Self-Organization of Wireless Ad Hoc Networks as Small Worlds Using Long Range Directional Beams

1 Self-Or ganizat ion of W ireless Ad Hoc Netw orks as Small W orlds Using Lo ng Range Direct ional Beams Abhik Banerjee ∗ , Rachit Agarwal † , V incent Gauthier † , Chai Kiat Y eo ∗ , Hossa m Afifi † and Bu Sung Lee ∗ ∗ CeMNet, Scho ol of Computer Engineering, Nanyang T echn ological Uni versity , Singapore Email: { abhi0018, asckyeo, ebslee } @ntu.edu. sg † Lab . CNRS SAMO V AR UMR 5157 , T elecom Sud Paris, Evry , France Email: { rachit.agarwal, vincent.gauthier , hossam. afifi } @it-sudparis.eu Abstract — W e study how long range directional beams can be used for self-organization of a wireless network to exhibit small w orld properties. Using simulation results for randomized beamf orming as a guidelin e, we identify crucial design issues fo r algorithm design. Sub sequently , we propose an algorithm f or deterministic creation of sm all worlds. W e define a new centrality measure that estimates the structural importance of nodes b ased on traffic fl ow in the n etwork, which is used to identify the optimum nodes f or beamf orming. This results in significant reduction in path length w hile maintain ing connectivity . I . I N T RO D U C T I O N Initially studied in th e context o f social networks [1], th e small world p henom enon refers to the ability of such networks to maintain high clustering and low av erage path length between nodes. Subsequen t research has explored ways to redesign networks so as to exhibit small world characteristics. W atts & Strogatz [2] showed that regular networks can be reorganized to exhibit small w orld beha viour by r andomly rewiring a small set o f links. Small worlds are an attractive mod el fo r reorganizing a wireless ad hoc network so as to ensure perfor mance gu ar- antees. As wireless n etworks typica lly suffer f rom issues of connectivity , maintaining high clustering guaran tees reliability . Further, r eorganization o f a n etwork with the path leng th bound ed as the lo garithm of the network size ensure s p er- forman ce scalability . Howe ver , due to the spatial nature of wireless networks, lin ks between n odes canno t be rando mly rewired as they a re constrain ed by the transmission range. In [3], Helmy used simulation r esults to study the behaviour of wireless networks as a result o f rando m add ition of distance limited short cuts. Instead of random rewiring or addition of links, determ inistic placemen t of sho rt cuts was studied by [4] and [5] in wh ich th ey consider hybr id sensor networks that include a small set of wire d link s. Sim ilarly , Gu idoni et al. in [6] propo sed using high capacity nod es in a heterog enous sensor network for short cut cre ation. In this pape r , we stud y h ow small world beh avior can be realized in a wireless n etwork b y the use of directional beam forming at nod es. Our primary motivation for using directio nal antennas stems f rom the fact that they can be used to tran smit over longer transmission ran ges than o mnidirec tional antennas while using the same transmission power . This implies th at short cuts can be created between nodes withou t the need for additional in frastructur e. This distinguishes u s from existing literature th at fo cus on addition of new links between nod es. Instead, e xisting o mnidirectio nal links are rewired as long range dire ctional on es. Anoth er unique p roperty o f using directional antennas is that the shap e o f th e beam implies that lo ng range link s can be created with mo re than just one node. T hus, th e fraction of long range links in the n etwork is a ctually hig her than the numb er of nodes be amform ing. The foc us of th is paper is to study the issues surrou nding the u se of directional beamfor ming for sm all world cre ation in a wireless ad ho c network and pro pose ways to achieve an optimal design. T o our knowledge, the only other work wher e directional antennas were used for shor t cut cr eation was in [7]. Howe ver, like oth er existing papers, the propo sed model considers addition of link s u sing multiple radios. The first p art of the paper provides a simulation based analysis of using r andom ized directional beamform ing for small world cr eation in wireless ad ho c networks. Our results show that, while significan t red uction ca n b e ach iev ed in the av erage path length, this is accompanied by loss in co nnec- ti vity . Subseq uently , we f ocus on distributed algorithm desig n for determin istic sm all world creation . Our design ce nters on a new definition of centrality that is