On Capacity Scaling in Arbitrary Wireless Networks

On Capacity Scaing in Arbitrary Wireess Networks Urs Niesen, Piyush Gupta, and Devavrat Shah 1 Abstract arxiv:07112745v3 [csit] 3 Aug 2009 In recent work, Özgür, Lévêque, and Tse 2007) obtained a compete scaing characterization of throughput scaing for random extended wireess networks ie, n nodes are paced uniformy at random in a square region of area n) They showed that for sma path-oss exponents α 2, 3] cooperative communication is order optima, and for arge path-oss exponents α > 3 muti-hop communication is order optima However, their resuts both the communication scheme and the proof technique) are strongy dependent on the reguarity induced with high probabiity by the random node pacement In this paper, we consider the probem of characterizing the throughput scaing in extended wireess networks with arbitrary node pacement As a main resut, we propose a more genera nove cooperative communication scheme that works for arbitrariy paced nodes For sma path-oss exponents α 2, 3], we show that our scheme is order optima for a node pacements, and achieves exacty the same throughput scaing as in Özgür et a This shows that the reguarity of the node pacement does not affect the scaing of the achievabe rates for α 2, 3] The situation is, however, markedy different for arge path-oss exponents α > 3 We show that in this regime the scaing of the achievabe per-node rates depends cruciay on the reguarity of the node pacement We then present a famiy of schemes that smoothy interpoate between muti-hop and cooperative communication, depending upon the eve of reguarity in the node pacement We estabish order optimaity of these schemes under adversaria node pacement for α > 3 Index Terms Arbitrary node pacement, capacity scaing, cooperative communication, hierarchica reaying, muti-hop communication, wireess networks I INTRODUCTION Consider a wireess network with n nodes paced on [0, n] 2 usuay referred to as an extended network), with each node being the source for one of n source-destination pairs and the destination for another pair The performance of this network is captured by ρ n), the argest uniformy achievabe rate of communication between these source-destination pairs Whie the scaing behavior of ρ n) as the number of nodes n goes to infinity is by now we understood for random node pacement, itte is known for the case of arbitrary node pacements In this paper, we are interested in anayzing the impact of such arbitrary node pacement on the scaing of ρ n) A Reated Work The probem of determining the scaing of ρ n) was first anayzed by Gupta and Kumar in [1] They show that, under random pacement of nodes in the region, certain modes of communication motivated by current technoogy, and random source-destination pairing, the maximum achievabe per-node rate ρ n) can scae at most as On 1/2 ) Moreover, it was shown that muti-hop communication can achieve essentiay the same order of scaing Since [1], the probem has received a considerabe amount of attention One stream of work [2] [8] has progressivey broadened the conditions on the channe mode and the communication mode, under which U Niesen and D Shah are with the Laboratory of Information and Decision Systems, Department of EECS at the Massachusetts Institute of Technoogy Emai: {uniesen,devavrat}@mitedu P Gupta is with the Mathematics of Networks and Communications Research Department, Be Labs, Acate-Lucent Emai: pgupta@researchbe-abscom The work of U Niesen and D Shah was supported in parts by DARPA grant ITMANET) 18870740-37362-C and NSF grant CNS-0546590; the work of P Gupta was supported in part by NSF Grants CCR-0325673 and CNS-0519535

2 muti-hop communication is order optima Specificay, with a power oss of r α for signas sent over distance r, it has been estabished that under high signa attenuation α > 3 and random node pacement, the best achievabe per-node rate ρ n) for random source-destination pairing scaes essentiay ike Θn 1/2 ) and that this scaing is achievabe with muti-hop communication Another stream of work [8] [12] has proposed progressivey refined muti-user cooperative schemes, which have been shown to significanty out-perform muti-hop communication in certain environments In an exciting recent work, Özgür et a [8] have shown that with nodes paced uniformy at random, and with ow signa attenuation α 2, 3], a cooperative communication scheme can perform significanty better than muti-hop communication More precisey, they show that for α 2, 3], the best achievabe per-node rate for random source-destination pairing scaes as ρ n) = On 1 α/2+ε ) and cooperative communication achieves a per-node rate of Ωn 1 α/2 ε ) here, ε > 0 is an arbitrary but fixed constant) That is, cooperative communication is essentiay order optima in the attenuation regime α 2, 3] In summary, for random extended networks with random source-destination pairing, the optima communication scheme exhibits the foowing threshod behavior: for α 2, 3] the cooperative communication scheme is order optima, whie for α > 3 the muti-hop communication scheme is order optima B Our Contributions The characterization of the scaing of ρ n) as a function of the path-oss exponent α mentioned in the ast paragraph depends criticay on the reguarity induced with high probabiity by pacing the nodes uniformy at random However, a wireess network encountered in practice might not exhibit this amount of reguarity Our interest is therefore in understanding the impact of the node pacement on the scaing of ρ n) To this end, we consider wireess networks with arbitrary ie, deterministic) node pacement with minimum-separation constraint) The impact of this arbitrary node pacement depends cruciay on the path-oss exponent α For sma path-oss exponents α 2, 3], we show that for random source-destination pairing, the rate of the best communication scheme is upper bounded as ρ n) = Oog 6 n)n 1 α/2 ) We then present a nove cooperative communication scheme that achieves for any path-oss exponent α > 2 a per-node rate of ρ HR n) n 1 α/2 o1) Thus, our cooperative communication scheme is essentiay order optima for any such arbitrary network with α 2, 3] In other words, in the sma path-oss regime, the scaing of ρ n) is the same irrespective of the reguarity of the node pacement The situation is, however, quite different for arge path-oss exponents α > 3 We show that in this regime the scaing of ρ n) depends cruciay on the reguarity of the node pacement, and muti-hop communication may not be order optima for any vaue of α In fact, for ess reguar networks we need more compicated cooperative communication schemes to achieve optima network performance Towards that end, we present a famiy of communication schemes that smoothy interpoate between cooperative communication and muti-hop communication, and in which nodes communicate at scaes that vary smoothy from oca to goba The amount of interpoation between the cooperative and muti-hop schemes depends on the eve of reguarity of the underying node pacement We estabish the optimaity of this famiy of schemes for a α > 3 under adversaria node pacement In summary, for α 2, 3] the reguarity of the node pacement has no impact on the scaing of ρ n) Cooperative communication is order optima in this regime and achieves the same scaing as in the case of random node pacement For α > 3 the reguarity of the node pacement strongy impacts the scaing of ρ n), and a communication scheme interpoating between muti-hop and cooperative communication depending on the reguarity of the node pacement is order optima under adversaria node pacement) In particuar, simpe muti-hop communication may not be order optima for any α > 3 This contrasts with the case of random node pacement where muti-hop communication is order optima for a α > 3 C Organization The remainder of this paper is organized as foows Section II describes in detai the communication mode Section III provides forma statements of our resuts Sections IV and V describe our new

3 cooperative communication scheme for the α 2, 3] regime) and interpoation scheme for the α > 3 regime) for arbitrary wireess networks Sections VI through XI contain proofs Finay, Sections XII and XIII contain discussions and concuding remarks II MODEL In this section, we introduce some notationa conventions and describe in detai the network and channe modes We use the foowing conventions: K i for different i denote stricty positive finite constants independent of n Vectors and matrices are denoted by bodface whenever the vector or matrix structure is of importance We denote by ) T and ) transpose and conjugate transpose, respectivey To simpify notation, we assume, when necessary, that fractions are integers and omit and operators Consider the square An) [0, n] 2 of area n, and et V n) An) be a set of V n) = n nodes on 1 An) We say that V n) has minimumseparation r min if r u,v r min for a u, v V n), where r u,v is the Eucidean distance between nodes u and v We use the same channe mode as in [8] Namey, the samped) received signa at node v is y v [t] = h u,v [t]x u [t] + z v [t] 1) u V n)\{v} for a v V n), and where {x u [t]} u,t are the samped) signas sent by the nodes in V n) Here {z v [t]} v,t are independent and identicay distributed iid) with distribution N C 0, 1) ie, circuary symmetric compex Gaussian with mean 0 and variance 1), and h u,v [t] = ru,v α/2 exp 1θ u,v [t]), for path-oss exponent α > 2 We assume that for each t N, the phases {θ u,v [t]} u,v are iid 2 with uniform distribution on [0, 2π) We either assume that for each u, v V n) the random process {θ u,v [t]} t is stationary ergodic in t, which is caed fast fading in the foowing, or that for each u, v V n) the random process {θ u,v [t]} t is constant in t, which is caed sow fading in the foowing In either case, we assume fu channe state information CSI) is avaiabe at a nodes, ie, each node knows a {θ u,v [t]} u,v at time t Whie the fu CSI assumption is quite strong, it can be shown that avaiabiity of a 2-bit quantized version of {θ u,v [t]} u,v at a nodes is sufficient for the achievabe schemes presented here see Section XII-A for the detais) We aso impose an average power constraint of 1 on the signa {x u [t]} t for every node u V n) Each node u V n) wants to transmit information at uniform rate ρn) to some other node w V n) We ca u the source and w the destination node of this communication pair The set of a communication pairs can be described by a traffic matrix λn) {0, 1} n n, where the entry in λn) corresponding to u, w) is equa to 1 if node u is a source for node w We say that λn) is a permutation traffic matrix if it is a permutation matrix ie, every node is a source for exacty one communication pair and a destination for exacty one communication pair) For a traffic matrix λn), et ρ n) be the highest rate of communication that is uniformy achievabe for each source-destination pair For a permutation traffic matrix λn), ρ n) can aso be understood as the maxima achievabe per-node rate 1 The setting considered here with n nodes paced on a square of area n is caed an extended network If the n nodes are paced on a square of unit area, we speak of a dense network Whie dense networks are not treated in detai in this paper, we briefy discuss impications of the resuts for the dense setting in Section XII-C 2 It is worth pointing out that recent work [13] suggests that, under certain assumptions on scattering eements, for α 2,3), and for very arge vaues of n, the iid phase assumption as a function of u, v V n) used here is too optimistic However, subsequent work by the same authors [14] shows that under different assumptions on the scatterers, the channe mode used here is sti vaid even for α 2,3), and for very arge vaues of n This indicates that the question of channe modeing for very arge networks in the ow path-oss regime is somewhat deicate and requires further investigation We point out that for α 3 this issue does not arise

4 III MAIN RESULTS This section presents the forma statement of our resuts The resuts are divided into two parts In Section III-A, we consider ow path-oss exponents, ie, α 2, 3] We present a cooperative communication scheme for arbitrary node pacement and for either fast or sow fading We show that this communication scheme is order optima for a node pacements when α 2, 3] In Section III-B, we consider high path-oss exponents, ie, α > 3 We present a communication scheme that interpoates between the cooperative and the muti-hop communication schemes, depending on the reguarity of the node pacement We show that this communication scheme is order optima under adversaria node pacement with reguarity constraint when α > 3 A Low Path Loss Regime α 2, 3] The first resut proposes a nove communication scheme, caed hierarchica reaying in the foowing, and bounds the per-node rate ρ HR n) that it achieves This provides a ower bound to ρ n), the argest achievabe per-node rate The hierarchica reaying scheme enabes cooperative communication on the scae of the network size In the random node pacement case, this cooperation coud be enabed in a custer around the source node cooperativey transmitting) and in a custer around its destination node cooperativey receiving) With arbitrary node pacement, such an approach does no onger work, as both the source as we as the destination nodes may be isoated The hierarchica reaying scheme circumvents this issue by reaying data between each source-destination pair over a densey popuated region in the network A detaied description of this scheme is provided in Section IV, the proof of Theorem 1 is contained in Section VII Theorem 1 Under fast fading, for any α > 2, r min 0, 1), and δ 0, 1/2), there exists ) b 1 n) n O og δ 1/2 n) such that for any n, node pacement V n) with minimum separation r min, and permutation traffic matrix λn), we have ρ n) ρ HR n) b 1 n)n 1 α/2 The same concusion hods for sow fading with probabiity at east )) 1 exp 2 Ω og 1/2+δ n) = 1 o1) as n Theorem 1 shows that the per-node rate ρ HR n) achievabe by the hierarchica reaying scheme is at east n 1 α/2 βn), where the oss term βn) converges to zero as n at a rate arbitrariy cose to O og 1/2 n) ) by choosing δ sma) The performance of the hierarchica reaying scheme can intuitivey be understood as foows As mentioned before, the scheme achieves cooperation on a goba scae This eads to a muti-antenna gain of order n On the other hand, communication is over a distance of order n 1/2, eading to a power oss of order n α/2 Combining these two factors resuts in a per-node rate of n 1 α/2 We note that Theorem 1 remains vaid under somewhat weaker conditions than having minimum separation r min 0, 1) Specificay, we show that the resut of Özgür et a [8] can be recovered through Theorem 1 as the random node pacement satisfies these weaker conditions We discuss this in more detai in Section XII-D The next theorem estabishes optimaity of the hierarchica reaying scheme in the range of α 2, 3] for arbitrary node pacement The proof of the theorem is presented in Section VIII

