Network-Decomposed Hierarchical Cooperation in Ad Hoc Networks With Social Relationships
aa r X i v : . [ c s . I T ] S e p IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 1
Network-Decomposed HierarchicalCooperation in Ad Hoc Networks With SocialRelationships
Cheol Jeong,
Member, IEEE, and Won-Yong Shin,
Senior Member, IEEE
Abstract
In this paper, we introduce a network-decomposed hierarchical cooperation (HC) protocol andcompletely characterize the corresponding throughput–delay trade-off for a large wireless ad hocnetwork formed in the context of social relationships . Instead of randomly picking source–destinationpairings, we first consider a distance-based social formation model characterized by the social groupdensity γ and the number of social contacts per node, q , where the probability that any two nodes indistance d away from each other are socially connected is assumed to be proportional to d − γ , whichis a feasible scenario. Then, using muiltihop and network-decomposed HC protocols under our socialformation model, we analyze a generalized throughput–delay trade-off according to the operatingregimes with respect to parameters γ and q in both a dense network of unit area and an extendednetwork of unit node density via a non-straightforward network transformation strategy. Our mainresults reveal that as γ increases, performance on the throughput–delay trade-off can remarkably beimproved, compared to the network case with no social relationships. It is also shown that in thedense network, the network-decomposed HC protocol always outperforms the multihop protocol,while the superiority of the network-decomposed HC depends on γ and the path-loss exponent inthe extended network. Index Terms
Ad hoc network, multihop (MH), network-decomposed hierarchical cooperation (HC), scalinglaw, social relationships, throughput–delay trade-off.
The work of W.-Y. Shin was supported by the Basic Science Research Program through the National Research Foundationof Korea (NRF) funded by the Ministry of Education (2017R1D1A1A09000835). The work of C. Jeong was supported bythe NRF through the Korea Government under Grant NRF-2017R1C1B1009145. (Corresponding author: Won-Yong Shin.)
C. Jeong is with the School of Intelligent Mechatronics Engineering, Sejong University, Seoul 05006, Republic of Korea.E-mail: [email protected]. Shin is with the Department of Computer Science and Engineering, Dankook University, Yongin 16890, Republicof Korea. E-mail: [email protected] received September 24, 2017; revised June 16, 2018; accepted August 21, 2018.
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I. I
NTRODUCTION
Communications between users (i.e., a source and a destination) over a wireless networkusually take place based on friendship , which is defined as online or offline social relationshipsamong users. In other words, a source and its destination(s) are not just randomly paired inreal-world communications, and rather a source tends to select its destination(s) along withfriendships. In [1]–[5], it was observed that social interactions among users indeed dependheavily on the geographic proximity of them. In [4], a close relationship between geographicdistance and probability distribution of friendship was demonstrated by experimental resultsbased on the LiveJournal social network. More specifically, it was shown that the proba-bility of befriending a particular user is inversely proportional to the positive power of thegeographic distance [4]. In [5], the degree of friendship related to the issue of space wasfurther studied on Twitter—the number of friends according to distance follows a doublepower-law distribution on Twitter, indicating that the probability of befriending a particularTwitter user is significantly reduced beyond a certain geographic distance between users.Moreover, there have been extensive studies on understanding the nature of friendships withrespect to the geographic distance in large-scale online social networks such as Twitter [6],Facebook [7], Flickr [8], LiveJournal [4], and Foursquare [9], while validating the small-world phenomenon and scale-free degree distributions. On the other hand, it has widely beenknown that social relationships influence users’ interactions with each other in physical space;thus, users’ social ties are closely related to the interactions of users’ communication devicessubject to diverse physical coupling (see, e.g., [10], [11] and references therein). For example,users at close proximity in a social group can share their pictures and videos, play gameswith friends, or exchange files using device-to-device (D2D) communication [12]. The D2Dcommunications between firefighters or between police officers for public safety are anotherexamples of wireless social networks in which users are socially tied and their geographicdistances are short. Another application of wireless social networks includes content-centric(caching) communications [13], [14] that content objects