comp uted ind ividually at nodes based o n in formation o btained fr om overhearing traf fic flows in the network . W e show th at significan t r eduction in path le ngth can be achieved while maintaining con nectivity . Our fo cus here is that of a con nected network with the primary motivation of limitin g the gr owth in path length with incr easing size o f the network without suffering losses in conn ectivity . A compan ion p aper in the same worksh op [8] considers a disconnected network an d how bio- inspired mechanisms can be used to e nsure connectivity by using directional beams. I I . S M A L L W O R L D W I R E L E S S N E T W O R K S U S I N G D I R E C T I O N A L A N T E N N A S In this section, we d o a simulation based an alysis of using directional anten nas for small worlds in wireless networks. W e use the results to iden tify crucial design aspects. A. Network Mo del W e conside r a wireless ad hoc network of N n odes a ll of which c onsist of a single beam forming antenn a. In itially , all nodes transm it u sing omnidir ectional beam s with range r . 2 (a) (b) A B A B Fig. 1. Effec t of beam width on the connec ti vity . Using a wider beam with shorter beam length allo ws a bidirect ional path between nodes A and B . Subsequen tly , a fraction p of the n odes in the network ar e random ly ch osen w hich u se lo ng r ange directional beams. Usage of directiona l anten na by a node can be classified into different categories dependin g o n the modes o f tran mission and rece ption [9]. W e con sider that when a node crea tes a lon g range directional beam, it op erates in the mo de o f dir ectional transmission an d o mnidirectio nal r eception ( DTOR). For the purpo se of analysis, we use th e sector model fo r d irectional antennas [11] . W e compa re the simulation results using the sector mod el to a mo re realistic unifor m lin ear ar ray (ULA) antenna [10] mo del. A direction al beam is chara cterized by the b eam length, width and th e bea m d irection. For the UL A, a lon ger beam length can b e achieved by incre asing the num ber of antenn a elements used [10]. By keeping the tran smission power con- stant, increasing the number of elements results in a narrower and longer main lobe, while using a single antenna resu lts in an omnidirec tional beam . When u sing the secto r mod el as an ab straction, a constant tra nsmission power implies that the area covered by the beam stays constant. A b eam of width θ , therefore, results in a beam length r ( θ ) = r q 2 π θ . Th e directed nature of the beam leads c an lead to the prob lem of asymm etric paths between nod es, as discussed in [9], [11 ]. As illu strated in Fig. 1, bid irectional conn ectivity between two nodes essentially requir es the p resence of a circ ular p ath. T o accou nt for the a bove tr adeoffs, we choose a bea mwidth that optimizes connectivity and beamlength dependin g on the density of nodes. T o incorporate the tradeoff between increased length and conne cti vity , we divide the sector into separate regions of width r . Sub sequently , we weigh the beam len gth r ( θ ) with th e prob ability that at least one no de is located in the first and the last regions, r C = r ( θ ) p nf p nl (1) where p nf and p nl are the probab ilities th at at least on e node is lo cated in the first and last secto ral regions respectively . The term p nf is in dicativ e of the probability that the no de maintains con nectivity to the no des in its omn idirection al neighbo urho od while p nl indicates the prob ability that at least one node benefits from the incr eased b eam length. The optimum bea m width θ ∗ among a set of values for θ is chosen 0 0.2 0.4 0.6 0.8 1 -3 -2.5 -2 -1.5 -1 -0.5 0 Ratio log10(p) L(p)/L(0) - Sector Model C(p)/C(0) - Sector Model L(p)/L(0) - ULA C(p)/C(0) - ULA (a) Path Length and Clustering Coef ficient 0 0.2 0.4 0.6 0.8 1 -3 -2.5 -2 -1.5 -1 -0.5 0 Fraction of Asymmetric Paths log_{10}(p) Directional - Sector Directional - ULA (b) Unidirectiona l Connecti vity Fig. 2. Small W orld charact eristic s as a function of varyi ng probabil ity of rewi ring for N = 300 . L (0) and C (0) den ote the avera ge pat h length and clustering coef ficient for the initial network with all nodes using omnidirec tional antenna s. as the on e that maximizes the weighted beam leng th r C , i.e. θ ∗ = arg max θ " r r 2 π θ # p nf p nl (2) The values for p nf and p nl are obtain ed based o n the no de density in the n etwork. Given that the number o f nod es in the omn idirectional n eighbo urhoo d of a n ode is n , the correspo nding v alues are o btained as p nf = 1 − (1 − A f π r 2 ) n and p nl = 1 − (1 − A l π r 2 ) n where A f and A l are the area of th e first and last r egions respectively . Recall that the area under the b eam is equal to that of the omnid irectional are a when the same transmission power is used. While the sector mo del is ideal for analysis, we also r un simulations on a mo re realistic model of dir ectional antennas to comp are the perf ormance . The model we use is that of a unifor m linear array (ULA) [10], in which the antenna elements are arranged linear ly . The beam patter n of a ULA is characte rized by the n umber of elements u sed m and the boresight direction θ b . An important d ifference with th e sector model is that the bea m pattern o f ULA is chara cterized by one or more side lo bes in ad dition to the main lobe. The maximum gain o btained in the direction of θ b is equa l to m . In order to map th e sector model used earlier to ULA, we equate the nu mber o f an tenna elements m = r ( θ ∗ ) where r ( θ ∗ ) is the nor malized beam length corre sponding to θ ∗ obtained in equation (2). B. S imulation Results For our simulations, we conside r a network co nsisting of nodes usin g om nidirection al an tennas distributed rand omly in a rectang ular r egion. W e study the im pact of using random ly 3 1 2 3 4 5 6 7 8 9 10 4 6 8 10 12 14 16 18 Avg. Path Length D log D Omnidirectional Directional - Sector Directional - ULA Fig. 3. Growt h of APL with increa se in the size of the simulat ion region. oriented directional beams o n the average p ath length ( APL) and conn ectivity of this network. For th e first set o f simulations, we vary the fr action of nodes p that use directional beams while the rest of the nodes continue to use omn idirectiona l beam s. The omnidire ctional transmission range is no rmalized to 1 with nodes distributed random ly in a 10 x 10 region . Fig. 2 shows th e im pact o n path length reductio n and connectivity for N = 30 0 . The results for the sector m odel, shown u sing lines, are compared to those using the ULA mo del, shown with d ots. T he b eam length r ( θ ∗ ) ob tained using equation (2) results in a ra tio r ( θ ∗ ) D ≈ 0 . 2 . D deno tes the m aximum distance betwe en any two node s in the n etwork, i.e. the diameter of th e network since we normalize r to 1 . In Fig. 2(a), the r atio of the reduced path length L ( p ) and clustering co efficient C ( p ) to the initial values L (0 ) and C (0) are shown. W e note that th e values for L ( p ) L (0) and C ( p ) C (0) are quite close to each o ther f or low values o f p when u sing the sector mo del. This is contrary to the desired re sults for sm all world network s as the path length reduction is accomp anied b y lo ss in conn ectivity . Th e adverse impact of C ( p ) on conne cti vity is seen in Fig. 2(b ) as a high percenta ge o f nodes have asymme tric path s. Howe ver, when a mor e realistic ULA mo del is used, better results are o btained in terms of both the path length imp rovement and con nectivity . Th e pre sence of side lobes im plies that a beamfor ming no de retains conn ectivity to a g reater fraction of nodes in its omnidir ectional neig hbour hood, resulting in higher values of the cluster ing coefficient. This also acco unts for shor ter path length s since, in contrast to the sector model, these n odes in the omnid irectional ne ighbou rhood can now be reached in a sing le hop. T o un derscore the suitability o f d irectional b eamform ing for small world creation , we v iew it in the context of existing literature. Helm y [ 3] obtained results showing the impac t of distance limited short cuts o n the p ath length red uction. In our results when u sing th e sector model, fo r the cor respond ing value of r ( θ ∗ ) D , we o bserve that p ath length red uction is compara ble for sm all values of p th ough the red uction is less for higher values. For the more realistic U LA mo del, th ough , the path le ngth imp rovements can actually exceed those o f [3 ]. W e also run add itional sim ulations to stud y the g rowth in path length co mpared to the logarithm of th e n etwork size, which is measured in ter ms of D as this is a spatial network. Fig. 3 shows tha t th e APL gr ows in O (lo g D ) when p = 1 . C. Discussion and I nsights An im portant featu re of our results in the previous section is the tradeo ff between p ath length improvement and con- nectivity in the network. Desp ite choo sing beam wid th so as to maintain connectivity , we observe tha t a high fraction of beamf orming nodes d rastically redu ces the pr obability of having a bidirec tional p ath between nodes. Based on the results from sector mode l alon e, it is difficult to iden tify an optimum value of p such that the n etwork exhibits small world behaviour . Ho wev er , look ing at the resu lts u sing the realistic ULA model, we observe that the constraints imposed by the sector model can be relaxed to an extent. W e o bserve in Fig. 2 that a 3 0% of redu ction in the average path length can be achieved with p = 0 . 