5 Theorem 2 Under either fast or sow fading, for any α 2, 3], r min 0, 1), there exists b 2 n) = O og 6 n) ) such that for any n, node pacement V n) with minimum separation r min, and for λn) chosen uniformy at random from the set of a permutation traffic matrices, we have with probabiity 1 o1) as n ρ n) b 2 n)n 1 α/2 Note that Theorem 2 hods ony with probabiity 1 o1) for different reasons for the sow and fast fading case For fast fading, this is due to the randomness in the seection of the permutation traffic matrix In other words, for fast fading, with high probabiity we seect a traffic matrix for which the theorem hods For the sow fading case, there is additiona randomness due to the fading reaization Here, with high probabiity we seect a traffic matrix and we experience a fading for which the theorem hod Comparing Theorems 1 and 2, we see that for α 2, 3] the proposed hierarchica reaying scheme is order optima, in the sense that ogρ HR n)) im n ogn) ogρ n)) = im = 1 α/2 n ogn) Moreover, the rate it achieves is the same order as is achievabe in the case of randomy paced nodes Hence in the ow path-oss regime α 2, 3], the heterogeneity caused by the arbitrary node pacement has no effect on achievabe communication rates B High Path Loss Regime α > 3 We now turn to the high path-oss regime α > 3 In the case of randomy paced nodes, muti-hop communication achieves a per-node rate of ρ MH n) = Ωn 1/2 ) with probabiity 1 o1) and is order optima for α > 3 For arbitrariy paced nodes, the situation is quite different as Theorem 3 shows The proof of Theorem 3 is contained in Section IX Theorem 3 Under either fast or sow fading, for any α > 3, for any n, there exists a node pacement V n) with minimum separation 1/2 such that for λn) chosen uniformy at random from the set of a permutation traffic matrices, we have as n with probabiity 1 o1) ρ n) 2 2+5α n 1 α/2, ρ MH n) 4 α n α/2, Comparing Theorem 3 with Theorem 1 shows that under adversaria node pacement with minimumseparation constraint the hierarchica reaying scheme is order optima even when α > 3 Moreover, Theorem 3 shows that there exist node pacements satisfying a minimum separation constraint for which hierarchica reaying achieves a rate of at east a factor of order n higher than muti-hop communication for any α > 3 In other words, for those node pacements cooperative communication is necessary for order optimaity aso for any α > 3, in stark contrast to the situation with random node pacement, where muti-hop communication is order optima for a α > 3 Theorem 3 suggests that it is the eve of reguarity of the node pacement that decides what scheme to choose for path-oss exponent α > 3 So far, we have seen two extreme cases: For random node pacement, resuting in very reguar node pacements with high probabiity, ony oca cooperation is necessary and muti-hop is an order-optima communication scheme For adversaria arbitrary node pacement, resuting in a very irreguar node pacement, goba cooperation is necessary and hierarchica reaying is an orderoptima communication scheme We now make this notion of reguarity precise, and show that, depending on the reguarity of the node pacement, an appropriate interpoation between muti-hop and hierarchica reaying is required for α > 3 to achieve the optima performance We refer to this interpoation scheme as cooperative muti-hop communication in the foowing

6 Before we state the resut, we need to introduce some notation Consider again a node pacement V n) An) with minimum separation r min 0, 1) Divide An) into squares of sideength dn) n, and fix a constant µ 0, 1] We say that V n) is µ-reguar at resoution dn) if every such square contains at east µd 2 n) nodes Note that every node pacement is triviay 1-reguar at resoution n; a random node pacement can be shown to be µ-reguar at resoution ogn) with probabiity 1 o1) as n for any µ < 1; and nodes that are paced on each point in the integer attice inside An) are 1-reguar at resoution 1 The cooperative muti-hop scheme enabes cooperative communication on the scae of reguarity dn) Neighboring squares of sideength dn) cooperativey communicate with each other To transmit between a source and its destination, we use muti-hop communication over those squares In other words, we use cooperative communication at sma scae dn), and muti-hop communication at arge scae n For reguar node pacements, ie, dn) = 1, the cooperative muti-hop scheme becomes the cassica muti-hop scheme For very irreguar node pacement, ie, dn) = n 1/2, the cooperative muti-hop scheme becomes the hierarchica reaying scheme discussed in the ast section The next theorem provides a ower bound on the per-node rate ρ CMH n) achievabe with the cooperative muti-hop scheme The proof of the theorem can be found in Section X Theorem 4 Under fast fading, for any α > 2, r min 0, 1), µ 0, 1), and δ 0, 1/2) there exists ) b 3 n) n O og δ 1/2 n) such that for any n, node pacement V n) with minimum separation r min, and permutation traffic matrix λn), we have ρ n) ρ CMH n) b 3 n)d 3 α n)n 1/2, where d n) min{h : V n) is µ reguar at resoution h} The same concusion hods for sow fading with probabiity 1 o1) as n Theorem 4 shows that if V n) is reguar at resoution d n) then a per-node rate of at east ρ CMH n) d 3 α n)n 1/2 βn) is achievabe, where, as before, the oss term βn) converges to zero as n at a rate arbitrariy cose to O og 1/2 n) ) The performance of the cooperative muti-hop scheme can intuitivey be understood as foows The scheme achieves cooperation on a scae of d 2 n) This eads to a muti-antenna gain of order d 2 n) On the other hand, communication is over a distance of order dn), eading to a power oss of order d α n) Moreover, each source-destination pair at a distance of order n 1/2 must transmit their data over order n 1/2 d 1 n) many hops, eading to a muti-hop oss of n 1/2 dn) Combining these three factors resuts in a per-node rate of d 3 α n)n 1/2 The next theorem shows that Theorem 4 is tight under adversaria node pacement under a constraint on the reguarity The proof of the theorem is presented in Section XI Theorem 5 Under either fast or sow fading, for any α > 3, there exists b 4 n) = O og 6 n) ), such that for any n, and d n), there exists a node pacement V n) with minimum separation 1/2 and 1/2-reguar at resoution d n) such that for λn) chosen uniformy at random from the set of a permutation traffic matrices, we have ρ n) b 4 n)d 3 α n)n 1/2, with probabiity 1 o1) as n As an exampe, assume that d n) = n η for some η 0 Then Theorem 4 shows that for any node pacement of reguarity d n) and α > 3, ρ CMH n) n 3 α)η 1/2 βn),

7 where βn) converges to zero as n at a rate arbitrariy cose to O og 1/2 n) ) In other words ogρ CMH n)) im 3 α)η 1/2 n ogn) Moreover, by Theorem 5 there exist node pacements with same reguarity such that for random permutation traffic with high probabiity ρ n) is essentiay) of the same order, in the sense that ogρ n)) im 3 α)η 1/2 n ogn) In particuar, for η = 0 ie, reguar node pacement), and for η = og ogn)/ ogn) ie, random node pacement), we obtain the order n 1/2 scaing as expected For η = 1/2 ie, competey irreguar node pacement), we obtain the order n 1 α/2 scaing as in Theorems 1 and 3 IV HIERARCHICAL RELAYING SCHEME This section describes the architecture of our hierarchica reaying scheme On a high eve, the construction of this scheme is as foows Consider n nodes V n) paced arbitrariy on the square region An) with a minimum separation r min Divide An) into squareets of equa size Ca a squareet dense, if it contains a number of nodes proportiona to its area For each source-destination pair, choose such a dense squareet as a reay, over which it wi transmit information see Figure 1) u 1 MAC u 2 BC u 3 w 1 w 2 w 3 Fig 1 Sketch of one eve of the hierarchica reaying scheme Here {u i, w i)} 3 i=1 are three source-destination pairs Groups of sourcedestination pairs reay their traffic over dense squareets, which contain a number of nodes proportiona to their area shaded) We time share between the different dense squareets used as reays Within a these reay squareets the scheme is used recursivey to enabe joint decoding and encoding at each reay Consider now one such reay squareet and the nodes that are transmitting information over it If we assume for the moment that a the nodes within the same reay squareet coud cooperate then we woud have a mutipe access channe MAC) between the source nodes and the reay squareet, where each of the source nodes has one transmit antenna, and the reay squareet acting as one node) has many receive antennas Between the reay squareet and the destination nodes, we woud have a broadcast channe BC), where each destination node has one receive antenna, and the reay squareet acting again as one node) has many transmit antennas The cooperation gain from using this kind of scheme arises from the use of mutipe antennas for these mutipe access and broadcast channes To actuay enabe this kind of cooperation at the reay squareet, oca communication within the reay squareets is necessary It can be shown that this oca communication probem is actuay the same as the origina probem, but at a smaer scae Hence we can use the same scheme recursivey to sove this

8 subprobem We terminate the recursion after severa iterations, at which point we use simpe TDMA to bootstrap the scheme The construction of the hierarchica reaying scheme is presented in detai in Section IV-A A back-ofthe-enveope cacuation of the per-node rate it achieves is presented in Section IV-B A detaied anaysis of the hierarchica reaying scheme is presented in Sections VI and VII A Construction Reca that Ab) [0, b] 2 is the square region of area b The scheme described here assumes that n nodes are paced arbitrariy in An) with minimum separation r min 0, 1) We want to find some rate, say ρ 0, that can be supported for a n source-destination pairs of a given permutation traffic matrix λn) The scheme that is described beow is recursive and hence hierarchica) in the foowing sense In order to achieve rate ρ 0 for n nodes in An), it wi use as a buiding bock a scheme for supporting rate ρ 1 for a network of n 1 n 2γn) nodes over Aa 1 ) square of area a 1 ) with a 1 n γn) for any permutation traffic matrix λn 1 ) of n 1 nodes Here the branching factor γn) is a function such that γn) as n We wi optimize over the choice of γn) ater The same construction is used for the scheme over Aa 1 ), and so on In genera, our scheme does the foowing at eve 0 of the hierarchy or recursion) In order to achieve rate ρ for any permutation traffic matrix λn ) over n n 2 γ n) nodes in Aa ), with a n γ n), use a scheme achieving rate ρ +1 over n +1 nodes in Aa +1 ) for any permutation traffic matrix λn +1 ) The recursion is terminated at some eve Ln) to be chosen ater We now describe how the hierarchy is constructed between eves and + 1 for 0 < Ln) Each source-destination pair chooses some squareet as a reay over which it transmits its message This reaying of messages takes pace in two phases a mutipe access phase and a broadcast phase We first describe the seection of reay squareets, then the operation of the network during the mutipe access and broadcast phases, and finay the termination of the hierarchica construction 1) Setting up Reays: Given n nodes in Aa ), divide the square region Aa ) into γn) equa sized squareets Denote them by {A k a +1 )} γn) k=1 Ca a squareet dense if it contains at east n /2γn) = n +1 nodes In other words, a dense squareet contains a number of nodes of at east a 1/2 +1 fraction of its area We show that since the nodes in Aa ) have constant minimum separation r min, a squareet can contain at most Oa +1 ) ie Oa /γn))) nodes, and hence that there are at east Θ2 γn)) dense squareets Each source-destination pair chooses a dense squareet such that both the source and the destination are at a distance Ω a +1 ) from it We ca this dense squareet the reay of this source-destination pair We show that the reays can be chosen such that each reay squareet has at most n +1 communication pairs that use it as reay, and we assume this worst case in the foowing discussion

9 2) Mutipe Access Phase: Source nodes that are assigned to the same dense) reay squareet send their messages simutaneousy to that reay We time share between the Θ2 γn)) different reay squareets If the nodes in the reay squareet coud cooperate, we woud be deaing with a MAC with at most n +1 transmitters, each with one antenna, and one receiver with at east n +1 antennas In order to achieve this cooperation, we use a hierarchica ie, recursive) construction For this recursive construction, assume that we have access to a communication scheme to transmit data according to a permutation traffic matrix λn +1 ) between n +1 nodes ocated in a square of area a +1 We now show how this scheme at scae a +1 can be used to construct a scheme for scae a see Figure 2) u 1 x 11 x 1m y 11 y 1m ŷ 11 q 1 ŷ 1m ŷ 11 ŷ m1 ˆx 11 ˆx m1 v 1 P y x {λ k } m k=1 u m x m1 x mm y m1 y mm q m ŷ m1 ŷ mm ŷ 1m ŷ mm ˆx 1m ˆx mm v m Fig 2 Description of the mutipe access phase at eve in the hierarchy with m n +1 The first system bock represents the wireess channe, connecting source nodes {u i} n +1 i=1 with reay nodes {v i} n +1 i=1 The second system bock are quantizers {qi}n +1 i=1 used at the reay nodes The third system bock represents using n +1 times the communication scheme at eve + 1 organized as n +1 permutation traffic matrices {λ k n +1 )} n +1 k=1 ) to transpose the matrix of quantized observations {ŷij}n +1 i,j=1 In other words, before the third system bock, node v 1 has access to {ŷ 1j} n +1 j=1, and after the third system bock, node v1 has access to {ŷi1}n +1 i=1 The fourth system bock are matched fiters used at the reay nodes Suppose there are n +1 source nodes u 1,,u n+1 ocated anywhere in Aa )) that reay their message over the n +1 reay nodes v 1,,v n+1 ocated in the same dense squareet of area a +1 ) Each source node u i divides its message bits into n +1 parts of equa ength Denote by x ij the encoded part j of the message bits of node u i x ij is reay a arge sequence of channe symbos; to simpify the exposition, we sha, however, assume it is ony a singe symbo) The message parts corresponding to {x ij } n +1 i=1 wi be reayed over node v j, as wi become cear in the foowing Sources {u i } n +1 i=1, transmit {x ij} n +1 i=1 at time j for j {1, n +1 } Let y kj be the observed channe output at reay v k at time j Note that y kj depends ony on channe inputs {x ij } n +1 i=1 In order to decode the message parts corresponding to {x ij} n +1 i=1 at reay node v j, it needs to obtain the observations {y ij } n +1 i=1 from a other reay nodes In other words, a reays need to exchange information For this, each reay v k quantizes its observation {y kj } n +1 j=1 at an appropriate rate K independent of n to obtain {ŷ kj } n +1 j=1 Quantized observation ŷ kj is to be sent from reay v k to reay v j Thus, each of the n +1 reay nodes now has a message of size K for every other reay node This communication demand within the reay squareet can be organized as n +1 permutation traffic matrices {λ j n +1 )} n +1 j=1 between the n +1 reay nodes Note that these reay nodes are ocated in the same square of area a +1 In other words, we are now faced with the origina probem, but at smaer scae a +1 Therefore, using n +1 times the assumed scheme for transmitting according to a permutation traffic matrix for n +1 nodes in Aa +1 ), reay v j can obtain a quantized observations {ŷ ij } n +1 i=1 Now v j uses n +1 matched fiters on {ŷ ij } n +1 i=1 to obtain estimates {ˆx ij } n +1 i=1 of {x ij } n +1 i=1 In other words, each node v j computes 3 n +1 h u ˆxij = i,v k [j] k h u i,v k [j] 2ŷkj k=1 for every i {1,, n +1 } Using these estimates it then decodes the messages corresponding to {x ij } n +1 i=1 3 Note that, since we assume fu CSI, node v j has access to the channe gains {h ui,v k [j]} i,k at any time t j In particuar, this is the case at the time the matched fitering is performed

