Research on Wireless Multi-hop Networks: Current State and Challenges

Research on W ireless Multi-hop Networks: Current State and Challenges Guoqiang Mao School of Electrical and Information Engineering The Uni versity of Sydne y National ICT Australia Email: guoqiang.mao@sydney .edu.au Abstract —Wir eless multi-hop networks, in various f orms and under various names, ar e being increasingly used in military and civilian applications. Studying connectivity and capacity of these networks is an important problem. The scaling beha vior of connectivity and capacity when the network becomes sufficiently large is of particular inter est. In this position paper , we briefly over view recent de velopment and discuss r esearch challenges and opportunities in the area, with a f ocus on the network connectivity . I . I N T R O D U C T I O N W ireless multi-hop networks, in v arious forms, e.g. wireless sensor networks, underwater sensor networks, vehicular net- works, mesh networks and UA V (Unmanned Aerial V ehicle) formations, and under various names, e.g. ad-hoc networks, hybrid networks, delay tolerant networks and intermittently connected networks, are being increasingly used in military and civilian applications. There are three defining features that characterize a wireless multi-hop network: 1) W ireless devices are self-organized or assisted by some infrastructure to form a network. The former case cor- responds to ad-hoc networks whereas the latter case corresponds to infrastructure-based multi-hop networks. Depending on the applications, the forms of the infras- tructure can be quite flexible, e.g. a subset of de vices connected via wired connections, a subset of de vices with more powerful transmission capability such that they form a wireless backbone for the network, or in a U A V formation, the infrastructure may assume the form of a subset of U A Vs with satellite links. 2) Communication is mostly via wireless multi-hop paths. This feature sets wireless multi-hop networks apart from the traditional one-hop networks, i.e. cellular networks and wireless LANs. Therefore, there is a unique set of challenging problems specific to wireless multi-hop networks. This research is partially supported by ARC Discov ery project DP110100538. This material is based on research partially sponsored by the Air Force Research Laboratory , under agreement number F A2386-10-1-4102. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory or the U.S. Government. 3) Packets are forw arded collaboratively from the source to the destination. Studying connectivity and capacity of wireless multi-hop net- works is an important problem [1]–[3]. The scaling behavior of connectivity and capacity when the network becomes suffi- ciently large is of particular interest. In this paper , we briefly ov ervie w recent de velopment and discuss research challenges and opportunities in the area, with a focus on the network connectivity . A network is said to be connected if f (if and only if) there is a (multi-hop) path between any pair of nodes. Further , a network is said to be k -connected if there are k mutually independent paths between any pair of nodes that do not ha ve any node in common except the starting and the ending nodes. k -connecti vity is often required for robust operations of the network. The rest of the paper is organized as follows: Section II discusses connecti vity of large-scale random networks; Section III discusses connectivity of giant component; Section IV discusses recent development, research challenges and oppor- tunities in mobile networks and Section V concludes the paper . I I . C O N N E C T I V I T Y O F L A R G E - S C A L E R A N D O M N E T W O R K S A. Unit disk model and connectivity Extensiv e research has been done on connecti vity problems using the well-known random geometric gr aph and the unit disk model , which is usually obtained by randomly and uni- formly distributing n nodes in a giv en area and connecting any two nodes iff their Euclidean distance is smaller than or equal to a given threshold r ( n ) , known as the transmission range [3], [4]. Significant outcomes have been achiev ed for both asymptotically infinite n [1], [3], [5]–[9] and finite n [10]–[12]. Research on the connectivity of lar ge-scale random ad- hoc networks under the unit disk model is spearheaded by Penrose [13], [14] and Gupta and Kumar [1]. Specifically , Penrose [13], [14] and Gupta and Kumar [1] proved using different techniques that if the transmission range is set to r ( n ) = q log n + c ( n ) π n , a random network formed by uniformly placing n nodes on a unit-area disk in < 2 is asymptotically almost sur ely (a.a.s.) connected as n → ∞ iff c ( n ) → ∞ . An ev ent ξ n depending on n is said to occur a.a.s. if its probability tends to one as n → ∞ . Penrose’ s result is based on the fact that in