are cached by numerous nodes overa network, in which each request is served by nearby content source nodes in a friendshiprelation. For this reason, there has been a growing interest in analyzing the impact of socialgroups on the performance of wireless networks. In [12], [15]–[17], traffic offloading, resourceallocation, and medium access control (MAC) protocols were designed for device-to-devicewireless communications in a social-aware perspective. In [18], the impact of social selfishnesson the performance of epidemic routing was also investigated in delay tolerant networks.Social context-aware small cell networks were designed in [19] by optimizing the overallallocation of resources. In [20], it was presented how to form multihop D2D connectionsbased on a community-based approach in D2D communications. Moreover, the throughputscaling laws of large wireless ad hoc networks were also studied by incorporating the notionof social characteristics into their network models [21]–[24], where the throughput scalingresults depend on the number of nodes and the geographic distance unlike the case with nosocial relationships. In this paper, we aim to characterize a fundamental throughput–delaytrade-off of a large wireless ad hoc network, where users communicate with others in thecontext of social relationships . A. Related Work
In [25], it was shown that the aggregate throughput of a large wireless network having n source-destination (S–D) pairs randomly distributed in a unit area (i.e., a dense network)scales as Θ( p n/ log n ) , which is achieved by the nearest-neighbor multihop (MH) protocol. EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 3
This throughput scaling was improved to Θ( √ n ) using percolation theory [26]. MH protocolswere further studied and analyzed in various aspects [27]–[29]. There has been a great deal ofresearch to improve the aggregate throughput of dense networks up to a linear scaling in [30]–[38]. It was shown that an almost linear throughput scaling, i.e., Θ( n − ǫ ) for an arbitrarilysmall ǫ > , can be achieved by hierarchical cooperation (HC) protocols [30], [31]. Theimpact and benefits of infrastructure support in improving the throughput scaling in hybridnetworks, consisting of both wireless ad hoc nodes and infrastructure nodes, were studiedin [32]–[34]. Novel techniques such as networks with node mobility [35] and directionalantennas [36]–[38] were also introduced to achieve a linear throughput scaling. Besides thethroughput, delay is also a key performance metric in most wireless network applications.One can usually improve the per-node throughput at the cost of an increased delay of apacket. The trade-off between throughput and delay metrics of both static and mobile adhoc networks was examined in terms of scaling laws in some papers [39]–[42]. In [39], thethroughput–delay trade-off of a mobile ad hoc network adopting a two-hop relay protocolwas analyzed under a simple independent and identically distributed (i.i.d.) mobility model.In [40], [41], the throughput–delay trade-off was derived in another mobile network adoptinga random walk mobility model as well as in a static network. In [42], the throughput–delaytrade-off of a static ad hoc network was studied by modifying the original HC protocol in [30]in order to improve the delay performance for the same throughput.In all these previous studies, it was assumed that a source selects its destination at random for analytical convenience. In practice, however, this assumption is hardly realistic sincea source and its destination tend to be paired up along with one-to-one friendship in thepresence of social groups. Thus, existing achievable schemes and analytical frameworks thatshow the capacity scaling laws cannot be directly applicable to the performance analysis of adhoc networks with social groups. Recently, the notion of social relationships was taken intoaccount in studying the capacity scaling laws of large wireless ad hoc networks [21]–[24].In [21], the throughput scaling achieved by the MH protocol was analyzed again under asocial formation model such that each node has a social group consisting of a fixed numberof nodes and selects its destination uniformly among the nodes in its social group. This resultwas generalized in [22] by assuming social groups with different numbers of nodes and anon-uniform probability of selecting one destination in each social group. As an alternativeapproach to analyzing the network throughput scaling, it was assumed in [23] that the numberof friends (i.e., the friendship degree) follows a Zipf’s distribution [43] and the probability ofbefriending a particular user depends on both the geographic distance between nodes and thenode density. In [24], the capacity scaling of a hybrid network with social contact behaviorwas also investigated. B. Main Contributions
In this paper, we introduce a new HC protocol, termed a network-decomposed