1 while less than 2% of node pairs suffer from unidirection al connec ti vity . T his also provides us with additional insight for algorithm design. While the sector model is mo re tr actable f or algor ithm design, we co nclude tha t an algorithm that results in 20% of lin ks being un idirectiona l is permissible as that is co mpensated in a realistic setup. I I I . T R A FFI C A W A R E S M A L L W O R L D C R E AT I O N U S I N G D I R E C T I O N A L B E A M F O R M I N G W e outlin e an algor ithm design for realizing small world behaviour by deterministically choo sing the set of nod es to beamform and the beamfor ming parameters. Gi ven our primary objective of maximizin g th e reduction in path length across the network, we need to identify the set of nodes that are most ideally located to minimize path lengths across the network. This correlates directly to the tradition al n otion of between ness cen trality [12 ]. Creating short cu ts o n nod es with hig h values of betweenn ess is likely to reduce p ath lengths across th e network as they lie alo ng the m ajority of shortest paths b etween node s in the network. For distributed self-organization of the network as a small world network, a measure of betweenn ess n eeds to be identified that can no t only be comp uted in a distributed manner but on e that adapts to the routes taken by flows. Further, as nodes need to decid e on their beamfo rming behaviour individually , th ey also n eed to be able to estimate their rank o f their betweenness in the context o f the network. W e pro pose the Wireless Betweenness Centr ality (WFB) which is computed at no des based on neigh bourh ood traffic flow infor mation. A key aspect of WFB is that it exploits the W ireless Broad cast Advantage (WB A) as node s use implicitly av ailable infor mation fro m n eighbou rhood transmissions to compute their cen trality values. WFB values are compu ted re- cursively allowing network structu ral infor mation to pro pagate over mu ltiple hops. Subseque ntly , we pro pose an alg orithm that u ses WFB values at no des alo ng with in sights ob tained from simu lation results to realize small world beh aviour in th e network. A. W ir eless Flow Betweenness (WFB) The broadcast n ature o f the wireless medium results in implicit sh aring of informatio n among nodes in th e network. This feature is kn own as the w ireless broa dcast advantage (WB A) [1 3] a nd h as b een exploited in existing litera ture 4 for improvin g network perform ance. W e note that, as eac h node is aware of all transmissions in its neighbo urho od, it can estima te the set o f source -destination pairs co rrespon ding to the ob served flows. As a result, a simp le measure of betweenness of a no de may be obtain ed as, B e t ( v ) = g ( v ) P u ∈{N ( v ) ∪ v } g ( u ) (3) where g ( u ) deno tes the number of p ackets fo rwarded by a node u for distinct sourc e-destination p airs while N ( v ) denotes the set o f neighb ours of v . The d enomin ator of the above expression gives th e total nu mber of packets forwarded in the n eighbou rhoo d o f v . E ach node reco rds the num ber of packets forwarded by each of its ne ighbou rs. Howe ver , the value of B et ( v ) co mputed using (3) does n ot giv e any idea abo ut th e im portance of node v with regard to the stru cture of th e rest of th e network. In order to do so, we propose exploiting WB A to pro pagate infor mation over multiple ho ps. Our d esign pr oceeds by having nodes br oadcast their self comp uted v alues of centrality as part of packet transmissions. Whenev er a node tran smits a p acket, either as part of for warding it for anoth er flo w or as a source, it ap pends the value of centrality co mputed for itself to the packet. Any neighbo ur overhearin g this tr ansmission stores the c entrality value in add ition to u pdating th e numb er of packets fo rwarded by this nod e. W e elaborate on the recursive c omputatio n of W ireless Flow Betweenness (WFB) fo r a nod e v and its neig hbour s when the form er transmits a packet in a time slot t . Let the WFB of a no de v after ( t − 1) time slots be denoted as w t − 1 ( v ) . For eac h node u ∈ N ( v ) , v stores th e number o f packets