the abov e random network, as n → ∞ the longest edge of the minimum spanning tree conv erges in probability to the minimum transmission range required for the abov e random network to have no isolated nodes (or equiv alently the longest edge of the nearest neighbor graph of the abov e network) [3], [13], [14]. Gupta and Kumar’ s result is based on a key finding in the continuum percolation theory [15, Chapter 6]: Consider an infinite network with nodes distrib uted in < 2 following a Poisson distrib ution with density ρ ; and a pair of nodes separated by a Euclidean distance x are directly connected with probability g ( x ) , independent of the ev ent that another distinct pair of nodes are directly connected. Here, g : < + → [0 , 1] satisfies the conditions of non-increasing monotonicity and integral boundedness [15, pp. 151-152]. As ρ → ∞ , a.a.s. the above network in < 2 has only a unique unbounded component and isolated nodes. In [6], Philips et al. proved that the average node degree, i.e. the expected number of neighbors of an arbitrary node, must grow logarithmically with the area of the netw ork to ensure that the network is connected, where nodes are placed randomly on a square according to a Poisson point process with a constant density . This result by Philips et al. actually provides a necessary condition on the av erage node degree required for connectivity . In [5], Xue et al. showed that in a network with a total of n nodes randomly and uniformly distributed on a unit square, if each node is connected to c log n nearest neighbors with c ≤ 0 . 074 then the resulting random network is a.a.s. disconnected as n → ∞ ; and if each node is connected to c log n nearest neighbors with c ≥ 5 . 1774 then the network is a.a.s. connected as n → ∞ . In [8], Balister et al. adv anced the results in [5] and improv ed the lo wer and upper bounds to 0 . 3043 log n and 0 . 5139 log n respectiv ely . In a more recent paper [9] Balister et al. achieved much impro ved results by showing that there exists a constant c crit such that if each node is connected to b c log n c nearest neighbors with c < c crit then the network is a.a.s. disconnected as n → ∞ , and if each node is connected to b c log n c nearest neighbors with c > c crit then the network is a.a.s. connected as n → ∞ . In both [8] and [9], the authors considered nodes randomly distributed following a Poisson process of intensity one on a square of area n . In [7], Ra velomanana in vestigated the critical transmission range for connectivity in 3-dimensional wireless sensor networks and derived similar results as the 2-dimensional results in [1]. In [11], Bettstetter empirically inv estigated the minimum node de gree and connectivity of a finite network with n ( 100 ≤ n ≤ 2000 ) nodes randomly and uniformly placed on a square of area A . T ang et al. [12] proposed an empirical formula relating the probability of having a connected network to the transmission range for a finite network with n ( n ≤ 125 ) nodes randomly and uniformly distributed on a unit square. Bettstetter [10] studied the network connectivity considering different node placement models, i.e. uniform distribution, Gaussian distribution. Note that most results for finite n are empirical results. B. More gener al connection models and connectivity All the work described in the last subsection is based on the unit disk model. This model may simplify analysis but no real antenna has an antenna pattern similar to it. The log- normal shadowing connection model, which is more realistic than the unit disk model, has accordingly been considered for in vestigating network connecti vity in [16]–[21]. Under the log- normal shadowing connection model, two nodes are directly connected if the received power at one node from the other node, whose attenuation follo ws the log-normal model [22], is greater than or equal to a given threshold. In [16], Hekmat et al. proposed an empirical formula for computing the av erage size of the largest connected component through simulations, where a total of n nodes are randomly and uniformly distributed in a bounded area in < 2 . In [20], Bettstetter deri ved a lower bound on the minimum node den- sity ρ required to ensure that a network with nodes Poissonly distributed in an area in < 2 with density ρ is k -connected with a high probability . The analysis is based on the observ ation that the minimum node density required for a k -connected network is larger than that required for the network to hav e a minimum node degree k , and the assumption that the ev ent that a node has a degree greater than or equal to k is independent of the ev ent that another node has a degree greater than or equal to k . Using simulations, they showed that the bound is tight when the node density is sufficiently large. Using the same model as in [20], Bettstetter et