HC protocol,and characterize a general throughput–delay trade-off in a large wireless ad hoc networkformed in the context of social relationships . To this end, we first consider a distance-based social formation model parameterized by the social group density γ , the number ofsocial contacts per node, q , and the probability that a source selects its destination amongsocial contacts. To better illustrate our results, we identify three operating regimes partitionedaccording to the social group density γ . More specifically, we focus on designing the network-decomposed HC protocol so that the network under our social formation model operatesproperly, since the conventional HC protocols in [30], [42] do not guarantee the best trade-off performance under our network model. In the proposed protocol, the whole network is EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 4 divided into multiple non-overlapping subnetworks, each of which operates in parallel alongwith the HC protocol. A time-division multiple access (TDMA) operation is used to avoidan edge node problem that may occur with network decomposition. To be specific, whenthe network is decomposed into multiple subnetworks, there may exist some sources in asubnetwork, whose destinations are out of the subnetwork. To solve this problem, we applythe 4-TDMA strategy with which all the subnetworks are shifted to the right, shifted up,and shifted diagonally in consecutive three time slots and remain unshifted in one time slot.Using both the MH protocol in [21], revisited in the context of social relationships, and thenetwork-decomposed HC protocol and then computing the average distance between a sourceand its destination derived under the social formation model, we completely characterize thethroughput–delay trade-off according to each operating regime with respect to parameters γ and q in not only a dense network of unit area but also an extended network of unit nodedensity through a non-straightforward network transformation strategy in terms of γ . As mainresults, we show that the throughput–delay trade-off is significantly improved as γ increasesbeyond a certain value, compared to the network case with no social relationships. We alsoshow that in the dense network, the network-decomposed HC protocol always outperformsthe multihop protocol, while the superiority of the network-decomposed HC is determinedaccording to γ and the path-loss exponent in the extended network.Our main contributions are five-fold and summarized as follows: • We incorporate the notion of social relationships into a large wireless ad hoc network inorder to analyze a fundamental throughput–delay trade-off for not only dense networksbut also more challenging extended networks, whereas the throughput performance wasonly characterized in some prior studies dealing with wireless social networks [21]–[24]. • Unlike the conventional HC protocols [30], [42] that did not exploit one-to-one socialrelationships, we propose a network-decomposed HC protocol that is properly designedbased on the 4-TDMA strategy under the distance-based social formation model. • To better interpret our results along with achievable schemes, we also identify threeoperating regimes on the throughput–delay trade-off with respect to the social groupdensity and the number of social contacts. • In both dense and extended networks with our social formation model, we completelycharacterize a general throughput–delay trade-off achieved by the MH and network-decomposed HC protocols according to each operating regime. • Furthermore, we conduct numerical evaluation to validate that our analytical results showtrends consistent with computer simulation results.
C. Organization and Notations
The rest of the paper is organized as follows. The system model and preliminaries aregiven in Section II. The main results are presented in Section III. In Section IV, the network-decomposed HC protocol is described and the corresponding throughput–delay trade-off isderived. In Section V, numerical results are presented. Section VI summarizes the paper withsome concluding remarks.Throughout this paper, E [ · ] indicates the statistical expectation; and Pr {A} is the probabilityof an event A . We use the following asymptotic notation: i) f ( x ) = O ( g ( x )) means that thereexist positive constants C and c such that f ( x ) ≤ Cg ( x ) for all x > c , ii) f ( x ) = Ω( g ( x )) if g ( x ) = O ( f ( x )) , iii) f ( x ) = ω ( g ( x )) means that lim x →∞ g ( x ) f ( x ) = 0 , and iv) f ( x ) = Θ( g ( x )) if f ( x ) = O ( g ( x )) and g ( x ) = O ( f ( x )) . EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 5 s (a) γ is small. s (b) γ is large.Fig. 1. The social group members (marked with shaded circles) of a source node s when q = 8 . II. M
ODELS AND P ROBLEM D EFINITION
In this section, we first describe not only our network model but also our social formationmodel. We then present important lemmas that are required to derive our main results includingthe throughput–delay trade-off.