forwarded alo ng with their WFB values. When transmitting a packet in the t th time slo t, v up dates its WFB using the following expression an d appends it to the transm itted p acket, w t ( v ) = g t ( v ) w t − 1 ( v ) P u ∈{N ( v ) ∪ v } g t − 1 ( u ) w t − 1 ( u ) (4) Upon r eceiving the value of WFB transmitted by v , each of its neigh bours u ∈ N ( v ) recomp utes its own value of WFB as, w t ( u ) = g t ( u ) w t − 1 ( u )  g t ( v ) w t ( v ) + P u ′ ∈{{N ( u ) \ v }∪ u } g t − 1 ( u ′ ) w t − 1 ( u ′ )  (5) Note that, f or all node s a part fro m v , th e WFB recor ded till the ( t − 1) th time slot is used. Th is is becau se, tran smission from any no de u ′ ∈ {N ( u ) \ v } in the time slot t would imply a collision at u . Since we assum e successful r eception of the transmission of v at u , it precludes the above cond ition. It is easy to see how this r ecursive comp utation r esults in p ropagatio n of informa tion r egarding centrality across the network. After v transmits in the t th time slot, its neighbou rs u ∈ N ( v ) are schedu led in subsequen t time slots in which they broadcast their updates values of WFB leading to further recompu tation. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 (a) Initial O mnidirec tional Networ k 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 (b) After nodes use self computed va lues of WFB to decide on direc tional antenna beha viour . The solid edges correspond to directional beams. Fig. 4. Using WFB to ac hie ve small propertie s in a wire less network when 50% of all nodes transmit a pack et to rando mly chosen destina tions. B. Directional Beamforming using WFB Our next challeng e is to have nodes decide on u sing long range directio nal beams b ased on th eir individually computed values of WFB. From the po int of vie w of centrality , we observe th at a node with a high value of cen trality is likely to have ne ighbou rs with high values as well. Th is is p articularly true in our scenar io since the WFB value of a node is weigh ted with the values of its ne ighbou rs. Ho we ver , this is also true f or a generic view of a n etwork d ue to the fact as nodes with high values of closeness centrality are likely to be clustered towards the center of the network. In addition to identifying the nodes with high values of WFB, we ensure that the connec ti vity of beamfor ming n odes is maintained by spacing ou t such nod es. This ensure s that a beamfo rming n ode h as a high percen tage of omnidirection al no des in its n eighbo urhoo d which preserve bidirection al con nectivity . Based on th e ab ove observations, we prop ose an design in which a node v decides on using a directional beam if the av erage WFB of its neighbo urho od is c lose to its own value of WFB, as determined by a similarity constant β . Th us, a node v decid es to u se a directio nal b eam after t time slots if | w t avg ( N ( v )) − w t ( v ) | < β w t ( v ) (6) where w t avg ( N ( v )) is the average WFB of all n odes u ∈ N ( v ) . Once a node d ecides to use a directional beam , it broadc asts its decisio n over both its o riginal omn idirectional neighbo urho od as well as the new neighbo urhoo d d etermined by the dire ctional beam . Subsequ ently , any n ode overhear ing 5 1 2 3 4 5 6 7 5 6 7 8 9 10 11 12 13 14 15 Avg. Path Length D log D Omnidirectional Directional (a) Reduction in A vg. Path Length 0 0.002 0.004 0.006 0.008 0.01 5 6 7 8 9 10 11 12 13 14 15 Fraction of Asymmetric Paths D (b) Unidirectiona l connecti vity Fig. 5. Path Length and Connecti vity for self-or ganiz ation using WFB. either of th ese two bro adcasts decides against using a long range b eam even if the above condition is satisfied for th e node itself. This en sures that nodes u sing long ra nge links a re separated sufficiently so a s to maintain conn ectivity . Having decided on using a directional beam, a node needs to identify the optimal beam width and direc tion. The choice of beam width θ is done in the sam e manner as earlier , based on the n ode den sity usin g equatio n (2). Instead of u sing a rando m direction, tho ugh, a nod e orients its beam in a direction in which it record s the maximu m hop count. This ensures that the longest paths in the n etwork benefit from the self-organ ization. Fig. 4 illustrates the creation of lon g r ange directional beams in a network based o n the WFB values computed at nodes when 50 % of all nodes tran smit a packet to random ly chosen destinations. C. Simulation Results W e ev aluate the pro posed algorithm using MA TLAB simu- lations. W e con sider a dense network in which co nnectivity is guaran teed for the initial network setup in wh ich all n odes use o mnidirectio nal beams. The curr ent set of r esults only consider the sector model of directio nal an tennas. Nodes are random ly distributed over a squa re region. W e ob tain r esults for increase size o f the network area, or the diameter D . Fig. 5 shows the resu lts when 50% of nodes gen erate source packets to random ly cho sen destination nodes. W e o btain a beam leng th ratio r ( θ ∗ ) D ≈ 0 . 