10 3) Broadcast Phase: Nodes in the same reay squareet then send their decoded messages simutaneousy to the destination nodes corresponding to this reay We time share between the different reay squareets If the nodes in the reay squareet coud cooperate, we woud be deaing with a BC with one transmitter with at east n +1 antennas and with at most n +1 receivers, each with one antenna In order to achieve this cooperation, a simiar hierarchica construction as for the MAC phase is used As in the MAC phase, assume that we have access to a scheme to transmit data according to a permutation traffic matrix λn +1 ) between n +1 nodes ocated in a square of area a +1 We again use this scheme at scae a +1 in the construction of the scheme for scae a see Figure 3) v 1 x 11 x m1 x 11 x m1 ˆx 11 ˆx m1 q 1 ˆx 11 ˆx 1m y 11 y 1m w 1 {λ k } m k=1 P y ˆx v m x 1m x mm x 1m x mm q m ˆx 1m ˆx mm ˆx m1 ˆx mm y m1 y mm w m Fig 3 Description of the broadcast phase at eve in the hierarchy with m n +1 The first system bock represents transmit beamforming at each of the reay nodes {v i} n +1 i=1 The second system bock are quantizers {qi}n +1 i=1 used at the reay nodes The third system bock represents using n +1 times the communication scheme at eve + 1 organized as n +1 permutation traffic matrices {λ k n +1 )} n +1 k=1 ) to transpose the matrix of quantized beamformed channe symbos {ˆx ij} n +1 i,j=1 In other words, before the third system bock, node v1 has access to {ˆx i1} n +1 i=1, and after the third system bock, node v1 has access to {ˆx1j}n +1 j=1 The fourth system bock is the wireess channe, connecting reay nodes {v i} n +1 i=1 with destination nodes {w i} n +1 i=1 Suppose there are n +1 reay nodes v 1,,v n+1 ocated in the same dense squareet of area a +1 ) that reay traffic for n +1 destination nodes w 1,,w n+1 ocated anywhere in Aa )) Reca that at the end of the MAC phase, each reay node v j has assuming decoding was successfu) access to parts j of the message bits of a source nodes {u i } n +1 i=1 Node v j re-encodes these parts independenty; ca { x ij } n +1 i=1 the encoded channe symbos as before, we assume x ij is ony a singe symbo to simpify exposition) Reay node v j then performs transmit beamforming on { x ij } n +1 i=1 for the n +1 transmit antennas of {v k } n +1 k=1 to be sent at time T + j for some appropriatey chosen T > 0 not depending on j) Ca x kj the resuting channe symbo to be sent from reay node v k Then 4 x kj = i h v k,w i [T + j] k h v k,w i [T + j] 2 x ij In order to actuay send this channe symbo, reay node v k needs to obtain x kj from node v j Thus, again a reay nodes need to exchange information To enabe oca cooperation within the reay squareet, each reay node v j quantizes its beamformed channe symbos {x kj } n +1 k=1 at an appropriate rate K ogn) with K independent of n to obtain {ˆx kj} n +1 Now, quantized vaue ˆx kj is sent from reay v j to reay v k Thus, each of the n +1 reay nodes now has a message of size K ogn) for every other reay node This communication demand within the reay squareet can be organized as n +1 permutation traffic matrices {λ k n +1 )} n +1 k=1 between the n +1 reay nodes Note that these reay nodes are ocated in the same square of area a +1 Hence, we are again faced with the origina probem, but at smaer scae a +1 Using n +1 times the assumed scheme for transmitting according to a permutation traffic matrix for n +1 nodes in Aa +1 ), reay v k can obtain a quantized beamformed channe symbos {ˆx kj } n +1 j=1 Now each v k sends ˆx kj over the wireess channe at time instance T + j with T chosen to account for the preceding MAC phase and the oca cooperation in the BC phase) Ca y ij the received channe output at destination node 4 Note that, since we ony assume causa CSI, reay node v j does not actuay have access to {h vk,w i [T +j]} k,i at the time the beamforming is performed This probem can, however, be circumvented The detais are provided in the proofs see Lemma 10) k=1

11 w i at time instance T + j Using y ij, destination node w i can now decode part j of the message bits of its source node u i 4) Spatia Re-Use and Termination of Recursion: The scheme does appropriatey weighted timedivision among different eves 0 Ln) Within any eve 1, mutipe regions of the origina square An) of area n are being operated in parae The detais reated to the effects of interference between different regions operating at the same eve of hierarchy are discussed in the proofs The recursive construction terminates at some arge enough eve L = Ln) to be chosen ater) At this scae, we have n L nodes in area Aa L ) A permutation traffic matrix at this eve comprises n L source-destination pairs These transmissions are performed using TDMA Again, mutipe regions in the origina square of area n at eve L are active simutaneousy B Achievabe Rates Here we present a back-of-the-enveope cacuation of the per-node rate ρ HR n) achievabe with the hierarchica reaying scheme described in the previous section The compete proof is stated in Section VII We assume throughout that ong bock codes and corresponding optima decoders are used for transmission Instead of computing the rate achieved by hierarchica reaying, it wi be convenient to instead anayze its inverse, ie, the time utiized for transmission of a singe message bit from each source to its destination under a permutation traffic matrix λn) Using the hierarchica reaying scheme, each message traves through L eves of the hierarchy Ca τ n) the amount of time spent for the transmission of one message bit between each of the n source-destination pairs at eve in the hierarchy We compute τ n) recursivey At any eve 1, there are mutipe regions of area a operating at the same time Due to the spatia re-use, each of these regions gets to transmit a constant fraction of time It can be shown that the addition of interference due to this spatia re-use eads ony to a constant oss in achievabe rate Hence the time required to send one message bit is ony a constant factor higher than the one needed if region Aa ) is considered separatey Consider now one such region Aa ) By the time sharing construction, ony one of its Θ2 γn)) dense reay squareets of area a +1 is active at any given moment Hence the time required to operate a reay squareets is a Θ2 γn)) factor higher than for just one reay squareet separatey Consider now one such reay squareet, and assume n +1 source nodes in Aa ) communicate each n +1 message bits to their respective destination nodes through a MAC phase and BC phase with the hep of the n +1 reay nodes in this reay squareet of area a +1 In the MAC phase, each of the n +1 sources simutaneousy sends one bit to each of the n +1 reay nodes The tota time for this transmission is composed of two terms i) Transmission of n +1 message bits from each of the n +1 source nodes to those many reay nodes Since we time share between Θ2 γn)) reay squareets, we can transmit with an average power constraint of Θ2 γn)) during the time a reay squareet is active, and sti satisfies the overa average power constraint of 1 With this bursty transmission strategy, we require a tota of a O n α/2 ) +1 = O n 2 +1 4 γ 1 α/2) n)n α/2 1) 2) γn)n +1 channe uses to transmit n +1 bits per source node The terms on the eft-hand side of 2) can be understood as foows: n +1 is the number of bits to be transmitted; a α/2 is the power oss since most nodes communicate over a distance of Θa 1/2 ); 2 γn) is the average transmit power; n +1 is the mutipe-antenna gain, since we have that many transmit and receive antennas ii) We show that constant rate quantization of the received observations at the reays is sufficient Hence the n +1 bits for a sources generate On +1 ) transmissions at eve + 1 of the hierarchy Therefore, On +1 τ +1 n)) 3)

12 channe uses are needed to communicate a quantized observations to their respective reay nodes Combining 2) and 3), accounting for the factor 2 γn) oss due to time division between reay squareets, we obtain that the transmission time for one message bit from each source to the reay squareet in the MAC phase at eve is ) τ MAC n) = O 2 γ 1+1 α/2) n)n α/2 1 + τ +1 n) 4) Next, we compute the number of channe uses per message bit received by the destination nodes in the BC phase Simiar to the MAC phase, each of the n +1 reay nodes has n +1 message bits out of which one bit is to be transmitted to each of the n +1 destination nodes Since there are n +1 reay nodes, each destination node receives n +1 message bits As before the required transmission time has two components i) Transmission of the encoded and quantized message bits from each of the n +1 reay nodes to a other reay nodes at eve + 1 of the hierarchy We show that each message bit resuts in O + 1) og n ) quantized bits Therefore, O n +1 + 1) og n ) bits need to be transmitted from each reay node This requires O n +1 + 1) ogn)τ +1 n) ) 5) channe uses ii) Transmission of n +1 message bits from the reay nodes to each destination node As before, we use bursty transmission with an average power constraint of Θ2 γn)) during the fraction Θ2 γ 1 n)) of time each reay squareet is active this satisfies the overa average power constraint of 1) Using this bursty strategy requires a O n α/2 ) +1 = O n 2 +1 4 γ 1 α/2) n)n α/2 1) 6) γn)n +1 channe uses for transmission of n +1 bits per destination node As in the MAC phase, n +1 in the eft hand side of 6) can be understood as the number of bits to be transmitted, a α/2 as the power oss for communicating over distance Θa 1/2 ), 2 γn) as the average transmit power, and n +1 as the mutipe-antenna gain Combining 5) and 6), accounting for a factor 2 γn) oss due to time division between reay squareets, the transmission time for one message bit from the reays to each destination node in the BC phase at eve is ) τ BC n) = O 2 γ 1+1 α/2) n)n α/2 1 + + 1) ogn)τ +1 n) 7) From 4) and 7), we obtain the foowing recursion τ n) = τ MAC n) + τ BC n) ) = O 2 γ 1 α/2)+1 n)n α/2 1 + + 1) ogn)τ +1 n) ) = O 2 L γn)n α/2 1 + L ogn)τ +1 n), 8) where we have used α > 2 This recursion hods for a 0 < L At eve L, we use TDMA among n L nodes in region Aa L ) with a permutation traffic matrix λn L ) Each of the n L source-destination pairs uses the wireess channe for 1/n L fraction of the time at power On L ), satisfying the average power constraint Assuming the received power is ess than 1 for a n so that we operate in the power imited regime), we can achieve a rate of at east Ωa α/2 L ) between any source-destination pair Equivaenty τ L n) = Oa α/2 L ) = O n α/2 γ Lα/2 n) ) = O n α/2 γ L n) ) 9)

13 Combining 8) and 9), we have τ 0 n) = O n α/2 1 2 L γn) + L ogn)τ 1 n) ) = = O n α/2 1 L ogn) )L 2 L γn) + L ogn) )L ) τ L n) = O n α/2 1 L ogn) ) L 2 L γn) + nγ L n) )) 10) The term ) L L ogn) 2 L γn) + nγ L n) ) is the oss factor over the desired order n α/2 1 scaing, and we now choose the branching factor γn) and the hierarchy depth L Ln) to make it sma Fix a δ 0, 1/2) and set With this Ln) og 1/2 δ n), γn) n 1/Ln)+1) Ln) ogn) ) Ln) n 2og 1/2 δ n)og ogn), 2 Ln) γn) n og 1/2 δ n)+og δ 1/2 n), nγ Ln) n) n ogδ 1/2 n) Since δ > 0, the n ogδ 1/2 n) term dominates in 10), and we obtain τ 0 n) bn)n α/2 1, where bn) n Oog δ 1/2 n)) Hence the per-node rate of the hierarchica reaying scheme is ower bounded as ρ HR n) = 1/τ 0 n) bn)n 1 α/2, with bn) n Oogδ 1/2 n)) Note that to minimize the oss term, we shoud choose δ > 0 to be sma V COOPERATIVE MULTI-HOP SCHEME In this section, we provide a brief description of the cooperative muti-hop scheme The detais of the construction and the anaysis of its performance can be found in Section X Reca that a node pacement V n) is µ-reguar at resoution dn) if every square [idn), i + 1)dn)] [jdn), j +1)dn)] for some i, j N contains at east µd 2 n) nodes Given such a node pacement V n), divide it into squares of sideength dn) Consider four adjacent squares, combined into a bigger square of sideength 2dn) By the reguarity assumption on V n), this bigger square contains at east 4µd 2 n) nodes Hence we can appy the hierarchica reaying scheme introduced in the ast section to support any permutation traffic within this bigger square at a per-node rate of bn)d 2 n)) 1 α/2 = bn)d 2 α n), where bn) is essentiay of order n og 1/2 n) By propery choosing the permutation traffic matrices within every possibe such bigger square of sideength 2dn), this creates a equivaent communication graph with n/d 2 n) nodes each corresponding to a square of sideength dn) in An), and with edges between nodes