al. obtained in [21] a lower bound on the minimum node density required for an almost surely connected network using essentially the same technique as that in [20]. The analysis relies on the assumption that the ev ent that a node is isolated and the ev ent that another node is isolated are independent, hereafter referred to as the independence assumption. Orriss et al. [17] considered nodes uniformly and randomly distributed on a plane and communicating with each other follo wing the log-normal shadowing model in the framework of cellular networks. They in vestigated the distribution of the number of base stations that communicate with a giv en mobile and found that the number of base stations able to communicate with a gi ven mobile and lying within a specified range of the mobile follows a Poisson distribution. In [19], Miorandi et al. presented an analytical procedure for computing the node isolation probability in the presence of channel randomness, where nodes are distributed following a Poisson point process in < 2 (which extends their earlier work in [18]). They further obtained an estimate of the probability that there is no isolated node in the network based on the abo ve independence assumption. The pre vious results in [16]–[21] dealing with a necessary condition on the critical transmission power for connectivity under the log-normal shadowing model all rely on the independence assumption that the node isolation ev ents are independent. Realistically howe ver , one may expect the event that a node is isolated and the event that another node is isolated will be correlated whenever there is a non- zero probability that a third node may exist which may have direct connections to both nodes. In the unit disk model, this may happen when the transmission range of the two nodes ov erlaps. In the log-normal model, any node may have a non- zero probability of having direct connections to both nodes. This observation and a lack of rigorous analysis on the node isolation ev ents to support the independence assumption raised a question mark over the v alidity of the results of [16]–[21]. Other work in the area includes [23]–[26], which studies from the percolation perspectiv e, the impact of mutual in- terference caused by simultaneous transmissions, the impact of physical layer cooperative transmissions, the impact of directional antennas and the impact of unreliable links on connectivity respectively . C. Random connection model and connectivity In the more recent work [27]–[29], the authors considered a network where all nodes are distributed on a unit square A ,  − 1 2 , 1 2  2 following a Poisson distribution with kno wn density ρ and a pair of nodes are directly connected following a random connection model , viz. a pair of nodes separated by a Euclidean distance x are directly connected with probability g r ρ ( x ) , g  x r ρ  , where g : [0 , ∞ ) → [0 , 1] , independent of the e vent that another pair of nodes are directly connected. Here r ρ = s log ρ + b C ρ (1) and b is a constant. The function g is required to satisfy the properties of non-increasing monotonicity and integral boundedness [15], [30, Chapter 6]. Further, it is required that g satisfies the more restrictiv e requirement that g ( x ) = o x  1 x 2 log 2 x  (2) in order for the impact of the truncation effect , which accounts for the dif ference between an infinite network and a finite (or asymptotically infinite) network, on connectivity to be asymptotically v anishingly small [29]. Based on the above model, it is shown that as ρ → ∞ , the probability that the abov e network has no isolated nodes and the probability that the above network forms a connected network both conv erge to e − e − b as ρ → ∞ . As a ready consequence of these results, the abo ve network is a.a.s. connected if f b → ∞ as ρ → ∞ ; and is a.a.s. disconnected iff b → −∞ as ρ → ∞ . The abov e results extend the earlier work by Penrose [13], [14] and Gupta and Kumar [1] from the unit disk model to the more generic random connection model and bring theoretical research in the area closer to reality . It can be readily shown that the results on the random connection model include the work of Penrose [13], [14] and Gupta and Kumar [1] on the unit disk model and the work on the log-normal model [16]– [21] as two special cases. D. Challenges There remain significant challenges ahead. Most results in the area rely on three main assumptions: a) the connection function g is isotropic, b) the connections are independent, c) nodes are Poissonly or uniformly distributed. W e conjecture that assumption a) is not a critical as- sumption, i.e. under some mild conditions, e.g. nodes are independently and randomly oriented, assumption a) can be remov ed while the abo ve results, particularly the ones obtained assuming a random connection model, are still valid. It how- ev