A. System and Channel Models
We consider the following two types of network configurations: 1) a dense network of unitarea [25], [30], [40], [41] and 2) an extended network of unit node density [26], [30]. Inwhat follows, the dense network model is assumed, and our main results will be extendedto the extended network configuration later (refer to Sections III-B and IV-C). It is assumedthat our network is composed of n nodes, which are uniformly and independently distributedin a square of given unit area, and one central processor, which enables the network to beglobally synchronized and coordinated. Specifically, the central processor plays a role ofnot only controlling interference based on the TDMA strategy but also determining packetdelivery routes using geo-located information of nodes. Updated information is delivered tothe central processor from nodes only when new S–D pairings are established. Each nodeacts as a source and has exactly one corresponding destination node. Now, let us turn tothe channel modeling. Under wireless networks in light-of-sight (LOS) environments, thecomplex channel between nodes i and k is denoted by h ki = e jθk,i d α/ k,i where θ k,i is the randomphase uniformly distributed in [0 , π ) , d k,i denotes the distance between nodes i and k , and α ≥ is the path-loss exponent. The phase θ k,i and the path-loss model are based on afar-field assumption, i.e. the distance between any two nodes is assumed to be much largerthan the carrier wavelength. It is also assumed that one-to-one friendship relations on onlinesocial networks are available at the central processor so that sources and destinations arepaired up in the presence of social groups. Moreover, the central processor updates the socialrelationships between any two nodes, which takes place intermittently since the period ofsuch changes to friendship would be much longer than the entire communication period. B. Social Formation Model
In prior studies on the large-sale network analysis, each source node selects its destinationnode in a random fashion without taking into account the geographic distance between the In [44], it was shown that the capacity scaling of wireless networks is fundamentally limited by the relation between n and the wavelength due to laws of physics. Based on this investigation, the throughput scaling achieved by HC wasrediscovered in [45]. In this paper, however, we adopt the i.i.d. phase assumption as in [30], [31], [34] for analyticaltractability. Moreover, we deal simply with the LOS channel rather than multipath fading channels to focus on analyzingthe effect of social relationships in our network. This is due to the fact that throughput scaling laws do not fundamentallychange in the presence of fading if all nodes have their own traffic demands [28], [29], [46], [47]. EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 6 two nodes, i.e., S–D pairings are randomly picked so that each node is the destination ofexactly one source. It is thus likely that the source is far away from its destination. On theother hand, in our work, nodes are assumed to be geographically related with each otherin a social context, which is more feasible in practice. In online social networks, it wasobserved that the probability of friendship formation (or social relationships) between twonodes (users) is proportional to the inverse of the power of the Euclidean distance between thetwo nodes (users) [3], [4]. Each node is allowed to have multiple friends by forming a socialgroup for the node. A source node and its destination node are chosen only out of its socialgroup members by the central processor that is aware of one-to-one friendship relations. Inour distance-based social formation model, we assume that the Euclidean distance betweena source and its social group members follows a power-law distribution as in [3], [4]. Asillustrated in Fig. 1, a source node s selects other node v i as its contact with a probabilityproportional to d − γs,v i , where γ > is the social group density. Each node has a social groupthat consists of q contacts selected independently, where q ∈ { , · · · , n − } (i.e., q = O ( n ) ).The probability that the set of nodes { v i , . . . , v i q } forms a social group of the node s is givenby Pr { Social group of s = { v i , . . . , v i q }} = d − γs,v i · · · d − γs,v iq N γ,q , (1)where N γ,q = P ≤ i < ···