14 corresponding to neighboring squares With the above communication procedure and appropriate spatia re-use, each such edge has a capacity of d 2 n)bn)d 2 α n) = bn)d 4 α n) The resuting communication graph is depicted in Figure 4 Fig 4 Communication graph in bod) resuting from the construction of the cooperative muti-hop scheme The entire square has sideength n, and the dashed squares have sideength dn) Each bod) edge in the communication graph corresponds to using the hierarchica reaying scheme between the nodes in the adjacent squares of sideength dn) Now, to send a message from a source node in V n) to its destination node, we first ocate the squares of sideength dn) they are ocated in We then route the message over the edges of the communication graph constructed above in a muti-hop fashion By the construction of the communication graph, each such edge is impemented using the hierarchica reaying scheme In other words, we perform mutihop communication over distance n with hop ength dn), and each such hop is impemented using hierarchica reaying over distance dn) Since each edge in the communication graph has a capacity of bn)d 4 α n) and has to support roughy n 1/2 dn) source-destination pairs, we obtain a per-node rate of per source-destination pair ρ CMH n) bn)d 4 α n)n 1/2 d 1 n) = bn)d 3 α n)n 1/2 VI ANALYSIS OF THE HIERARCHICAL RELAYING SCHEME In this section, we anayze in detai the hierarchica reaying scheme Throughout Sections VI-A to VI-C, we consider communication at eve, 0 < L = Ln), of the hierarchy A constants K i are independent of Reca that at eve, we have a square region Aa ) of area a n γ n) containing n n 2 γ n) nodes V n ) We divide Aa ) into γn) squareets of area a +1 Reca that a squareet of area a +1 in eve of the hierarchy is caed dense if it contains at east n +1 nodes We impose a power constraint of

15 P n) = Θ2 γn)) during the time any particuar reay squareet is active Since we time share between Θ2 γn)) reay squareets, this satisfies the overa average power constraint by choosing constants appropriatey) Since other regions of area a are active at the same time as the one under consideration, we have to dea with interference To this end, we consider a sighty more genera noise mode that incudes the experienced interference at the reay squareets More precisey, we assume that, for a u V n ), the additive noise term {z u [t]} t is independent of the signa {x u [t]} t and of the channe gains {h u,v [t]} v,t ; that the noise term is stationary and ergodic across time t, but with arbitrary dependence across nodes u; and that the noise has zero mean and bounded power N 0 independent of n Note that we do not require the additive noise term to be Gaussian In the above, N 0 accounts for both noise which has power 1 in the origina mode), as we as interference We show in Section VII that these assumptions are vaid Reca the foowing choice of γn) and Ln): Ln) og 1/2 δ n), γn) n 1/Ln)+1), with δ 0, 1/2) independent of n This choice satisfies γn) γñ) if n ñ, γ Ln) n) n for a n, 2 Ln) γn) as n, The first condition in 12) impies that the number of squareets γn) we divide An) into increases in n The second condition impies the squareet area a Ln) at the ast eve of the hierarchy is bigger than 1 As we sha see, the third condition impies that the number of dense squareets at the ast eve and hence at every eve) grows unbounded as n see Lemma 6 beow) Throughout Section VI, we consider the fast fading channe mode Sow fading is discussed in Section VII-B A Setting up Reays The first emma states that the minimum-separation requirement r min 0, 1) impies that a constant fraction of squareets must be dense We point out that this is the ony consequence of the minimumseparation requirement used to prove Theorem 1 Thus Theorem 1 remains vaid if we just assume that Lemma 6 beow hods directy See aso Section XII-D for further detais Lemma 6 For any V n ) Aa ) with V n ) n and with minimum separation r min 0, 1), each of its squareets of area a +1 contains at most K 1 a /γn) nodes, and there are at east K 2 2 γn) dense squareets Proof Put a circe of radius r min /2 around each node By the minimum-separation requirement, these circes do not intersect Each node covers an area of πr 2 min/4 Increasing the sideength of each squareet by r min, this provides a tota area of a /γn) + r min ) 2 a γn) 1 + r min) 2 in which the circes around these nodes are packed Here we have used that γ +1 n) n by 12), and therefore γn) n/γ n) = a Hence there can be at most K 1 a /γn) nodes per squareet with K 1 4 1 + r min) 2 πrmin 2 11) 12)

16 Note that, since r min < 1, we have K 1 > 1 Let dn ) be the number of dense squareets in Aa ), and therefore γn) dn ) is the number of squareets that are not dense By the argument in the ast paragraph, each dense squareet contains at most K 1 a /γn) nodes, and those squareets that are not dense contain ess than n +1 nodes by the definition of dense squareets Hence dn ) must satisfy dn )K 1 a /γn) + γn) dn ) ) n +1 V n ) n Thus, using a = 2 n, n +1 = n /2γn), we have dn )K 1 2 + γn) dn ))/2 γn) As K 1 2 > 1, this yieds with dn ) 1 1/2 2 γn) γn) = K K 1 2 2 2 γn), 1/2 2K 1 K 2 1 2K 1 Consider V n ) Aa ) with V n ), and choose arbitrary K 2 2 γn) dense squareets of area a +1 as guaranteed by Lemma 6) Ca those squareets {A k a +1 )} K 22 γn) k=1 For each sour-destination pair, we now seect one such dense squareet to reay traffic over To avoid bottenecks, this seection has to be done such that a reay squareets carry approximatey the same amount of traffic Moreover, for technica reasons, the distances from the source and the destination to the reay squareet cannot be too sma Formay, the seection of reay squareets can be described by the schedues S {0, 1} n K 2 2 γn) with s u,k = 1 if source node u reays traffic over dense squareet k, and S {0, 1} K 22 γn) n with s k,w = 1 if destination node w receives traffic from dense squareet k With sight abuse of notation, et r u,ak a +1 ) be the distance between node u V n ) and the cosest point in A k a +1 ), ie, Define the sets and Sn ) r u,ak a +1 ) { S {0, 1} n K 2 2 γn) : 0 n u=1 s u,k n +1 k, min r u,v 13) v A k a +1 ) 0 K 2 2 γn) k=1 s u,k 1 u, s u,k = 1 impies r u,ak a +1 ) } 2a +1 u, k Sn ) { S {0, 1} K 2 2 γn) n : S T Sn ) } 14) The sets Sn ) and Sn ) are the coection of schedues satisfying the conditions mentioned in the ast paragraph More precisey, the first condition in 14) ensures that at most n +1 source-destination pairs reay over the same dense squareet, the second condition ensures that each source-destination pair chooses at most one reay squareet, and the third condition ensures that sources and destinations are at east at distance 2a +1 from the chosen reay squareet Next, we prove that any node pacement that satisfies Lemma 6 aows for a decomposition of any permutation traffic matrix λn ) into a sma number of schedues beonging to Sn ) and Sn )

17 Lemma 7 There exist K 3 such that for a n arge enough independent of ), and every permutation traffic matrix λn ) {0, 1} n n we can find K 3 2 schedues {S i) n )} K 32 i=1 Sn ), { S i) n )} K 32 i=1 Sn ) satisfying λn ) = K 3 2 i=1 S i) n ) S i) n ) Proof Pick an arbitrary source-destination pair in λn ), and consider the squareets containing the source and the destination node Since each squareet has side ength a +1, there are at most 50 squareets at distance ess than 2a +1 from either of those two squareets As 2 Ln) γn) as n by 12), there exists K independent of ) such that for n K we have 50 K 2 2 1 γn) Since there are at east K 2 2 γn) dense squareets by Lemma 6, there must exist at east K 2 2 1 γn) dense squareets that are at distance at east 2a +1 from both the squareets containing the source and the destination node In order to construct a decomposition of λn ), we use the foowing procedure Sequentiay, each of the n source-destination pairs chooses one of the at east) K 2 2 1 γn) dense squareets at distance at east 2a +1 that has not aready been chosen by n +1 other pairs If any source-destination pair can not seect such a squareet, then stop the procedure and use the source-destination pairs matched with dense squareets so far to define matrices S 1) n ) and S 1) n ) Now, remove a the matched sourcedestination pairs, forget that dense squareets were matched to any source-destination pair and redo the above procedure, going through the remaining source-destination pairs Let K 3 4/K 2 We caim that by repeating this process of generating matrices S i) n ) and S i) n ), we can match a source-destination pairs to some dense squareet with at most K 3 2 such matrices Indeed, a new pair of matrices is generated ony when a source-destination pair can not be matched to any of its avaiabe at east) K 2 2 1 γn) dense squareets If this happens, a these dense squareets are matched by n +1 = n /2γn) pairs Hence at east K 2 2 2 n source-destination pairs are matched in each round Since there are n tota pairs, we need at most n K 2 2 2 n = K 3 2 matrices S i) n ) and S i) n ) For a permutation traffic matrix λn ), communication proceeds as foows Write λn ) = K 3 2 i=1 S i) n ) S i) n ) as in Lemma 7 Spit time into K 3 2 equa ength time sots In sot i, we use S i) n ) S i) n ) as our traffic matrix Consider without oss of generaity i = 1 in the foowing Write S 1) n ) S 1) n ) = K 2 2 γn) k=1 S 1,k) n +1 ) S 1,k) n +1 ), where S 1,k) n +1 ) S 1,k) n +1 ) is the traffic reayed over the dense squareet A k a +1 ) We time share between the schedues for k {1,, K 2 2 γn)} Consider now any such k In the worst case, there are exacty n +1 communication pairs to be reayed over A k a +1 ), and the reay squareet A k a +1 ) contains exacty n +1 nodes We sha assume this worst case in the foowing We focus on the transmission according to the traffic matrix S 1,1) n +1 ) S 1,1) n +1 ) Let V n +1 ) be the nodes in A 1 a +1 ), and et Un +1 ) and Wn +1 ) be the source and destination nodes of S 1,1) n +1 ) S 1,1) n +1 ), respectivey In other words, the source nodes Un +1 ) communicate to their respective destination nodes Wn +1 ) using the nodes V n +1 ) as reays

18 B Mutipe Access Phase Each source node in Un +1 ) spits its message into n +1 equa ength parts Part j at every node u Un +1 ) is to be reayed over the j-th node in V n +1 ) Each part is separatey encoded at the source and separatey decoded at the destination After the source nodes are done transmitting their messages, the nodes in the reay squareet quantize their samped) observations corresponding to part j and communicate the quantized vaues to the j-th node in the reay squareet This node then decodes the j-th message parts of a source nodes Note that this induces a uniform traffic pattern between the nodes in the reay squareet, ie, every node needs to transmit quantized observations to every other node Whie this traffic pattern does not correspond to a permutation traffic matrix, it can be written as a sum of n +1 permutation traffic matrices A fraction 1/n +1 of the traffic within the reay squareet is transmitted according to each of these permutation traffic matrices This setup is depicted in Figure 2 in Section IV-A Assuming for the moment that we have a scheme to send the quantized observations to the dedicated node in the reay squareet, the traffic matrix S 1,1) n +1 ) between Un +1 ) and V n +1 ) describes then a MAC with n +1 transmitters, each with one antenna, and one receiver with n +1 antennas We ca this the MAC induced by S 1,1) n +1 ) in the foowing Before we anayze the rate achievabe over this induced MAC, we need an auxiiary resut on quantized channes y 1 q 1 ŷ 1 w f x P y x y m q m ŷ m Pˆx ŷ ˆx ϕ ŵ Fig 5 Sketch of the quantized channe f and ϕ are the channe encoder and decoder, respectivey; {q k } m k=1 are quantizers; P y x and Pˆx ŷ represent stationary ergodic channes with the indicated margina distributions Consider the quantized channe in Figure 5 Here, f is the channe encoder, ϕ the channe decoder, {q k } m k=1 quantizers A these have to be chosen P y x and Pˆx ŷ, on the other hand, represent fixed stationary ergodic channes with the indicated margina distributions We ca R the rate of the channe code f, ϕ) and {R k } m k=1 the rates of quantizers {q k} m k=1 Lemma 8 If there exist distributions P x and {Pŷk y k } m k=1 such that R < Ix; ˆx) and R k > Iy k ; ŷ k ), k, then R, {R k } m k=1) is achievabe over the quantized channe Proof The proof foows from a simpe extension of Theorem 1 in Appendix II of [8] Lemma 9 Let the additive noise {z v } v V n+1 ) be uncorreated over v) For the MAC induced by S 1,1) n +1 ) with per-node average power constraint P n) n 1, a rate of +1 aα/2 ρ MAC n) K 4 P n)n +1 a α/2 per source node is achievabe, and the number of bits required at each reay node to quantize the observations is at most K 5 bits per n +1 tota message bits 5 sent by the source nodes Proof The source nodes send signas with a power of essentiay) n 1 +1 aα/2 for a fraction P n)n +1 a α/2 1 of time and are sient for the remaining time To ensure that interference is uniform, the time sots during which the nodes send signas are chosen randomy as foows Generate independenty for each region Aa ) a Bernoui process {B[t]} t N with parameter P n)n +1 a α/2 /1 + η) 1 for some sma η > 0 The nodes in Aa ) are active whenever B[t] = 1 and remain sient otherwise Since the bockength of the codes used is assumed to be arge, this satisfies the average power constraint of P n) with high probabiity for any η > 0 Since we are interested ony in the scaing of capacity, we ignore the additiona 5 Tota message bits refers to the sum of a message bits transmitted by the n +1 source nodes

