +( ) l () < ( ( e λα(i+) l ) j+ ) 4 3(i+)j. Proof: We assume the length of the open path in L is (i + )j l. If the coorinate of the square box containing the en point of the open path is (m,n ) (without loss of generality, we assume m n ). It is obvious that the largest Eucliean istance between the origin an the possible in this box will be less or equal to (m + ) +(n + ) l. Then we maximize (m + ) +(n + ) for m,n N uner the constraint m + n (i +)j, an solve problem as: when m =(i +)j an n =, (m + ) +(n + ) ((i reaches its maximum value +)j + ( ) +. ) Subsequently, we efine two events as follows. ) E A : There exists an open path in L with length (i + )j l ; ) E B : The origin stretch OS() is larger than ((i +)j + ) +( ) l. Using Law of Total Probability, we have Prob(E A ) = Prob(E A E B ) Prob(E B )+Prob(E A E B ) Prob(E B ) Prob(E A E B ) Prob(E B ). Since when OS() is larger than ((i +)j + ) +( ) l, there must exist an open path in L with length (i +)j l, we have Prob(E A E B ) =, an thus further obtain Prob(E A ) Prob(E B ). From Lemma, we know Prob(E A ) < ( ( e λα(i+) l ) j+ ) 4 3(i+)j. Then, we prove the lemma. To simplify the long expression, we introuce a new term ((i c(i, j) enote as c(i, j) = +)j + ( ) +.From ) Lemma 3, for a given l satisfying Eq. (3), we can only fin the upper boun of the probability that OS( ) is larger than the iscrete values c(i, j) l (i, j N). Then, we will ask, for any given value L, what is the upper boun of Prob(OS( ) > L )? Then, we will investigate if we can fin appropriate l, i an j, such that l satisfies Eq. (3) an L = c(i,j ) l. If this is feasible, we can irectly use Eq. () to obtain the upper boun of Prob(OS( ) >L ). Notice that when l satisfies Eq. (3), the following conition c(i,j) hols: <c(i, j) (i+) +i l < c(i,j) i. Then, we have the +i following lemma. Lemma 4: The union of all such intervals c(i,j) (, c(i,j) ) (i, j N) is ( (i+) +i i, ), i.e., +i i,j N ( c(i,j) (i+) +i, c(i,j) i +i )=(, ). Proof: To prove the lemma, we first let j fixe an investigate i N ( c(i,j), c(i,j) (i+) +i i ). To fin this result, we +i then consier the union of two following consecutive intervals: c(i,j) (, c(i,j) ) an ( c(i+,j), c(i+,j) ). (i+) +i i +i (i+) +(i+) (i+) +(i+) It is easy to prove the union of above two consecutive c(i+,j) intervals is (, c(i,j) (i+) +(i+) i ). From this result, we +i know c(i, j) ( (i +) + i, c(i, j) i + i )=( j, 8j +j + 4 ). i N () Then, we consier the union of two following consecutive intervals: ( j, 8j +j + 4 ) an ( (j +), 8(j +) +(j +)+ 4 ). Following the similar steps as above, we can prove ( ) j, 8j +j + ( ) =,. (3) 4 j N Lemma 4 valiates that for any value L (, ), we can fin appropriate l, i an j, such that l satisfies Eq. (3) an L = c(i,j ) l. In orer to fin the upper boun of the ccf of origin stretch OS( ), we will investigate how to fin the appropriate l, i an j to satisfy the above conition. Lemma 5: For any given value L (, ), we can fin the appropriate j, i an l by the following equations, such that l satisfies Eq. (3) an L = c(i,j ) l. j = ceil( L ). (4) where ceil(x) rouns X towars positive infinity; i = floor( L +4j + j + Δ L )+. (5) 4j where Δ= L 4 +(4j +j +)L j an floor(x) rouns X towars negative infinity; an l = L c(i,j ). (6) Proof: We start from fining the appropriate j. From the proof of Lemma 4, we know for any given value L (, ), there must exist j N such that L ( j, 8 j + j + 4 ). Then, it is not ifficult to fin that j = ceil( L ) satisfies this conition. Then we will locate the appropriate i such that c(i,j ) <L (i +) + i < c(i,j ). (7) i + i From Eq. (7), we can fin Eq. (5) is a solution of i. Apparently, for the j an i obtaine from Eq. (4) an (5), l = L c(i,j ) will satisfy Eq. (3). Lemma 5 provies an approach to fin appropriate appropriate l, i an j, such that l satisfies Eq. (3) an