3 while the fraction of nodes p lies b etween 0 . 04 and 0 . 07 . A path length redu ction of u p to 3 0% is achieved as the ne twork size in creases. Further, the effect on co nnectivity is negligib le a s less than 0 . 4% of all nod e pairs hav e asymm etric p aths. The path length improvements are equ iv alent to those obtained by Helmy [3] for co rrespond ing values of r ( θ ∗ ) D . Further, the pa th length improvements are muc h h igher than corr espondin g v alues of p for randomized beamfor ming shown earlier . W e observe that the r eduction in path length is lo wer than that shown in Fig. 3 ev en though the growth in th e a verage path length is in O (log D ) . The difference in performan ce is explained b y the fact that the resu lts shown in Fig . 3 cor respond to p = 1 which is mu ch h igher tha n what is o btained h ere. T he r esults shown are for β = 0 . 2 . Using high er values of β results in greater red uction in path length but is accomp anied by loss in connectivity as a g reater number of ne ighbor ing nodes use directional beam s. Howe ver , h igher values of β can be used for realistic beamfo rming mod els such as ULA as they result in greater conn ectivity . W e leave this for futur e invest igation. I V . C O N C L U S I O N In this paper, we explore the use of directional beam formin g for self-organization o f a d ense wireless ad hoc n etwork as a small world . W e p rovide a simulation based analysis of relev ant issues in such a scenar io fo llowed by an algorithm design for deterministic small world creation. Our results show that significant path length redu ction can be ach iev ed with negligible e ffect o n the co nnectivity . As part of our fu ture work, we would like to extend the algo rithm design to ach iev e greater reduction in pa th length. R E F E R E N C E S [1] S. Milgram, “The small world problem, ” P sychol ogy T oday , vol. 1, no. 61, 1967. [2] D. W atts and S. Strogatz, “Collect i ve dynamics of sm all-w orld networks, ” Natur e , vol. 393, pp. 440-442, 1998. [3] A. Helmy , “Small worlds in wireless netw orks, ” IEE E Communic ations Letter s , 7(10):490-492, Oct 2003. [4] R. Chitrad urga and A. Helmy , “ Analysis of wired short cuts in wirele ss sensor networks, ” in IEE E/ACS Interna tional Confer ence on P ervasive Service s , Jul. 2004. [5] G. Sharma and R. Mazumdar , “ A case for hybrid sensor networ ks, ” in Pr oc. ACM Int. Symp. on Mobile Ad Hoc Networkin g and Comput ing (MobiHoc5 ) , 2005, pp. 366-377. [6] D. Guidoni, R.A. Mini and A.A. Loureiro, “On the Design of Resilient Heteroge neous W ireless Sensor Networks based on Small W orld Con- cepts, ” Computer Networks: Special Issue on Resilient and Survivable Network s (COMNET) 54 (8) (2010) pp. 1266-128 1. [7] Chetan K . V erma, T . Bheemarjuna Reddy , B. S. Manoj, and Ramesh R. Rao, ”A Realistic Small-W orld Model for Wire less Mesh Networks”, IEEE Communicati ons Letter s , Oct. 2010. [8] R. Agarwal , A. Banerjee , V . Gauthier , M. Becker , C. K. Y eo, and B. S. Lee, “Self-organ izati on of N odes Using Bio-Inspired T echniques for Achie ving Small W orld Propertie s, ” accepted for publica tion in Proc. IEEE GLOBE COM W orkshop on Complex Communication Networks (CCNet) , Dec. 2011. [9] P . Li, C. Zhang, and Y . Fang, Asymptotic connecti vity in wireless ad hoc networks using directiona l antenna s, in IEEE/ACM T ransactions on Network ing , vol. 17, no. 4, pp. 1106-1117, Aug. 2009. [10] C. Bettste tter , C. Hartmann, and C. Moser , Ho w does randomized beamforming improv e the connecti vity of ad hoc networks? , in Proc. IEEE Int. Conf. on Communicati ons (ICC) , May 2005. [11] Z . Y u, J. T eng, X. Bai, D. Xuan and W . Jia, “Connected Cov erage in Wire less Netwo rks with Direc tional Antennas, ” in Proc. of IEEE Int. Conf . on Computer Communic ations (INFOCOM) , 2011. [12] L . C. Freeman, “ A set of measures of centralit y based on betweenne ss, ” Sociomet ry (1977), pp. 35-41. [13] J. E. W ieselthier , G.D. Nguyen and A. Ephremides, “Energy-Ef ficient Broadca st and Multi cast Trees in Wire less Netw orks, ” Mobile Netw orks Applicat ion , vol. 7, no. 6, pp. 481-492, 2002.