19 1/1 + η) term in the foowing to simpify notation Ceary, we ony need to consider the fraction of time during which B[t] = 1 Let y be the received vector at the reay squareet, ŷ the componentwise) quantized observations We use a matched fiter at the reay squareet, ie, ˆx u = h u h u ŷ, where coumn vector h u = {h u,v } v V n+1 ) are the channe gains between node u Un +1 ) and the nodes in the reay squareet V n +1 ) The use of a matched fiter is possibe since we assume fu CSI is avaiabe at a the nodes We now use Lemma 8 to show that we can design quantizers {q v } v V n+1 ) of constant rate and achieve a per-node communication rate of at east K 4 P n)n +1 a α/2 The first channe in Lemma 8 see Figure 5) wi correspond to the wireess channe between a source node u and its reay squareet V n +1 ) The second channe in Lemma 8 wi correspond to the matched fiter used at the reay squareet To appy Lemma 8, we need to find a distribution for x u and for ŷ v y v Define r u r u,a1 a +1 )/ 2a 1 with r u,a1 a +1 ) as in 13), to be the normaized distance of the source node u Un +1 ) to the reay squareet A 1 a +1 ) For each u Un +1 ) et x u N C 0, r u αn 1 +1 aα/2 ) independent of xũ for u ũ, and et ŷ v = y v + z v for z v N C 0, 2 ) independent of y and for some 2 > 0 Note that the channe input x u has power that depends on the normaized distance r u ie, ony nodes u Un +1 ) that are at maxima distance 2a from the reay squareet transmit at fu avaiabe power) This is to ensure that a signas are received at roughy the same strength by the reays We proceed by computing the mutua informations Iy v ; ŷ v {hũ,ṽ }) and Ix u ; ˆx u {hũ,ṽ }) as required in Lemma 8 the conditioning on {hũ,ṽ } being due to the avaiabiity of fu CSI) Note first that by construction of S 1,1) n +1 ) see 14)), we have for u Un +1 ) and v V n +1 ) and hence From this, and since h u,v 2 = ru,v α, we obtain r u,a1 a +1 ) r u,v 2r u,a1 a +1 ), 2 3α/2 a α/2 2 3α/2 n +1 a α/2 1 2 2a r u r u,v 1 2a 15) h u,v 2 r α u 2 α/2 a α/2, h u 2 r α u 2 α/2 n +1 a α/2 We start by computing Iy v ; ŷ v {hũ,ṽ }) We have ŷ v = h u,v x u + z v + zṽ, and hence ŷ v has mean zero and variance E ŷ v 2 ) = u Un +1 ) u Un +1 ) h u,v 2 r α un 1 +1 aα/2 + N 0 + 2 n +1 2 α/2 a α/2 n 1 +1 aα/2 + N 0 + 2 = 2 α/2 + N 0 + 2, 16)

20 where we have used 16) Hence Iy v ; ŷ v {hũ,ṽ }) = hŷ v {hũ,ṽ }) hŷ v y v, {hũ,ṽ }) og 2πeE ŷ v 2 ) ) og2πe 2 ) og 2πe2 α/2 + N 0 + 2 ) ) og2πe 2 ) = og 1 + 2 α/2 + N ) 0 17) 2 We now compute Ix u ; ˆx u {hũ,ṽ }) We have Conditioned on {hũ}ũ Un+1 ), and E ũ Un +1 )\{u} ˆx u = h u x u + ũ Un +1 )\{u} h u h ũ h u x ũ + h u z + z) h u h u x u N C 0, hu 2 r ) un α 1 +1 aα/2, h u h ũ h u x ũ + h u h u z + z) 2 ) {hũ} = n 1 +1 aα/2 ũ Un +1 )\{u} r α ũ h u h ũ 2 h u 2 + N 0 + 2, where we have used the assumption that {z v } v V n+1 ) are uncorreated in the second ine Using 16), this is, in turn, upper bounded by r ũ α h u h ũ 2 + N 0 + 2 2 3α/2 r α u n 2 +1 aα ũ Un +1 )\{u} Simiary, we can ower bound the received signa power as E h u 2 x u 2) 2 3α/2 Since Gaussian noise is the worst additive noise under a power constraint [15], and appying Jensen s inequaity to the convex function og1 + 1/x), we obtain ) ) 2 3α/2 Ix u ; ˆx u {hũ,ṽ }) E og 1 + 2 3α/2 r un α 2 +1 aα ũ Un +1 )\{u} rα ũ h uhũ 2 + N 0 + 2 ) 2 3α/2 og 1 + 2 3α/2 r u αn 2 +1 aα ũ Un +1 )\{u} rα ũ E h uhũ 2) 18) + N 0 + 2 We have for u ũ, and hence using 15) E r u α ũ Un +1 )\{u} E h u h ũ 2) = Eh u h ũh ũ h u) = h u,v 2 hũ,v 2 r α ũ h uhũ 2 ) v V n +1 ) = v V n +1 ) = r α u,vr α ũ,v, 19) ũ Un +1 )\{u} v V n +1 ) 2 α n 2 +1 a α r α ur α u,v r α ũr α ũ,v

21 Therefore we can continue 18) as Ix u ; ˆx u {hũ,ṽ }) 1 2 og 1 + ) 2 3α/2 K 2 α/2 + N 0 + 2 4 20) Using 17) and 20) in Lemma 8, and observing that we ony communicate during a fraction P n)n +1 a α/2 1 of time yieds a per source node rate ρ MAC n) arbitrariy cose to and a quantizer of rate arbitrariy cose to og K 4 P n)n +1 a α/2 1 + 2 α/2 + N ) 0 2 bits per observation at each reay node Since by 20) mutua information Ix u ; ˆx u {hũ,ṽ }) is at east K 4 for every u Un +1 ) during the fraction of time we actuay communicate, this impies that there are at most 1/K 4 observations at each reay node per n +1 tota message bits Thus the number of bits per reay node required to quantize the observations is at most K 5 1 og 1 + 2 α/2 + N ) 0 K 4 2 bits per n +1 tota message bits sent by the source nodes C Broadcast Phase At the end of the MAC phase, each node in the reay squareet received a part of the message sent by each source node In the BC phase, each node in the reay squareet encodes these messages together for n +1 transmit antennas The encoded message is then quantized and communicated to a the nodes in the reay squareet These nodes then send the quantized encoded message to the destination nodes Wn +1 ) Note that this again induces a uniform traffic pattern between the nodes in the reay squareet, ie, every node needs to transmit quantized encoded messages to every other node Whie this traffic pattern does not correspond to a permutation traffic matrix it can be written as a sum of n +1 permutation traffic matrices A fraction 1/n +1 of the traffic within the reay squareet is transmitted according to each of these permutation traffic matrices This setup is depicted in Figure 3 in Section IV-A Assuming for the moment that we have a scheme to send the quantized encoded messages to the corresponding nodes in the reay squareet, the traffic matrix S 1,1) n +1 ) between V n +1 ) and Wn +1 ) describes then a BC with one transmitter with n +1 antennas and n +1 receivers, each with one antenna We ca this the BC induced by S 1,1) n +1 ) in the foowing Lemma 10 For the BC induced by S 1,1) n +1 ) with per-node average power constraint P n) n 1 +1 aα/2, a rate of ρ BC n) K 6 P n)n +1 a α/2 is achievabe per destination node, and the number of bits required to quantize the observations is at most K 7 + 1) ogn) bits at each reay node per n +1 tota message bits 6 received by the destination nodes Proof Consider a node v V n +1 ) in the reay squareet, say the first one From the MAC phase, this node received the first part of the messages of each source node u Un +1 ) We woud ike to jointy encode these message parts at the reay node using transmit beamforming, and then transmit the 6 Tota message bits refers to the sum of a message bits received by the n +1 destination nodes

22 corresponding encoded signa using a the nodes in the reay squareet However, this cannot be done directy, because at the encoding time, the future channe state at transmission time is unknown We circumvent this probem by reordering the signas to be transmitted at the reay nodes as foows Let {ˆθ v,w } v V n+1 ),w Wn +1 ) {0, π/2, π, 3π/2} n2 +1 be a quantized channe state The part of the messages at node one in the reay squareet is encoded for n +1 transmit nodes with an assumed channe gain of ĥ v,w [t] = rv,w α/2 exp 1ˆθ v,w [t]), where the {ˆθ v,w [t]} v,w,t are cyced as a function of t through a possibe vaues in {0, π/2, π, 3π/2} n2 +1 The components of the encoded messages are then quantized and each component sent to the corresponding node in the reay squareet Once a nodes in the reay squareet have received the encoded message, they send in each time sot a sampe of the encoded messages corresponding to the quantized channe state cosest in Eucidean distance) to the actua channe reaization in that time sot By ergodicity of {θ u,v [t]} t, each quantized channe state is used approximatey the same number of times More precisey, as the message ength grows to infinity, we can send sampes of the encoded message parts a 1/1 + η) fraction of time with probabiity approaching 1 for any η > 0 Since we have no constraint on the encoding deay in our setup, we can choose η arbitrariy sma, and given that we are ony interested in scaing aws, we wi ignore this term in the foowing to simpify notation Note that the destination nodes can reorder the received sampes since we assume fu CSI In the foowing, we et {ˆθ v,w } v,w be the random quantized channe state induced by {θ v,w } v,w through the above procedure Denote by {ĥv,w} v,w the corresponding channe gains As in the MAC phase, the nodes in the reay squareet send signas at a power essentiay) n 1 +1 aα/2 a fraction P n)n +1 a α/2 1 of time and are sient for the remaining time To create interference at uniform power, this is done in the same randomized manner as in the MAC phase Generate independenty for each region Aa ) a Bernoui process {B[t]} t N with parameter P n)n +1 a α/2 /1 + η) for some sma η > 0 The nodes in Aa ) are active whenever B[t] = 1 and remain sient otherwise As before, we ignore the additiona 1/1 + η) term Again we ony need to consider the fraction of time during which B[t] = 1 Consider the message part at a reay node for destination node w Wn +1 ) We encode this part independenty; ca x w the encoded message part The reay node then performs transmit beamforming to construct the encoded message for a its destination nodes, ie, x = w Wn +1 ) ĥ w h w x w, where row vector h w = {h v,w } v V n+1 ) contains the channe gains to node w, and where we have used ĥv,w = h v,w The reay node then quantizes the vector of encoded messages componentwise and forwards the quantized version ˆx to the other nodes in the reay squareet These nodes then send ˆx over the channe to the destination nodes The received signa at destination node w is thus y w = h w ˆx + z w With this, we have the setup considered in Lemma 8 with different variabe names) The first channe in Lemma 8 see Figure 5) wi correspond to the transmit beamforming used at the reay squareet The second channe in Lemma 8 wi now correspond to the wireess channe between the reay squareet V n +1 ) and a destination node w To appy Lemma 8, we need to find a distribution for x w and for ˆx v x v We aso need to guarantee that ˆx v satisfies the power constraint at each node v in the reay squareet For each w Wn +1 ) et x w N C 0, Kn 1 ) for some K to be chosen ater) independent of x w +1 aα/2