7 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM. L = c(i,j ) l. After combining the results in Lemma 3, 4 an 5, we obtain a main result of this section illustrate in Theorem. IV. HOW DO β-nodes IMPROVE THE CONNECTIVITY OF α-nodes? In this section, we will stuy how s improve the connectivity of s. First, we consier unity transmission range of s, i.e., r β =. From the results in Section III, we investigate the sufficient conition that all s in hybri network H(α, β) are sufficiently use with probability nearly. Then, uner the conition that all s in hybri network H(α, β) are sufficiently use, β-component size is the summation of the number of s, which are connecte by each irectly. Base on this stuy, we provie the impact of s ensity on the connectivity of s, that is, how β-component size scales with the ensity λ β. Further, we will explore how s transmission range, r β affect the connectivity of s. Specifically, we stuy a hybri network architecture which can utilize large r β to obtain aitional benefit on the connectivity of s. A. Sufficient Conitions for Sufficient Use of β-noes As we efine in Section II-C, all s in hybri network H(α, β) are sufficiently use iff any two s cannot be connecte through the relay of s. Here, we consier unity s transmission range an erive the sufficient conitions for sufficient use of s. Proof of Theorem : As we efine before, U OS ( ) is the upper boun of origin stretch OS( ). From Section III, we know the event E boun that s connecte by a β- noe irectly, is boune within the isk centere at with raius U OS ( ) happens with probability nearly. Then, uner the conition that event E boun happens, if the Eucliean istance between the ajacent s is larger than U OS ( )+, all s in hybri network H(α, β) are sufficiently use. Otherwise, if two ajacent s an can be connecte through the relay of s, there must exists one α s that is within the unit isk centere at an can be connecte by through the relay of s. This is equivalent to say, α s is not within the isk centere at with raius U OS ( ), an thus contraicts with the conition that event E boun happens. Then from m >U OS ( )+, we know λ β = m < (U OS()+). B. Impact of β-noes Density, λ β In this part, we consier the case that all s in hybri network H(α, β) are sufficiently use. If we ignore the wireline connection between the s, each connecte component containing a is an inepenent connecte component, which can be viewe as C in G(H λα,, ). Then in hybri network H(α, β), β-component size is the summation of the number of s connecte by each β- noe irectly. In other wor, let k i (i {,,,m}) be the number of s that are connecte by irectly. Then in hybri network H(α, β), β-component size is equal to C (α,β) =Σ m i= k i.asthe ensity λ ncreases, β- component size will become larger. Then, how oes C (α,β) scale with λ β? To answer this question, we first give the following lemma. Lemma 6: Let C be the number of s in C of G(H λα,, ). When G(H λα,, ) is in subcritical phase, the mean of C : E( C ), an the variance of C :Var( C ), are both finite. Proof: From Theorem 9.7 in [8], when G(H λα,, ) is in subcritical phase, there exist constants μ>, g > such that Prob( C g) e μg when g g. Hence, we have E( C )=Σ g=g Prob( C = g) =Σ g g= g Prob( C = g) +Σ g=g g Prob( C = g) Σ g g= g Prob( C = g)+σ g=g g e μg, an E( C )=Σ g=g Prob( C = g) Σ g g= g Prob( C = g)+σ g=g g e μg. Using the ratio test criteria, we know both Σ g=g g e μg an Σ g=g g e μg converge. Thus, both E( C ) an E( C ) are finite, an further Var( C )=E( C ) E ( C ) is finite. Base on the results in Lemma 6, we will investigate the variance of the inepenent an ientically