er remains to v alidate the conjecture. The above results ho we ver critically rely on assumption b), which is not necessarily valid in some networks due to channel correlation and interference, where the latter ef fect makes the connection between a pair of nodes dependent on the locations and activities of other nearby nodes. In [31], some preliminary work was conducted on the connectivity of CSMA networks considering the impact of interference. The w ork essentially uses a de-coupling approach to solve the challenges of connection correlation caused by interference and suggests that when some realistic constraints are considered, i.e. carrier - sensing, the connectivity results will be very close to those obtained under a unit disk model. This conclusion is in stark contrast with that obtained under an ALOHA multiple-access protocol [23]. The major obstacle in dealing with the impact of channel correlation is that there is no widely accepted model in the wireless communication community capturing the impact of channel correlation on connections. Finally , it is a logical move after the above work to consider connectivity of networks with nodes distributed follo wing a generic distribution other than Poisson or uniform. This remains a major challenge in the area. I I I . C O N N E C T I V I T Y O F G I A N T C O M P O N E N T A giant component is a component with a designated large percentage of nodes in the network, say p where 0 . 5 < p < 1 . A component is a maximal set of nodes where there is a path between any pair of nodes in the set. Results on connectivity of lar ge-scale random networks under both the unit disk model [1], [13], [14] and the more generic random connection model [27], [28] revealed the same scaling law . That is, when the number of nodes, denoted by n , in a network increases, the transmission range (or po wer) has to increase at a rate to maintain an average node degree of Θ (log n ) in order to achiev e connecti vity . F or two functions f ( x ) and h ( x ) , f ( x ) = Θ ( h ( x )) iff there exist a sufficiently large x 0 and two positiv e constants c 1 and c 2 such that for any x > x 0 , c 1 h ( x ) ≥ f ( x ) ≥ c 2 h ( x ) . For example, the critical transmission range for connectivity is r ( n ) = q log n + c ( n ) π n under the unit disk model for a random network formed by uniformly placing n nodes on a unit-area disk [1], [13], [14] 1 . In other words, a connected netw ork poses a very demanding requirement on the transmission range (or po wer). This in turn causes many undesirable effects on increased interference and reduced throughput. In [32], it was shown that the end-to-end throughput between a randomly chosen source-destination pair in the above network is Θ  W √ n log n  , where W is the link capacity . This result can be intuitiv ely explained using the results on connecti vity as follows: as the number of nodes n increases, the a verage distance, measured 1 By scaling, it can be sho wn that assuming an extended network model where nodes are distributed on a disk of area n with a constant density of 1 node per unit area, the critical transmission range for connectivity is r ( n ) = q log n + c ( n ) π . by the number of hops, between a randomly chosen pair of nodes is Θ  1 r ( n )  = Θ  q π n log n  . That is, for a typical node, for e very packet transmitted for itself, there are Θ  q π n log n  relay packets transmitted for other source-destination pairs. Further , the av erage node degree is nπ r 2 ( n ) = Θ (log n ) , which implies that in a neighborhood of a typical node, at any time there can only be one out of every Θ (log n ) nodes activ e. It follows that the end-to-end throughput between a typical source-destination pair is W Θ  √ πn log n  Θ(log n ) = Θ  W √ n log n  , hence comes the result in [32]. The above observ ation motiv ates a question: since the net- work connectivity is a very demanding requirement, whether there is any benefit in backing down from such a demanding requirement and requiring most nodes, instead all nodes, to be connected? Indeed in many applications, it is unnecessary for all nodes to always be connected to each other [33]. Examples of such applications include a wireless sensor network for habitat monitoring [34], [35] or en vironmental monitoring [36], [37] and a mobile ad-hoc network in which users can tolerate short off-service intervals [38]. In en vironmental monitoring, there are scenarios where the size of the monitored phenomenon is v ery lar ge (e.g. rain clouds) or the parameters (e.g. temperature, humidity) that are monitored change slo wly both in space and in time. When the number of nodes for monitoring the phenomenon or measuring the parameters is very large, ha ving a few disconnected nodes will not cause a statistically significant change in the monitored parameters. One example of such applications