23 for w w, and et ˆx v = x v + z v for z v N C 0, 2 ) independent of x and for some 2 > 0 We then have y w = hwĥ w h w x h w ĥ w w + h w x w + h w z + z w w Wn +1 )\{w} We proceed by computing the mutua informations Ix v ; ˆx v {hũ,ṽ }) and I x w ; y w {hũ,ṽ }) as required in Lemma 8 the conditioning in {hũ,ṽ } again being due to the avaiabiity of fu CSI) Note first that by construction of S 1,1) n +1 ), we have for any w Wn +1 ) and therefore for h v,w 2 h w 2 2 min r v,w max r v,w, v V n +1 ) v V n +1 ) minv V n+1) r v,w ) α n +1 maxv V n+1 ) r v,w ) α We start by computing Ix v ; ˆx v {hũ,ṽ }) ˆx v has mean zero and variance E ˆx v 2) = w Wn +1 ) h v,w 2 h w 2Kn 1 +1 aα/2 + 2 2 α n +1 Kn 1 +1 n aα/2 + 2 +1 2α n +1 21) n 1 +1 aα/2, 22) K 2 α 1 2 ), which is positive for 2 < 1, and where we have used 21) and that n 1 +1 aα/2 2 +1 γn) 1 by 12) Equation 22) shows that ˆx v satisfies the power constraint of node v in the reay squareet V n +1 ) Moreover, we obtain Ix v ; ˆx v {hũ,ṽ }) = hˆx v {hũ,ṽ }) hˆx v x v, {hũ,ṽ }) og 2πeE ˆx v 2)) og2πe 2 ) n 1 ) +1 og aα/2 23) It remains to compute I x w ; y w {hũ,ṽ }) Note that the encoding procedure guarantees that Moreover, for w w, 2 cosπ/4) 2 h w 4 h w ĥ w 2 h w 4 E h w ĥ w 2) = Eh w ĥ wĥ wh w ) = E h vw 2 ĥv w 2) v V n +1 ) = v V n +1 ) = E h w h w 2) E h vw 2 h v w 2)

24 From this, we get by a simiar argument as in Lemma 9 that I x w ; y w {hũ,ṽ }) K 6 24) Using 23) and 24) in Lemma 8, and observing that we ony communicate during a fraction P n)n +1 a α/2 of time, yieds a per destination node rate ρ BC n) arbitrariy cose to K 6 P n)n +1 a α/2 bits per channe use and a quantizer rate arbitrariy cose to n 1 ) +1 og aα/2 2 bits per encoded sampe Since by 24) mutua information I x w ; y w {hũ,ṽ }) is at east K 6 for every w Wn +1 ) during the fraction of time we actuay communicate, this impies that there are at most 1/K 6 encoded message sampes for each reay node per n +1 tota message bits received by the destination nodes Wn +1 ) Thus the number of bits required at each reay node to quantize the encoded message sampes is at most 1 n 1 ) +1 og aα/2 K 6 2 = 1 1 ) og γ 1+1 α/2) n)n α/2 1 K 6 22+1 1 ) n α/2 22+1 1 K 6 og K 7 + 1) ogn) bits per n +1 tota message bits received by the destination nodes, and where we have used γn) n by 12) VII PROOF OF THEOREM 1 The proof of Theorem 1 is spit into two parts In Section VII-A we prove the theorem for fast fading, and in Section VII-B for sow fading A Fast Fading In this section, we prove Theorem 1 under fast fading, ie, {θ u,v [t]} t is stationary and ergodic in t We first prove that the assumptions on the power constraint and the interference made in Section VI see Lemmas 9 and 10) during the anaysis of one eve of the hierarchica reaying scheme are vaid We then use the resuts proved there to anayze the behavior of the entire hierarchy, yieding a ower bound on the per-node rate achievabe with hierarchica reaying We first argue that the constraint P n) n 1 +1 aα/2 needed in Lemmas 9 and 10 is satisfied Consider the hierarchica reaying scheme as described in Section IV and fix a eve, 0 < L = Ln) in this hierarchy At eve we have a square of area a = n/γ n), with n = n/2 γ n) source-destination pairs Since we are time sharing between K 2 2 γn) reay squareets at this eve, we have an average power constraint of P n) K 2 2 γn)

25 during the time any particuar reay squareet is active Since α > 2 and since nγ Ln) n) as n, we have, for n arge enough independent of ), that P n) = K 2 2 γn) 2 n γn) γ Ln) n) n 2 +1 γn) γ n) = n 1 +1 aα/2 ) α/2 1 ) α/2 1 Therefore the power constraint in Lemmas 9 and 10 is satisfied We continue by anayzing the interference caused by spatia re-use Reca that the MAC and BC phases at eve induce permutation traffic within the dense squareets at eve + 1 The permutation traffic within those dense squareets at eve + 1 is transmitted in parae with spatia re-use We now describe in detai how this spatia re-use is performed Partition the squareets of area a +1 ie, at eve + 1) into four subsets such that in each subset a squareets are at distance at east a +1 from each other The traffic that the MAC and BC phases at eve induce in each of the reay squareets at eve + 1 is transmitted simutaneousy within a reay squareets in the same subset Consider now one such subset We show that at any reay squareet the interference from other reay squareets in the same subset is stationary and ergodic within each phase, additive ie, independent of the signas and channe gains in this reay squareet), and of bounded power N 0 1 independent of n We first argue that the interference is stationary and ergodic within each phase Note first that on any eve + 1 in the hierarchy, a reay squareets are either simutaneousy in the MAC phase or simutaneousy in the BC phase Furthermore, a reay squareets are aso synchronized for transmissions within each of these phases reca that the induced traffic in eve +1 is uniform and is sent sequentiay as permutation traffic) Hence it suffices to show that the interference generated by either the MAC or BC induced by some permutation traffic matrix is stationary and ergodic Since a codebooks for either of these cases are generated as iid Gaussian mutipied by a Bernoui process, and in the BC phase beamformed for stationary and ergodic fading, this is indeed the case The additivity of the interference foows easiy for the MAC phase, since codebooks are generated independenty of the channe reaization in this case Moreover, since the channe gains are independent from each other and a codebooks are generated as independent zero mean processes, the interference in the MAC phase is aso uncorreated over space) within each reay squareet For the BC phase, the codebook depends ony on the channe gains within each reay squareet at eve + 1 Since the channe gains within reay squareets are independent of the channe gains between reay squareets, this interference is additive as we We now bound the interference power Note that by the randomized time-sharing construction within the MAC and BC phases see Lemmas 9 and 10), in each reay squareet, at most n +1 nodes transmit at an average power of 1 In the MAC phase, a nodes use independenty generated codebooks with power at most 1, and thus the received interference power from another reay squareet at distance i a +1 is at most n +1 i α a α/2 +1 = i α 2 +1) n ) 1 α/2 i α, γ +1 n) by 12) In the BC phase, the nodes in each active reay squareet use beamforming to transmit to nodes within their own squareet Since the channe gains within a reay squareet are independent of the channe gains between reay squareets, the same cacuation as in 19) shows that we can upper bound the received interference power from another reay squareet at distance i a +1 by in the BC phase as we n +1 i α a α/2 +1 i α,

26 Now, by the way in which we perform spatia re-use, every active reay squareet has at most 8i active reay squareets at distance at east i a +1 Hence the tota interference power received at an active reay squareet is at most 8i2 α i α N 0 1 < i=1 since α > 2 With this, we have shown that the interference term has the properties required for Lemmas 9 and 10 to appy We now appy those two emmas to obtain a ower bound on the rate achievabe with hierarchica reaying Ca τ n) the number of channe uses to transmit one bit from each of n source nodes to the corresponding destination nodes at eve Lemma 7 states that for n arge enough independent of ), we reay over each dense squareet at most K 3 2 times Combining this with Lemma 9, we see that to transmit one bit from each source to its destination at this eve we need at most 4K 3 2 K 2 2 1 γn) K 4 P n) n 1 +1 aα/2 = K 32 2+3 n α/2 1 γ 1+1 α/2) n) channe uses for the MAC phase Here, the factor 4 accounts for the spatia re-use, K 3 2 accounts for reaying over the same reay squareets mutipe times, K 2 2 γn) accounts for time sharing between the reay squareets, and the ast term accounts for the time required to communicate over the MAC Simiary, combining Lemmas 7 and 10, we need at most K 4 K 3 2 2+3 n α/2 1 γ 1+1 α/2) n) K 6 channe uses for the BC phase Moreover, at eve + 1 in the hierarchy this induces a per-node traffic demand of at most K 5 bits from the MAC phase, and at most K 7 + 1) ogn) from the BC phase Thus we obtain the foowing recursion τ n) 8K 3 1 K 4 + 1 K 6 )n α/2 1 γn) 4γ 1 α/2 n) ) + K5 + K 7 + 1) ogn))τ +1 n) Kn α/2 1 γn)4 + K + 1) ogn)τ +1 n) Kn α/2 1 γn)4 L + KL ogn)τ +1 n) 25) for positive constants K, K independent of n and We use TDMA at scae a L with n L nodes and source-destination pairs Time sharing between a sourcedestination pairs, we have during the time we communicate for each node) an average power constraint of n L Since at this eve we communicate over a distance of at most 2a 1/2 L, we have ) τ L n) n L og 1 1 n L + 26) 2 α N 0 a α/2 L Since as n, we can upper bound 26) as for some constant K n L a α/2 L n L a 1 L = 2 L 0 τ L n) K a α/2 L = K n α/2 γ Lα/2 n) K n α/2 γ L n) 27)

27 Now, using the recursion 25) L times, and combining with 27), we obtain τ 0 n) Kn α/2 1 γn)4 L + KL ogn)τ 1 n) L 1 ) Kn ) α/2 1 γn)4 L KL ogn) =0 + KL ogn) )L τ L n) n α/2 1 KL ogn) ) ) L K4 L γn) + K nγ L n) 28) Using the definition of γn) and L = Ln) in 11), we have for n arge enough KLn) ogn) ) Ln) n 2 og 1/2 δ n) og ogn), 4 Ln) γn) n 2 og 1/2 δ n)+og δ 1/2 n), nγ Ln) n) n ogδ 1/2 n) Since δ > 0, the n ogδ 1/2 n) term dominates in 28), and we obtain τ 0 n) bn)n α/2 1, where as n Therefore with bn) n Oog δ 1/2 n)), ρ n) ρ HR n) = 1/τ 0 n) bn)n 1 α/2, bn) n Oogδ 1/2 n)), concuding the proof for the fast fading case B Sow Fading In this section, we prove Theorem 1 under sow fading, ie, {θ u,v [t]} t is constant as a function of t We sketch the necessary modifications for the scheme described in Section IV to achieve a per-node rate of at east bn)n 1 α/2 in the sow fading case Consider eve, 0 < Ln) in the hierarchy Instead of reaying the message of a source-destination pair over one reay squareet as in the scheme described in Section IV, we reay the message over many dense squareets that are at east at distance 2a +1 from both the source and the destination nodes We time share between the different reays The idea here is that the wireess channe between any node and its reay squareet might be in a bad state due to the sow fading, making communication over this reay squareet impossibe Averaged over many reay squareets, however, we get essentiay the same performance as in the fast fading case We first state a somewhat weaker) version of Lemma 7, appropriate for this setup Consider again the coection of schedues Sn ) and Sn ) satisfying the conditions that no reay squareet is seected by more than n +1 source-destination pairs and that a sources and destinations are at east at distance 2a+1 from their reay squareet see Section VI-A for the forma definition) The next emma shows that for each source-destination pair, we can find K 2 2 1 γn) distinct reay squareets satisfying the above conditions the requirement that these reay squareets are distinct is expressed by the orthogonaity condition of the schedues in Lemma 11 beow)

28 Lemma 11 For every n arge enough independent of ) and every permutation traffic matrix λn ) {0, 1} n n there are schedues {S i) n )} K 22 γ 2 n) i=1 Sn ), { S i) n )} K 22 γ 2 n) i=1 Sn ) satisfying λn ) = K 2 2 1 γ 2 n) S i) n K 2 2 1 ) S i) n ), γn) where {S i) n )} i, { S i) n )} i are coections of orthogona matrices in the sense that for i i, s i) ) u,k si u,k = 0, u,k k,u i=1 s i) k,u si ) k,u = 0 29) Proof The proof is simiar to that of Lemma 7 In order to construct {S i) n )} and { S i) n )}, consider the sequentia pass over a n source-destination pairs assume n is arge enough for Lemma 7 to hod) As before, for each source-destination pair, there are K 2 2 1 γn) dense reay squareets that are at distance at east 2a +1 Each pair chooses a of these K 2 2 1 γn) squareets, instead of just one as before Stop one round of this procedure as soon as any of the reay squareets is chosen by n +1 pairs Since by the end of one round at east one reay squareet is matched by n +1 source-destination pairs, there are at most n /n +1 = 2γn) such rounds Consider now the resut of one such round We construct K 2 2 1 γn) matrices S i) n ) and S i) n ), with the i-th pair of matrices describing communication over the i-th reay squareets chosen by sourcedestination pairs matched in this round Thus, this process produces a tota of 2γn)K 2 2 1 γn) = K 2 2 γ 2 n) such matrices The orthogonaity property foows since each source-destination pair reays over the same reay squareet ony once Given a decomposition of the scaed traffic matrix K 2 2 1 γn)λn) into K 2 2 γ 2 n) matrices, each source-destination pair tries to reay over K 2 2 1 γn) dense squareets We time share between these reay squareets Since each source-destination pair reays ony a K 2 2 1 γn)) 1 fraction of traffic over any of its reay squareets, the oss due to this time sharing is now K 2 2 γ 2 n) K 2 2 1 γn) = 2γn) as opposed to K 3 2 in Lemma 7 In other words, the oss is at most a factor 2γn) more than in Lemma 7 Using the definition of γn) in 11), we have γn) n ogδ 1/2 n) b 1 n) In other words, this additiona oss is sma Consider now a specific reay squareet If a source-destination pair can communicate over this reay squareet at a rate at east 1/64-th of the rate achievabe in the fast fading case given by Lemmas 9 and 10), it sends information over this reay Otherwise it does not send anything during the period of time it is assigned this reay We now show that, with probabiity 1 o1) as n, for every sourcedestination pair on every eve of the hierarchy at east one quarter of its reay squareets can support this rate As we ony communicate over a quarter of the reay squareets, this impies that we can achieve at east 1/256-th of the per-node rate for the fast fading case see Section VII-A), ie, that bn)n 1 α/2 is achievabe with probabiity 1 o1) as n Assume we have for each source-destination pair u, w) picked K 2 2 1 γn) dense squareets over which it can reay; ca those reay squareets {A u,w,k } K 22 1 γn) k=1 Consider the event B u,w,k that source node u can communicate at the desired rate to destination node w over reay squareets A u,w,k assuming, as before, that we can sove the communication probem within this squareet)