istribute (i.i..) ranom variable k i. In fact, k i is less than C, since C accounts the origin as an. Thus, it is easy to know both the mean of k i an the variance of k i are also finite, with E(k i )=E( C ) an Var(k i )=Var( C ). Then, let η( )=E(k i ) be the mean of k i for a given, i.e., η( )= E( C ). Then, we will have the following theorem. Theorem 3: With a ensity of <λ β <c ( ) s in hybri network H(α, β), β-giant component will emerge with probability nearly. An, β-giant component size scales linearly with λ β as C (α,β) = η(λα)n λ β + o(n). Proof: From Theorem, when λ β <c ( ),alls in the hybri network are sufficiently use with probability nearly. Then, as we iscuss before, if all s in the hybri network are sufficiently use, β-component size can be compute by C (α,β) = Σ m i= k i, in which k i (i {,,,m}) are i.i.. ranom variables with finite mean η( ) an finite variance. When λ β >, m,usingthe Weak Law of Large Numbers [], we know C (α,β) converges to mη( ) ± o(m) almost surely. Since m = λ β = λ βn, we have C (α,β) = η(λα)n λ β + o(n) =Θ(n), which inicates β-giant component emerges. Theorem 3 inicates that β-component size scales linearly with λ β with probability nearly, when <λ β <c ( ). Further, the result in Theorem 3 also shows that with a ensity of λ β ( <λ β <λ c α ) s in hybri network, β-giant component, which contains Θ(n) s, will emerge with probability nearly. Thus, aing base stations in the pure a hoc network will bring tremenous benefit on the connectivity of s. C. Impact of β-noes Transmission Range, r β As the transmission range of s, r ncreases from unit, more s will be insie the isk centere at a single with raius r β, that is, a single can connect together more connecte component of s, which are

8 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM. α a6 α a α p α q α a5 α a BS j() BS i() Fig. 9. stretch an stretch in hybri network. α a4 α a3 U OS() Fig.. The locations of aitional s (k =). Fig.. r β The upper boun of BS i (). separate without the help of s. In this section, we will stuy the impact of r β on the connectivity of s. First, we give a new efinition as follows. Definition 4 ( Stretch): In hybri network H(α, β), let C (α,βi) be the connecte component containing, an a subset of s that are connecte by irectly. stretch is efine as BS i ( )=max αj C α (α,βi ) j.for example, as epicte in Fig. 9, stretch is the Eucliean istance between α p an, an stretch is the Eucliean istance between α q an. Further, let U BSi ( ) be the upper boun of BS i ( ).This efinition of upper boun is also from the probabilistic perspective as before, i.e. BS i ( ) <U BSi ( ) with probability nearly. Then, we have the following theorem. Lemma 7: In hybri network H(α, β) for r β >, iftheβnoe ensity satisfies λ β <c 3 ( ),alls in the network are sufficiently use with probability nearly, where c 3 ( )= (r β +c ()). Proof: This proof is very similar to the proof in Theorem. When the Eucliean istance between the ajacent β- noes is large than U BSi ( )+, all s in hybri network H(α, β) are sufficiently use. From Fig., it is easy to know U BSi ( ) <r β + U OS ( ) with probability nearly. Therefore, when λ β < (r β +U OS()+),wehave λ β < (U BSi ()+), which inicates all s in the network are sufficiently use. Lemma 7 provies the sufficient conition that all β- noes in hybri network H(α, β) are sufficiently use with probability nearly, for the case r β >. Then, in orer to utilize the larger transmission range r β, we propose a new hybri network architecture below. First, we let r β = k c ( ) (k N). From Lemma 7, when ensity satisfies λ β < c(λα),alls in hybri network H(α, β) are sufficiently use with probability nearly. Then, a number of aitional s are place at some specific