is a wireless sensor network that was deployed underneath the Briksdalsbreen glacier in Norway to monitor the pressure, humidity , and temperature of ice to understand glacial dynamics in response to climate change [36]. In habitat monitoring, there are scenarios where the number of objects (e.g. zebras and cane toads [34]) that are monitored is large. Having a few nodes disconnected or lost may not significantly af fect the accuracy of the monitored parameter . In many mobile ad-hoc networks, having a number of nodes temporarily disconnected is also not critical, as long as users can tolerate short off-service intervals. For example, in a campus-wide wireless network, students and staff can share information using wireless devices (e.g. laptops and personal digital assistants) around the campus [38]. When a wireless device temporarily loses connection, it can store the data and complete the work after becoming connected later . In [39], [40], considering a network with a total of n nodes uniformly and i.i.d. on a unit square in < 2 , it was shown analytically that under both the unit disk model [40] and the log-normal model [39], the transmission range (or po wer) required for having a designated large percentage of nodes connected, say p where 0 . 5 ≤ p < 1 , is asymptotically vanishingly small compared to that required for having a connected network, irrespectiv e of the v alue of p . This result implies that significant energy savings can be achiev ed if we require only most nodes (e.g. 95% , 99% ) to be connected, instead of requiring all nodes to be connected; and giv en a network with most nodes connected, a sharp increase in the transmission range (or power) is required to connect the fe w remaining hard-to-reach nodes. It was further shown using simulations that under the unit disk model, in a network with 1000 nodes, the transmission range required for having 95% nodes connected is only 76% of that required for ha ving all nodes connected. Based on a conservati ve estimate that the required transmission power increases with the square of the required transmission range, an energy saving of at least 42% can be achie ved by sacrificing 5% of nodes. That energy sa ving will further increase with an increase in the number of nodes in the network. Other benefits of the reduced transmission range or power requirement is the reduced interference, hence better throughput. It remains to find the value of the transmission range (or power) required for guaranteeing a designated large percentage of nodes to be connected in a large scale network. This problem has some intrinsic connections to the problem of find- ing the percolation probability in the continuum percolation theory [15]. Further , it remains to quantitati vely characterize the benefit in capacity due to the reduced transmission range (or power) required for a giant component. Other researchers approached the problem caused by the demanding requirement of a connected network on the trans- mission range (or power) from a different perspecti ve and considered the use of infrastructure instead. Here the infras- tructure can be quite fle xible. It can be a subset of nodes connected through wired connections [41], or a subset of nodes with possibly more powerful transmission capability that forms a wireless backbone of the network [42], [43], or a subset of nodes with satellite links as one would possibly encounter in U A V formations [44]. The use of infrastructure does not change the wireless multi-hop nature of the end- to-end communication, instead the infrastructure assists the end-to-end communication by leapfrogging some long hops and reducing the number of hops between two nodes, hence improving the performance. Accordingly the concept of k-hop connected networks was proposed and inv estigated [45]–[48]. In a k-hop connected network, the maximum number of hops between any two nodes is smaller than or equal to k . Some research in the area was also conducted under the name of hybrid networks [41], [49]. Despite previous research in the area of hybrid networks or k-hop connected networks, no conclusive results have been obtained yet on the role of infrastructure in wireless multi- hop networks with many problems remain unanswered. Some examples include: for randomly deployed infrastructure nodes and “ordinary” nodes, ho w man y infrastructure nodes (v ersus ordinary nodes) are required for a k-hop connected network; for deterministically deployed infrastructure nodes and ran- domly deployed ordinary nodes, how many infrastructure nodes are required for a k-hop connected network and what is the optimum deployment of infrastructure nodes; how to combine the use of infrastructure-based communications and ad-hoc communications in one network in order to provide some performance guarantee, in terms of capacity or delay . These