29 Let {B i) u,w,k }4 i=1 be the events that the interference due to matched fitering in the MAC phase, the interference from spatia re-use in the MAC phase, the interference due to beamforming in the BC phase, and the interference from spatia re-use in the BC phase, are ess than 8 times the one for fast fading, respectivey From the proof of Lemmas 9, 10, and of Theorem 1 for the fast fading case in SectionVII-A, we see that 4 i=1 B i) u,w,k B u,w,k Due to spatia re-use, mutipe reay squareets wi be active in parae Let H denote the set of channe gains between active reay squareets Using essentiay the same arguments as for the fast fading case see Lemmas 9, 10, and Section VII-A) and from Markov s inequaity, we have PB i) u,w,k H) 7/8 for a i {1,, 4} and hence PB u,w,k H) 1/2 We now argue that the events { 4 i=1 Bi) u,w,k } K2 2 1 γn) are independent conditioned on H, by showing that these events depend on disjoint sets of channe gains and codebooks Assuming the codebooks are generated new for each communication round, then they are a independent Thus we ony have to consider the dependence on the channe gains Let U k and W k be the source and destination nodes communicating over reay squareet A u,w,k in round k, and et V k be the nodes in A u,w,k Let Ũk, W k be the source and destination nodes that are communicating at the same time as u, w) due to spatia re-use Let Ṽk be the reay nodes of Ũk and W k Now, B 1) u,w,k k=1 30) and B2) u,w,k depend for fixed H) on the channe gains between U k and V k B 3) u,w,k depends on the channe gains between V k and W k B 4) u,w,k depends again for fixed H) on the channe gains between Ṽk and W k Since these sets are disjoint for different k by the orthogonaity of the schedues see 29)), conditiona independence of the events in 30) foows To summarize, conditioned on the channe gains H between active reay squareets, the random variabes { 1 Bu,w,k } k are independent and have expected vaue E 1 Bu,w,k H) 1/2 The sum K 2 2 1 γn) k=1 1 Bu,w,k is the number of reay squareets over which the source-destination pair u, w) successfuy reays traffic We now show that with high probabiity at east one quarter of these reay squareets aow successfu transmission Appying the Chernoff bound yieds that ) ) P k 1 < K Bu,w,k 22 3 γn) H P k 1 < K Bu,w,k 22 2 γn)pb u,w,k H) H exp 2K2 γn)pb u,w,k H) ) exp K2 γn) ) for some constant K > 0 Since the right-hand side is the same for a H, this impies ) P k 1 < K Bu,w,k 22 3 γn) exp K2 γn) ) In each of the Ln) eves of the hierarchy there are at most n 2 source-destination pairs, and hence by the union bound with probabiity at east 1 Ln)n 2 exp K2 Ln) γn) ),

30 for every source-destination pair on every eve of the hierarchy at east one quarter of its reay squareets can support the desired rate By the choices of γn) and Ln) in 11), this probabiity is at east 1 Ln)n 2 exp K2 Ln) γn) ) ) 1 n 3 exp K2 Ln) 2 ogn)/2ln) 1 exp K2 og ogn) K2 1 2 og1/2+δ n) og n)) 1/2 δ ) 1 exp 2 Ωog1/2+δ n)) 1 o1) as n, and for some constant K This proves that the same order rate as in the fast fading case can be achieved with high probabiity for a eves 0 < Ln) It remains to argue that the same hods for eve = Ln) Note that since we assume phase fading ony, the received signa power is ony a function of distance and not of the fading reaization Since at eve Ln) we use simpe TDMA, this impies that we can aways achieve the same rate at eve Ln) as in the fast fading case Hence with probabiity 1 o1) as n, we achieve the same order rate at each eve 0 Ln) as for fast fading, proving Theorem 1 for the sow fading case VIII PROOF OF THEOREM 2 Here, we provide a generaization and sharpening of the converse in [8] Most of the arguments foow [8, Theorem 52] We start by proving a emma upper bounding the MIMO capacity Consider two subsets S 1, S 2 V n) such that S 1 S 2 = Assume we aow the nodes within S 1 and S 2 to cooperate without any restriction The maximum achievabe sum rate between the nodes in S 1 and S 2 is given by the MIMO capacity CS 1, S 2 ) between them The next emma upper bounds CS 1, S 2 ) in terms of the node distances between the two sets and the normaized channe gains h u,v h u,v ṽ S2 r α u,ṽ Lemma 12 Under either fast or sow fading, for every α > 2, S 1, S 2 V n) with S 1 S 2 =, we have { }) CS 1, S 2 ) 4 max 1, max h u,v 2 r v S 2 u,v α u S 1 v S 2 Proof Let H {h u,v } u S1,v S 2, H { h u,v } u S1,v S 2, be the matrix of normaized) channe gains between the nodes in S 1 and S 2 Consider first fast fading Under this assumption, we have CS 1, S 2 ) max E og det I + H QH)H )) QH) 0: Eq u,u) 1 u S 1 Define P S1,S 2 u S 1 as the tota received power in S 2 from S 1, and set ru,v α v S 2 P u,s2 P {u},s2 u S 1

31 with sight abuse of notation Then CS 1, S 2 ) = max E QH) 0: Eq u,u) P u,s2 u S 1 Define the event max E QH) 0: EtrQH)) P S1,S 2 og det I + H QH) H )) og det I + H QH) H )) 31) B { H 2 > b } for some b and where H denotes the argest singuar vaue of H In words, B is the event that the channe gains between S 1 and S 2 are good We argue that, for appropriatey chosen b, the event B has probabiity zero ie, the channe can not be too good ) By Markov s inequaity for any m We continue by upper bounding E H 2m ) We have for any k, and hence PB) b m E H 2m ), 32) H 2k tr H H ) k) Now, for any k m, we have by Jensen s inequaity E H 2m ) E tr H H ) k)) m/k ) 33) E tr H H ) k)) m/k ) Etr H H ) k)) m/k 34) Combining 32), 33), and 34) yieds PB) b m Etr H H ) k)) m/k 35) for any k m Now, the arguments in [8, Lemma 53] show that E tr H H ) k)) t k n max { }) k 1, max h u,v 2, v S 2 u S 1 where t k is the k-th Cataan number Combining with 35), this yieds { }) ) m PB) b 1 t 1/k k n 1/k max 1, max h u,v 2 v S 2 u S 1 Taking the imit as k and using that t 1/k k 4 yieds { }) ) m PB) b 1 4 max 1, max h u,v 2 v S 2 u S 1 Assume then taking the imit as m shows that { }) b > 4 max 1, max h u,v 2, 36) v S 2 u S 1 PB) = 0

32 Using this, we can upper bound 31) as )) CS 1, S 2 ) max E QH) 0: tr H QH) H EtrQH)) P S1,S 2 )) = max E 1 B ctr H QH) H QH) 0: EtrQH)) P S1,S 2 max E QH) 0: EtrQH)) P S1,S 2 bp S1,S 2 ) 1 B c H 2 trqh) Since this is true for a b satisfying 36), we obtain the emma for the fast fading case Under sow fading CS 1, S 2 ) max og det I + H QH ), Q 0: q u,u P u S 1 and the emma can be obtained by the same steps We now proceed to the proof of Theorem 2 Consider a vertica cut dividing the network into two parts By the minimum-separation requirement, an area of size on) can contain at most on) nodes, and hence we can find a cut such that each part is of size Θn) and contains Θn) nodes Ca the eft part of the cut S Since there are Θn) nodes in S and in S c, there are Θn) sources in S with their destination in S c with probabiity 1 o1) For technica reasons we add a node inside each square in V n) of the form [id, i + 1)d] [jd, j + 1)d] for some i, j N, where d 2 ogn) These additiona nodes have no traffic demands on their own, and simpy hep with the transmission This can ceary ony increase achievabe rates Moreover, this increases the number of nodes in V by ess than a factor 2 We now show that CS, S c ) = O og 6 n)n 2 α/2), 37) and hence by the cut-set bound, and since there are Θn) sources in S with their destination in S c, we have ρ n) = O og 6 n)n 1 α/2) We prove 37) using Lemma 12 To this end, we need to upper bound max v S c u S h u,v 2 The proof of [8, Lemma 53] shows that if 1) there are ess than ogn) nodes inside [i, i + 1] [j, j + 1] for any i, j {0,, n 1}, 2) there is at east one node inside [id, i + 1)d] [jd, j + 1)d] for any i, j, where d 2 og n, then max h u,v 2 K og 3 n), 38) and for α 2, 3] v S c u S u S v S c r α u,v K og 3 n)n 2 α/2, 39) for constants K, K For arbitrary node pacement with minimum separation, the first requirement is satisfied for n arge enough, since ony a constant number of nodes can be contained in each area of constant size By our addition of nodes into V n) described above, the second condition is aso satisfied Using Lemma 12 with 38) and 39) yieds 37), concuding the proof of Theorem 2

33 IX PROOF OF THEOREM 3 Consider a node pacement with n/2 nodes ocated uniformy on [0, n/4] [0, n] and n/2 nodes ocated on [ n/2, n] [0, n] with minimum separation r min = 1/2 A random traffic matrix λn) is such that at east n/4 communication pairs have their sources in the eft custer and destinations in the right custer with probabiity 1 o1) Assume we are deaing with such a λn) in the foowing In this setup, with muti-hop at east one hop has to cross the gap between the eft and the right custer Thus, even without any interference from other nodes, we can obtain at most ρ MH n) 4 α n α/2 Moreover, considering a cut between the two custers say, S and S c ), and appying Lemma 12 yieds that { ρ n) 16n 1 max 1, max u,v v S u S h 2}) r c u,v α 40) u S v S c Now note that for any u S, v S c, we have 1 n ru,v 2 n 4 Hence u,v u S h 2 = ru,v α u S ṽ S r α 2 3α, c u,ṽ and Combining this with 40) yieds for a α > 2 u S v S c r α u,v 4α 1 n 2 α/2 ρ n) 2 2+5α n 1 α/2 X PROOF OF THEOREM 4 We construct a cooperative muti-hop communication scheme and ower bound the per-node rate ρ CMH n) it achieves We use the hierarchica reaying scheme as buiding bock Assume the node pacement V n) is µ-reguar at resoution dn) for a n 1 We show that this impies that we can achieve a per-node rate of at east d 3 α n)n 1/2 βn) as n Taking the smaest such dn) then yieds the resut We consider three cases for the vaue of dn) namey, dn) = Θ n), dn) n o1), and dn) n o1) ) First, if dn) = Θ n) as n then the resut foows directy from Theorem 1 Considering a subsequence if necessary, we can therefore assume without oss of generaity that dn) = o n) in the foowing Second, consider dn) satisfying dn) n 1 2+α ogδ 1/2 n) 41) Divide An) into squares of sideength dn) Since dn) = o n), the number of such squares grows unbounded as n We now show that we can use muti-hop communication with a hop ength of dn) where each hops is impemented by squares cooperativey sending information to a neighboring square In other words, we perform cooperative communication at oca scae dn) and muti-hop communication at goba scae n Since V n) is µ-reguar at resoution dn), each such square contains at east µd 2 n) nodes Pick the top eft most square and construct the square of sideength 2dn) consisting of it together with its 3 neighbors Continue in the same fashion, partitioning a of An) into squares of sideength 2dn) Note

34 that each such bigger square contains at east 4µd 2 n) nodes by the definition of dn) We assume this worst case in the foowing Partition An) into 4 subsets of those bigger squares such that within each such subset each square is at distance at east 2dn) from any other square see Figure 6) We time share between those 4 subsets Consider in the foowing one such subset For every bigger square, we construct two permutation traffic matrices λ 1 4µd 2 n)) and λ 2 4µd 2 n)) In λ 1 the nodes in the top two squares have as destinations the nodes in the bottom two squares and the nodes in the bottom two squares have as destinations the nodes in the top two squares see Figure 6) Simiary, λ 2 contains communication pairs between eft and right squares We time share between λ 1 and λ 2 Fig 6 Sketch of the construction of the cooperative muti-hop scheme in the proof of Theorem 4 The dashed squares have sideength dn) The gray area is one of the 4 subsets of bigger squares that communicate simutaneousy The arrows indicate the traffic matrix λ 1 Communication according to λ i within bigger squares in the same subset occurs simutaneousy We are going to use hierarchica reaying within each bigger square This is possibe since each such square contains at east 4µd 2 n) nodes We have to show that the additiona interference from bigger squares in the same subset is such that Theorem 1 sti appies In particuar, we need to show that the interference has bounded power, say K Using the same arguments as in the proof of Theorem 1 in Section VII yieds that this is indeed the case the interference from other bigger squares here behaves the same way as the interference due to spatia re-use from other active reay squareets there) With this, we are now deaing with a hierarchica reaying scheme with area 4d 2 n), 4µd 2 n) nodes, and additive noise with power 1 + K Both the ower number of nodes and the higher noise power wi decrease the achievabe per-node rate by ony some constant factor, and hence Theorem 1 shows that under fast fading we can achieve a per-node rate of at east as n, where b 1 d 2 n) ) d 2 n)) 1 α/2 b 1 n)d 2 α n), ) b 1 n) n O og δ 1/2 n) Moreover, the same rate is achievabe under sow fading with probabiity 1 b 2 d 2 n)), where )) b 2 n) exp 2 Ω og 1/2+δ n) The setup is the same for a bigger squares within each of the 4 subsets We now shift the way we defined the bigger squares by dn) to the right and to the bottom With this, each new bigger square intersects with 4 bigger squares as defined before We use the same communication scheme within these new bigger squares and time share between the two ways of defining bigger squares