locations on circles Circ(,kc ( )) centere at (i {,,,m}) with raius kc ( ) (k {,,,k }). The locations of the aitional s are escribe as follows: For given i an k, let α a,α a,,α aq be all aitional s locate on Circ(,kc ( )). Without loss of generality, we assume α a,α a,,α aq are in clockwise irection. Then the location of α aj (j {,,,q}) isatthe intersection of Circ(,kc ( )) an Circ(α a(j ),c ( )). For example, for k =, i.e., r β = c ( ), as epicte in Fig., six aitional s α a,α a,,α a6 are place on the circle Circ(,c ( )). The locations of α aj (j {,,, 6}) is at the intersection of Circ(,c ( )) an Circ(α a(j ),c ( )). This hybri network architecture is ifferent from the previous one, because that a subset of s in the network are eploye at eterministic locations to improve the connectivity of other s. Therefore, we enote this hybri network as H(α, β), an then efine a new metric β-α-component as follows. Definition 5 (β-α-component): In hybri network H(α, β), theβ-α-component enote by C (α,β) is efine as the connecte component containing all s, all aitional s, an a subset of other s that are connecte by the s an aitional s. Similar to the efinition of β-component size, β-α-component size C (α,β) is efine as the number of s, which are containe by C (α,β) but not the aitional s. Then, we have the following theorem illustrating the benefit on connectivity of s by using this hybri network architecture. Theorem 4: In hybri network H(α, β), when ensity satisfies λ β < c(λα), with a ensity of Δλ α =(K ) λ β aitional s at the specific locations escribe η(λ above, β-α-component size satisfies C (α,β) >K α)n λ λβ α π with probability nearly. Here, K =+floor( arccos( ))+ π + floor( ). ) arccos( k k Proof: For k =, as epicte in Fig., we enote (k i ), (k i ),, (k i ) 6 respectively as the number of s connecte by each aitional irectly. Further, let (k i ) 7 be the number of s connecte by irectly when r β =. From our iscussion in Section IV-B, (k i ) j (j {,,, 7}) are i.i.. ranom variables with finite mean η( ) an finite variance. Obviously, in hybri network

9 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM. The factor K /(k +) k linearly with λ β with probability nearly, which emonstrates a tremenous benefit of using base stations to enhance the connectivity of a hoc noes. Further, we stuy a hybri network architecture which can utilize large base station transmission range r β, to obtain an aitional benefit on the connectivity of a hoc noes. The work in this paper provies funamental unerstaning of the connectivity in hybri networks, an also a provie guielines on esign of a new hybri network that contains low ensity of a hoc noes. Fig.. The factor K (k +) for ifferent k. H(α, β), the number of s connecte by an aitional six s irectly, is larger than Σ 7 j= (k i) j.from the Weak Law of Large Numbers [], β-α-component size C (α,β) is larger than 7mη( ) ± o(m) almost surely. Then we consier the general case for r β = k c ( ). Following the same logic as above, it is not ifficult to know β-α-component size C (α,β) is larger than K mη( )± o(m) almost surely, where K = + floor( π + floor( π arccos( k k ) arccos( ))+ ). Then, β-α-component size satisfies η(λ C (α,β) K α)n λ λβ α. At this time, λβ < c(λα) an Δ =(K ) λ β. K We plot the factor in Fig.. As we can see from the figure, when k increases from to 4, the factor K increases from.5 to 3.5. From Theorem 4, we know λ β +Δ = K λβ < K c ( ), an C (α,β) K η()n ( λ β +Δ ). Compare with the result in Section IV-B, C (α,β) = η(λα)n λ β when λ β < c ( ), we fin that through increasing r β from unit to k c ( ) (k N), we enlarge the linear scale region from (,c ( )) λβ = η(λα)n K to (, c ( )), while ecrease the require ensity, an thus obtain aitional