problems are important for wireless multi-hop networks, particularly for wireless vehicular networks in which both infrastructure-based communications and ad-hoc communica- tions will co-exist [50]. I V . D E V E L O P M E N T A N D C H A L L E N G E S I N M O B I L E N E T W O R K S In [51], Grossglauser and Tse studied the capacity of mobile ad-hoc networks. Particularly , they considered a network with a total of n nodes distributed on a unit-area disk, the trajec- tories of different nodes are i.i.d. and the nodes’ mov ement is such that the spatial distribution of nodes are stationary and ergodic with stationary uniform distribution on the disk. They showed that in the abov e network with unbounded delay r e- quir ement , the throughput between a randomly chosen source- destination pair can be kept constant even as n increases. This result is in stark contrast with its counter -part in static networks in which the throughout between a randomly chosen source-destination pair is sho wn to be Θ  W √ n log n  [32]. Follo wing the seminal work of Grossglauser and Tse, other researchers hav e conducted further research trying to quanti- tativ ely characterize the relationship between delay , mobility and capacity in mobile ad-hoc networks [45], [52]–[55] and the obtained results v ary greatly with the mobility models and network settings. A fundamental reason why mobility increases throughput is that in mobile networks message transmissions generally follow the store-carry-forward pattern versus the store-forward pattern found in static networks. As nodes move, new op- portunity may arise such that a mobile node can carry the message until it meets a node, which is in a better position than itself to transmit the message to the destination, or until it meets the destination directly . In this way , the number of relay nodes (number of hops) in v olved in transmitting a message to its destination can be greatly reduced and the required transmission range (or power) for a node to reach another node via a multi-hop path can also be greatly reduced, hence the benefit in improved capacity . The cost in achieving this benefit in capacity is the increased delay . By analogy , mobility can also improve connectivity . There are three fundamental differences between mobile networks and static networks [56]: in mobile networks • the wireless link between two directly connected nodes and the end-to-end path only exists temporarily; • two nodes may nev er be part of the same connected component but they are still able to communicate, i.e. exchange messages, with each other; and • while any one wireless link may be (or assumed to be) undirectional, the path connecting any two nodes is directional, i.e. there is a path from node v i to node v j within a designated time period does not necessarily mean there is a path from v j to v i within the same period. These are illustrated in Fig. 1. Particularly the last dif ference implies that it is important to consider the order of links in time when analyzing mobile networks, which has been incorrectly neglected in some pre vious work. Due to these dif ferences, many established concepts in static networks must be revisited for mobile networks. For example, a static wireless multi-hop network is said to be connected iff there is a path between an y pair of nodes in the network. Howe ver a more meaningful definition of connecti vity in mobile networks is to say that a mobile network is connected in time period [0 , T ] if an y node can e xchange a message with any other node within [0 , T ] . The above definition implies that the tradeoff between connectivity , mobility and delay is the prime issue when analyzing the connectivity of mobile networks. Despite intensiv e research on the properties of mobile networks, no conclusiv e results ha v e been obtained on the abo ve problem and it remains a major challenge in the area. V . S U M M A RY W ireless multi-hop networks hav e attracted significant re- search interest. This interest is expected to grow further with the proliferation of applications, particularly in the areas of wireless vehicular networks and sensor networks. In this pa- per , we briefly overvie wed recent de velopment and discussed research challenges and opportunities in the area mainly from the perspective of network connecti vity . W e also showed how the results on network connecti vity is related the study of other performance metrics, i.e. capacity and delay . R E F E R E N C E S [1] P . Gupta and P . R. Kumar, Critical P ower for Asymptotic Connectivity in W ireless Networks . Boston, MA: Birkhauser, 1998, pp. 547–566. [2] M. Haenggi, J. G. Andrews, F . Baccelli, O. Dousse, and M. Franceschetti, “Stochastic geometry and random graphs for the analysis and design of wireless networks, ” IEEE Journal on Selected Areas in Communications , vol. 27, no. 