35 Construct now a graph where each vertex corresponds to a square of sideength dn) and where two vertices are connected by an edge if they are adjacent in either the same od or new bigger square This graph is depicted in Figure 4 in Section V With the above construction, we can communicate aong each edge of this graph simutaneousy at a per-node rate of b 1 n) 16 d2 α n) in the fast fading case In the sow fading case, this statement hods with probabiity at east 1 n d 2 n) b 2d 2 n)) = 1 for constants K, K By assumption 41), and hence 1 exp og 1/2+δ dn) ) n d 2 n) exp 2 Ω og 1/2+δ d 2 n)) )) K 2 ogogn) 2 e K og 1/2+δ dn)) 1 ) 1/2+δ, 2 + α og1/2+δ n) 1 n d 2 n) b 2d 2 n)) 1 o1) as n, showing that with high probabiity we achieve the same order rate under sow fading as under fast fading The communication graph constructed forms a grid with n/d 2 n) nodes Using that each bigger square can contain at most K 1 d 2 n) nodes by the minimum-separation requirement, standard arguments for routing over grid graphs see [16]) show that in the fast fading case we can achieve a per-node rate of where ρ CMH n) bn)d 2 α n) dn) n bn)d 3 α n)n 1/2, ) bn) = n O og δ 1/2 n) Moreover, the same statement hods in the sow fading case with probabiity 1 o1) Finay, consider dn) such that dn) n 1 2+α ogδ 1/2 n) 42) Construct the same communication graph as before, but this time we use simpe muti-hop communication between adjacent squares of sideength dn) By time sharing between the at most K 1 d 2 n) nodes in each square, and since we communicate over a distance of at most 3dn), we achieve under either fast of sow fading a per-node rate between the squares of at east K d 2 α n) K n ogδ 1/2 n) for some constant K, and where we have used 42) Using the anaysis of grid graphs as before, we can achieve a per-node rate of at east for either the fast or sow fading case ρ CMH n) K n ogδ 1/2 n) dn) n bn)d 3 α n)n 1/2, )

36 XI PROOF OF THEOREM 5 Consider V n) with n/2 nodes ocated uniformy on [0, n d n))/2] [0, n] and n/2 nodes ocated uniformy on [ n/2, n] [0, n] such that r min = 1/2 This node pacement is 1/2-reguar at resoution d n) A random traffic matrix λn) is such that Θn) communication pairs have their sources in the eft custer and destinations in the right custer with probabiity 1 o1) Assume we are deaing with such a λn) in the foowing Considering a cut between the two custers and appying Lemma 12 sighty adapting the arguments in Section VIII), yieds that ρ n) = O og 6 n)d 3 α n)n 1/2) for α > 3 XII DISCUSSION We briefy discuss severa aspects of the proposed hierarchica reaying scheme Section XII-A comments on the fu CSI assumption and Section XII-B on the use of bursty communication Sections XII-C and XII-D outine how the resuts obtained here can be extended to the case of dense networks and networks without minimum separation between nodes Section XII-E compares our hierarchica reaying scheme to the hierarchica cooperation scheme presented in [8] A Fu CSI Assumption Throughout our anaysis, we have made a fu CSI assumption In other words, we assumed that the phase shifts {θ u,v [t]} u,v are avaiabe at time t at a nodes in the network As this assumption is quite strong, it is worth commenting on First, we make the fu CSI assumption in a the converse resuts in this paper This impies that a the converses aso hod under weaker assumptions on the CSI, and hence are vaid as we under a wide variety of more reaistic assumptions on the avaiabiity of side information Second, a achievabiity resuts can be shown to hod under weaker assumptions on the avaiabiity of CSI In fact, in a cases, a 2-bit quantization of the channe state {θ u,v [t]} u,v avaiabe at a nodes at time t is sufficient to obtain the same scaing behavior This foows by an argument simiar to the one used in the anaysis of the BC phase in Section VI-C, where it is shown that beamforming using a quantized channe state resuts ony in a constant factor rate oss B Burstiness of Hierarchica Reaying Scheme The hierarchica reaying scheme presented here is bursty in the sense that nodes communicate at high power during a sma fraction of time This eads to high peak-to-average power ratio, which is undesirabe in practice We chose burstiness in the time domain to simpify the exposition The same bursty behavior coud be achieved in a more practica manner by using CDMA with severa orthogona signatures or by using OFDM with many sub-carriers Each approach eads to many parae channes out of which ony few are used with higher power This avoids the issue of high peak-to-average power ratio in the time domain C Dense Networks Throughout this paper, we have ony considered extended networks, ie, n nodes paced on a square region of area n with a minimum separation of r u,v r min The resuts can, however, be recast for dense networks, where n nodes are arbitrariy paced on a square region of unit area with a minimum separation of r u,v r min / n It suffices to notice that by rescaing power by a factor n α/2 a dense network can essentiay be transformed into an extended network with path-oss exponent α see aso [8]) Hence the same resut for dense networks can be obtained from the resut for extended networks by considering the imit α 2 Appying this to Theorem 1, yieds a inear per-node rate scaing of the hierarchica reaying scheme

37 D Minimum-Separation Requirement The minimum-separation requirement r min 0, 1) on the node pacement is sufficient but not necessary for Theorem 1 to hod A weaker sufficient condition is that a constant fraction of squareets are dense, as shown in Lemma 6 to be a consequence of the minimum-separation requirement It is straightforward to show that this weaker condition is satisfied with high probabiity for nodes paced uniformy at random on [0, n] 2 This yieds a different proof of Theorem 51 in [8] E Comparison with [8] Both, the hierarchica reaying scheme presented here and the hierarchica scheme presented in [8], share that they use virtua mutipe-antenna communication and a hierarchica architecture to achieve essentiay goba cooperation in the network The schemes differ, however, in severa key aspects, which we point out here First, we note that we obtain a sighty better scaing aw Namey b 1 n)n 1 α/2 ρ n) b 2 n)n 1 α/2 with ) b 1 n) n O og δ 1/2 n), b 2 n) = O og 6 n) ), for any δ 0, 1/2) obtained here, compared to with b1 n)n 1 α/2 ρ n) b 2 n)n 1 α/2 b1 n) = Ω n ε), b2 n) = O n ε), for any ε > 0 in [8] For the ower bound ie, achievabiity), this is because the hierarchy here is not of fixed depth L as in [8], but rather of depth Ln) = og 1/2 δ n) for some constant δ 0, 1/2)), ie, changing with n For the upper bound ie, converse), this is due to a sharpening of the arguments in [8] Second, note that the muti-user decoding at the reay squareets during the MAC phase and the mutiuser encoding during the BC phase are very simpe in our setup In fact, using matched fiter receivers and transmit beamforming, we convert the muti-user encoding and decoding probems into severa singeuser decoding and encoding probems This differs from the approach in [8], in which joint decoding of a number of users on the order of the network size is performed Our resuts thus impy that these simper transmitter and receiver structures provide the same scaing as the more compicated joint decoding in [8] We note that the scheme proposed in [8] can be modified to aso use matched fiter receivers as suggested here Third, and probaby most important, the schemes differ in how they achieve the throughput gain from using mutipe antennas In [8], the nodes are ocated amost reguary with high probabiity This aowed the use of a scheme in which a source squareet directy communicates with a destination squareet In other words, the mutipe-antenna gain comes from setting up a virtua MIMO channe between the source and the destination In our setup, the arbitrary ocation of nodes prevents such an approach Instead, we use that at east some fixed fraction of squareets is amost reguar we caed them dense squareets) Sourcedestination pairs reay their traffic over such a dense squareet In other words, the mutipe-antenna gain comes from setting up a virtua mutipe-antenna MAC and BC Thus, the hierarchica reaying scheme presented here shows that consideraby ess structure on the node ocations than assumed in [8] suffices to achieve a mutipe-antenna gain essentiay on the order of the network size Note aso that the additiona degree of freedom offered by the choice of reay squareet for a given source-destination pair makes it possibe to extend the resut to hod aso for sow fading channes

38 XIII CONCLUSIONS We considered the probem of the scaing of achievabe rates in arbitrary extended wireess networks We generaized the hierarchica cooperative communication scheme presented in [8] for a fast fading channe mode and with random node pacements We proposed a different hierarchica cooperative communication scheme, which aso works for arbitrary node pacement with a minimum-separation requirement) and for either fast or sow fading For sma path-oss exponent α 2, 3], we showed that our scheme is order optima and achieves the same rate irrespective of the node pacement In particuar, this rate is equa to the one achievabe under random node pacement In other words, the reguarity of the node pacement has no impact on achievabe rates for sma path-oss exponent The situation is, however, quite different for arge path-oss exponent α > 3 We argued that in this regime the reguarity of the node pacement directy impacts the scaing of achievabe rates We then presented a cooperative communication scheme that smoothy interpoates between muti-hop and hierarchica cooperative communication depending on the reguarity of the node pacement We showed that this scheme is order optima for a α > 3 under adversaria node pacement with reguarity constraint This contrasts with the situation for more reguar networks ike the ones obtained with high probabiity through random node pacement), in which muti-hop communication is order optima for a α > 3 Thus, for ess reguar networks, the use of more compicated cooperative communication schemes can be necessary for optima operation of the network XIV ACKNOWLEDGMENTS The authors woud ike to thank the anonymous reviewers and the Associate Editor Gerhard Kramer for their comments We woud aso ike to acknowedge hepfu discussions with Oivier Lévêque, Ayfer Özgür, and Greg Worne REFERENCES [1] P Gupta and P R Kumar The capacity of wireess networks IEEE Transactions on Information Theory, 462):388 404, March 2000 [2] L Xie and P R Kumar A network information theory for wireess communication: Scaing aws and optima operation IEEE Transactions on Information Theory, 505):748 767, May 2004 [3] A Jovičić, P Viswanath, and S R Kukarni Upper bounds on transport capacity of wireess networks IEEE Transactions on Information Theory, 5011):2555 2565, November 2004 [4] O Lévêque and İ E Teatar Information-theoretic upper bounds on the capacity of arge extended ad hoc wireess networks IEEE Transactions on Information Theory, 513):858 865, March 2005 [5] F Xue, L Xie, and P R Kumar The transport capacity of wireess networks over fading channes IEEE Transactions on Information Theory, 513):834 847, March 2005 [6] L Xie and P R Kumar On the path-oss attenuation regime for positive cost and inear scaing of transport capacity in wireess networks IEEE Transactions on Information Theory, 526):2313 2328, June 2006 [7] M Franceschetti, O Dousse, D Tse, and P Thiran Cosing the gap in the capacity of wireess networks via percoation theory IEEE Transactions on Information Theory, 533):1009 1018, March 2007 [8] A Özgür, O Lévêque, and D Tse Hierarchica cooperation achieves optima capacity scaing in ad hoc networks IEEE Transactions on Information Theory, 5310):3549 3572, October 2007 [9] P Gupta and P R Kumar Towards an information theory of arge networks: An achievabe rate region IEEE Transactions on Information Theory, 498):1877 1894, August 2003 [10] L Xie and P R Kumar An achievabe rate for the mutipe-eve reay channe IEEE Transactions on Information Theory, 514):1348 1358, Apri 2005 [11] G Kramer, M Gastpar, and P Gupta Cooperative strategies and capacity theorems for reay networks IEEE Transactions on Information Theory, 519):3037 3063, September 2005 [12] S Aeron and V Saigrama Wireess ad hoc networks: Strategies and scaing aws for the fixed SNR regime IEEE Transactions on Information Theory, 536):2044 2059, June 2007 [13] M Franceschetti, M D Migiore, and P Minero The capacity of wireess networks: Information-theoretic and physica imits In Aerton Conference on Communication, Contro, and Computing, September 2007 [14] M Franceschetti, M D Migiore, and P Minero The degrees of freedom of wireess networks: Information theoretic and physica imits In Aerton Conference on Communication, Contro, and Computing, September 2008 [15] S Ihara On the capacity of channes with additive non-gaussian noise Information and Contro, 371):34 39, Apri 1978 [16] S R Kukarni and P Viswanath A deterministic approach to throughput scaing in wireess networks IEEE Transactions on Information Theory, 506):1041 1049, June 2004