benefit on the connectivity of s. This result provies a guieline on constructing the hybri network architecture. Further, from Fig., as K k increases, the factor increases slower. Taking k near woul be an optimal choice, though we still nee to consier the receiver complexity an the transmission power etc. of the base stations. Therefore, the result also provies us with a reference on how to choose the value of r β. V. CONCLUSIONS In this paper, we have stuie how base stations in largescale hybri network improve the connectivity of a hoc noes in subcritical phase. This problem is two-fol: the effect of base station ensity λ β an the effect of base station transmission range r β on network connectivity. For r s unit, through investigating the upper boun of origin stretch in pure a hoc networks, we fin the sufficient conition, i.e., λ β <c ( ), that any two base stations cannot be connecte through the relay of a hoc noes with probability nearly. Base on this result, we fin that uner <λ β <c ( ), the number of connecte a hoc noes is Θ(n) an scales REFERENCES [] P. Gupta an P. R. Kumar, Critical power for asymptotic connectivity in wireless networks, in Proc. of IEEE Conference on Decision an Control, December 998, pp. 6. [] F. Xue an P. R. Kumar, The number of neighbors neee for connectivity of wireless networks, Wireless Networks, vol., no., pp. 69 8, March 4. [3] P. Santi an D. M. Blough, The critical transmitting range for connectivity in sparse wireless a hoc networks, IEEE Transactions on Mobile Computing, vol., no., pp. 5 39, January 3. [4] P. Santi, The critical transmitting range for connectivity in mobile a hoc networks, IEEE Transactions on Mobile Computing, vol. 4, no. 3, pp. 3 37, June 5. [5] O. Dousse, F. Baccelli, an P. Thiran, Impact of interferences on connectivity in a hoc networks, IEEE/ACM Transactions on Networking, vol. 3, no., pp , April 5. [6] Z. Kong an E. M. Yeh, Connectivity an latency in large-scale wireless networks with unreliable links, in Proc. of IEEE Infocom, April 8, pp [7] R. Meester an R. Roy, Continuum Percolation. Cambrige University Press, 996. [8] M. Penrose, Ranom Geometric Graphs. Oxfor University Press, 3. [9] P. Gupta an P. R. Kumar, The capacity of wireless networks, IEEE Transactions on Information Theory, vol. 46, no., pp , March. [] O. Lévêque an I. E. Telatar, Information-theoretic upper bouns on the capacity of large extene a hoc wireless networks, IEEE Transaction on Information Theory, vol. 5, no. 3, pp , March 5. [] P. L, C. Zhang, an Y. Fang, Capacity an elay of hybri wireless broaban access networks, IEEE Journal on Selecte Areas in Communications, vol. 7, no., pp. 7 5, February 9. [] B. Liu, Z. Liu, an D. Towsley, On the capacity of hybri wireless networks, in Proc. of IEEE Infocom, March 3, pp [3] U. C. Kozat an L. Tassiulas, Throughput capacity of ranom a hoc networks with infrastructure support, in Proc. of ACM MobiCom, September 3, pp [4] A. Agarwal an P. R. Kumar, Capacity bouns for a hoc an hybri wireless networks, in Proc. of ACM SIGCOMM, July 4, pp [5] X. Mao, X.-Y. Li, an S. Tang, Multicast capacity for hybri wireless networks, in Proc. of ACM MobiHoc, May 8, pp [6] P. Balister, B. Bollobás, an M. Walters, Continuum percolation with steps in the square or the isc, Ranom Structures an Algorithms, vol. 6, no. 4, pp , April 5. [7] Y. Shi, Y. T. Hou, an A. Efrat, Algorithm esign for base station placement problems in sensor networks, in Proc. of ACM QShine, August 6. [8] Y. Shi an Y. T. Hou, Approximation algorithm for base station placement in wireless sensor networks, in Proc. of IEEE SECON, June 7, pp [9] G. Grimmett, Percolation. Springer, 999. [] R. Durrett, Probability: Theory an Examples. Duxbury Press, 996.