7, pp. 1029–1046, 2009. [3] M. D. Penrose, Random Geometric Graphs , ser . Oxford Studies in Probability . Oxford University Press, USA, 2003. [4] ——, “On k-connectivity for a geometric random graph, ” Random Structur es and Algorithms , vol. 15, no. 2, pp. 145–164, 1999. [5] F . Xue and P . Kumar , “The number of neighbors needed for connectivity of wireless networks, ” W ir eless Networks , vol. 10, no. 2, pp. 169–181, 2004. [6] T . K. Philips, S. S. Panwar , and A. N. T antawi, “Connectivity properties of a packet radio network model, ” IEEE T ransactions on Information Theory , vol. 35, no. 5, pp. 1044–1047, 1989. [7] V . Ravelomanana, “Extremal properties of three-dimensional sensor networks with applications, ” IEEE T ransactions on Mobile Computing , vol. 3, no. 3, pp. 246–257, 2004. [8] P . Balister , B. Bollobas, A. Sarkar, and M. W alters, “Connectivity of random k-nearest-neighbour graphs, ” Advances in Applied Probability , vol. 37, no. 1, pp. 1–24, 2005. [9] ——, “ A critical constant for the k nearest neighbour model, ” Advances in Applied Pr obability , vol. 41, no. 1, pp. 1–12, 2009. [10] C. Bettstetter , “On the connectivity of ad hoc networks, ” The Computer Journal , vol. 47, no. 4, pp. 432–447, 2004. [11] ——, “On the minimum node degree and connectivity of a wireless multihop network, ” in 3rd ACM International Symposium on Mobile Ad Hoc Networking and Computing , pp. 80–91. [12] A. T ang, C. Florens, and S. H. Low , “ An empirical study on the connectivity of ad hoc networks, ” in IEEE Aerospace Conference , v ol. 3, pp. 1333–1338. [13] M. Penrose, “The longest edge of the random minimal spanning tree, ” The Annals of Applied Pr obability , vol. 7, no. 2, pp. 340–361, 1997. [14] ——, “ A strong law for the longest edge of the minimal spanning tree, ” The Annals of Applied Pr obability , vol. 27, no. 1, pp. 246–260, 1999. [15] R. Meester and R. Roy , Continuum P ercolation . Cambridge University Press, 1996. [16] R. Hekmat and P . V . Mie ghem, “Connecti vity in wireless ad-hoc networks with a log-normal radio model, ” Mobile Networks and Ap- plications , vol. 11, no. 3, pp. 351–360, 2006. ! Figure 1. An illustration of connecti vity in a mobile ad-hoc network. A solid line represents a connection between two nodes. The network is disconnected at any time instant but there is a path from any node to any other node in the network. For example, nodes v 1 and v 6 are nev er part of the same connected component but a message from v 1 can still reach v 6 through the following path: t 1 : v 1 → v 2 , t 2 : v 2 → v 3 , t 3 : v 3 → v 4 , t 4 : v 4 → v 6 . Further, a message from v 6 can reach v 1 at t 2 but a message from v 1 can only reach v 6 at t 4 . [17] J. Orriss and S. K. Barton, “Probability distrib utions for the number of radio transceivers which can communicate with one another, ” IEEE T ransactions on Communications , vol. 51, no. 4, pp. 676–681, 2003. [18] D. Miorandi and E. Altman, “Coverage and connecti vity of ad hoc networks presence of channel randomness, ” in IEEE INFOCOM , vol. 1, pp. 491–502. [19] D. Miorandi, “The impact of channel randomness on coverage and connectivity of ad hoc and sensor networks, ” IEEE T ransactions on W ireless Communications , vol. 7, no. 3, pp. 1062–1072, 2008. [20] C. Bettstetter , “Failure-resilient ad hoc and sensor networks in a shadow fading en vironment, ” in IEEE/IFIP International Conference on Depend- able Systems and Networks . [21] C. Bettstetter and C. Hartmann, “Connectivity of wireless multihop networks in a shadow fading environment, ” W ir eless Networks , vol. 11, no. 5, pp. 571–579, 2005. [22] T . S. Rappaport, W ireless Communications: Principles and Practice . Prentice Hall, 2002. [23] O. Dousse, F . Baccelli, and P . Thiran, “Impact of interferences on con- nectivity in ad hoc networks, ” IEEE/A CM T ransactions on Networking , vol. 13, no. 2, pp. 425–436, 2005. [24] D. Goeckel, L. Benyuan, D. T owsle y , W . Liaoruo, and C. W estphal, “ Asymptotic connectivity properties of cooperative wireless ad hoc networks, ” IEEE J ournal on Selected Areas in Communications , vol. 27, no. 7, pp. 1226–1237, 2009. [25] P . Li, C. Zhang, and Y . Fang, “ Asymptotic connectivity in wireless ad hoc networks using directional antennas, ” IEEE/A CM T ransactions on Networking , vol. 17, no. 4, pp. 1106–1117, 2009. [26] Z. Kong and E. M. Y eh, “Connectivity and latency in large-scale wireless networks with unreliable links, ” in IEEE INFOCOM , pp. 11–15. [27] G. Mao and B. D. Anderson, “Connectivity of large scale networks: Distribution of isolated nodes, ” submitted to IEEE Tr ansactions on Mobile Computing, available at http://arxiv .org/abs/1103.1994 , 2011. [28] ——, “Connectivity of large scale networks: Emergence of unique unbounded component, ” submitted to IEEE Tr ansactions on Mobile Computing, available at http://arxiv .or g/abs/1103.1991 , 2011. [29] ——, “T owards a better understanding of large scale network models, ” accepted to appear in IEEE/ACM T ransactions on Networking , 2011. [30] M. Franceschetti and R. Meester, Random Networks for Communication . Cambridge Univ ersity Press, 2007. [31] T . Y ang, G. Mao, and W . Zhang, “Connectivity of wireless csma multi- hop networks, ” in IEEE International Conference on Communications , pp. 1–5. [32] P . Gupta and P . Kumar, “The capacity of wireless networks, ” IEEE T ransactions on Information Theory , vol. 46, no. 2, pp. 388–404, 2000. [33] D. M. Blough, M. Leoncini, G. Resta, and P . Santi, “The k-neighbors approach to interference bounded and symmetric topology control in ad hoc networks, ” IEEE T ransactions on Mobile Computing , vol. 5, no. 9, pp. 1267 – 1282, 2006. [34] W . Hu, V . N. T ran, N. Bulusu, C. Chou, S. Jha, and A. T aylor, “The design and e v aluation of a hybrid sensor network for cane-toad monitoring, ” in Proc. 4th Int. Symp. Inf. Pr ocess. Sens. Netw . , pp. 503 – 508. [35] P . Juang, H. Oki, Y . W ang, M. Martonosi, L. Peh, and D. Rubenstein, “Energy-ef ficient computing for wildlife tracking: Design tradeoffs and early experiences with zebranet, ” ACM SIGARCH Comput. Ar chit. News , vol. 30, no. 5, pp. 96 – 107, 2002. [36] P . P adhy , K. Martinez, A. Riddoch, J. K. Hart, and H. L. R. Ong, “Glacial en vironment monitoring using sensor networks, ” in Proc. 1st REAL WSN , pp. 10 – 14. [37] D. Ingraham, R. Beresford, K. Kaluri, M. Ndoh, and K. Srini vasan, “W ireless sensors: Oyster habitat monitoring in the bras d’or lakes, ” in IEEE 1st International Conference on Distributed Computing In Sensor Systems , pp. 399 – 400. [38] X. Y u and S. Chandra, “Delay-tolerant collaborations among campus wide wireless users, ” in IEEE INFOCOM , pp. 2101 – 2109. [39] X. T a, G. Mao, and B. D. Anderson, “On the giant component of wireless multi-hop networks in the presence of shadowing, ” IEEE T ransactions on V ehicular T echnology , vol. 58, no. 9, pp. 5152–5163, 2009. [40] ——, “On the properties of giant component in wireless multi-hop networks, ” in IEEE INFOCOM , pp. 2556 – 2560. [41] B. Liu, Z. Liu, and D. T owsley , “On the capacity of hybrid wireless networks, ” in IEEE INFOCOM , vol. 2, pp. 1543–1552. [42] M. Franceschetti, O. Dousse, D. N. C. Tse, and P . Thiran, “Closing the gap in the capacity of wireless networks via percolation theory , ” IEEE T ransactions on Information Theory , vol. 53, no. 3, pp. 1009–1018, 2007. [43] P . Li and Y . Fang, “The capacity of heterogeneous wireless networks, ” in IEEE INFOCOM , 2010, pp. 1–9. [44] G. Mao, S. Drake, and B. D. O. Anderson, “Design of an extended kalman filter for uav localization, ” in Information, Decision and Contr ol , 2007, pp. 224–229. [45] Q. W ang, X. W ang, and X. Lin, “Mobility increases the connectivity of k-hop clustered wireless networks, ” in MobiCom , 2009, pp. 121 – 132. [46] G. Mao, Z. Zhang, and B. Anderson, “Probability of k-hop connection under random connection model, ” IEEE Communication Letters , vol. 14, no. 11, pp. 1023 – 1025, 2010. [47] S. C. Ng, W . Zhang, Y . Y ang, and G. Mao, “ Analysis of access and connectivity probabilities in vehicular relay networks, ” IEEE Journal on Selected Areas in Communications–Special Issue V ehicular Commu- nications and Networks , vol. 29, no. 1, pp. 140 – 150, 2011. [48] X. T a, G. Mao, and B. D. O. Anderson, “Evaluation of the probability of k-hop connection in homogeneous wireless sensor networks, ” in IEEE Globecom , 2007, pp. 1279 – 1284. [49] O. Dousse, P . Thiran, and M. Hasler, “Connectivity in ad-hoc and hybrid networks, ” in IEEE INFOCOM , vol. 2, pp. 1079–1088. [50] Y . Y ang, H. Hu, J. Xu, and G. Mao, “Relay technologies for wimax and lte-advanced mobile systems, ” IEEE Communications Magazine , v ol. 47, no. 10, pp. 100–105, 2009. [51] M. Grossglauser and D. N. C. Tse, “Mobility increases the capacity of ad hoc wireless networks, ” IEEE/ACM T ransactions on Networking , vol. 10, no. 4, pp. 477–486, 2002. [52] G. Sharma, R. Mazumdar , and B. Shroff, “Delay and capacity trade- offs in mobile ad hoc networks: A global perspectiv e, ” IEEE/ACM T ransactions on Networking , vol. 15, no. 5, pp. 981–992, 2007. [53] M. J. Neely and E. Modiano, “Capacity and delay tradeoffs for ad hoc mobile networks, ” IEEE T ransactions on Information Theory , v ol. 51, no. 6, pp. 1917–1937, 2005. [54] A. E. Gamal, J. Mammen, B. Prabhakar, and D. Shah, “Throughput- delay trade-off in wireless networks, ” in IEEE INFOCOM , vol. 1, 2004. [55] S. T oumpis and A. J. Goldsmith, “Large wireless networks under fading, mobility , and delay constraints, ” in IEEE INFOCOM , vol. 1, 2004, pp. 609–619. [56] G. Mao and B. D. Anderson, “Graph theoretic models and tools for the analysis of dynamic wireless multihop networks, ” in IEEE WCNC , 2009, pp. 1–6.