In this subsection, we present important lemmas to analyze the throughput–delay trade-off for the MH and network-decomposed HC protocols in the subsequent sections. In purewireless ad hoc networks with no social group, since the sources and destinations are pairedup one-to-one in a random fashion, the average distance between an S–D pair is given by O (1) with high probability (w.h.p.) in a dense network. On the other hand, in the contextof social relationships, the average distance of an S–D pair depends on both the number ofcontacts in a social group, q , and the social group density γ since a source node selects one ofits social contacts in the belonging social group as its destination node randomly. The averagedistance between an S–D pair is specified according to the parameters q and γ below. Lemma 1: If q scales faster than a constant independent of n (i.e., q = ω (1) ), then theaverage distance between a source s and its destination v , E [ d s,v ] , in a dense network is givenby E [ d s,v ] = Θ(1) . (2)If q = Θ(1) , then E [ d s,v ] in a dense network is given by E [ d s,v ] = Θ (1) for ≤ γ < (cid:16)(cid:0) log nn (cid:1) γ − (cid:17) for ≤ γ ≤ (cid:18)q log nn (cid:19) for γ > , (3)where γ > denotes the social group density. EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 7
Proof:
From [21, Section IV], when the MH protocol is employed in a dense network,the average number of hops in any given S–D routing path, E [ X ] , is shown to be E [ X ] = Θ (cid:16) r ( n ) (cid:17) for q = ω (1)Θ (cid:16) nn − q +1 1 r ( n ) (cid:17) for q = Θ(1) , ≤ γ < (cid:16) nn − q +1 1 r − γ ( n ) (cid:17) for q = Θ(1) , ≤ γ ≤ (cid:16) nn − q +1 (cid:17) for q = Θ(1) , γ > , where r ( n ) = Θ (cid:18)q log nn (cid:19) is the transmission range per hop. By multiplying E [ X ] by r ( n ) ,the average distance E [ d s,v ] can be expressed as E [ d s,v ] = Θ (1) for q = ω (1)Θ (cid:16) nn − q +1 (cid:17) for q = Θ(1) , ≤ γ < (cid:16) nn − q +1 (cid:0) log nn (cid:1) γ − (cid:17) for q = Θ(1) , ≤ γ ≤ (cid:18) nn − q +1 q log nn (cid:19) for q = Θ(1) , γ > , which thus leads to (2) and (3). This completes the proof of this lemma.Since the distance between two nodes in an extended network is simply increased by afactor of √ n compared to a dense network, the average distance in the extended network isalso increased by a factor of √ n .A natural question arises to examine whether the distance between an S–D pair may deviatefrom its mean. In the following lemma, we show that the distance of an S–D pair does notscale faster than the average distance E [ d s,v ] w.h.p. Lemma 2:
The distance between a source s and its destination v , d s,v , does not scale ata faster rate than the average distance E [ d s,v ] within a factor of n ǫ w.h.p. for an arbitrarilysmall ǫ > . That is, the probability Pr { d s,v < n ǫ E [ d s,v ] } is given by − n ǫ . Proof:
By Markov inequality, we have Pr { d s,v < n ǫ E [ d s,v ] } = 1 − Pr { d s,v ≥ n ǫ E [ d s,v ] }≥ − E [ d s,v ] n ǫ E [ d s,v ] = 1 − n ǫ , which tends to one as n goes to infinity. This completes the proof of this lemma.Hence from Lemmas 1 and 2, one can straightforwardly replace the average distance E [ d s,v ] by the distance d s,v of a certain S–D pair as long as a factor of n ǫ can be ignored. D. Performance Metrics
In this subsection, we formally define the throughput and delay used throughout the paper.
Definition 1 (Throughput):
A per-node throughput R ( n ) is said to be achievable w.h.p.if all sources can transmit at the average rate of R ( n ) bits/s/Hz to their destinations withprobability approaching one as n increases. The achievable aggregate throughput is given by T ( n ) = Ω( nR ( n )) accordingly. One can use a tighter inequality with the second moment of random variable d s,v (e.g., Chebyshev’s inequality), whichhowever does not fundamentally change our main results. EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 8
Definition 2 (Delay):
The end-to-end delay of a packet is the time that it takes for thepacket to reach its destination after it leaves the source. The delay D ( n ) is the expectationof the average delay over all S–D pairs and is expressed as D ( n ) = 2 n n/ X i =1 D i ( n ) , where D i ( n ) is the sample mean of delay (over packets that reach their destinations) for S–Dpair i .Note that the queuing delay at the source node is not included in our work. Althoughthe queuing delay may account for a large portion of the overall delay of a packet, it wasshown in [41] that when the queuing delay is taken into account, the throughput–delay trade-off for the MH protocol remains unchanged in order sense since the average delay at eachserver, corresponding to a routing cell, is bounded by some constant independent of n . Hence,Theorems 1 and 3 in Section III will remain the same even with the queuing delay. On theother hand, for the HC protocol, the analysis of delay scaling with the queuing delay is leftopen even under network models without social relationships.III. M AIN R ESULTS
In this section, we present main results of this paper by characterizing the throughput–delaytrade-off achieved by the MH and network-decomposed HC protocols with the distance-basedsocial formation model in Section II-B. The throughput–delay trade-off depends on the path-loss exponent α and the parameters of the social formation model such as the social groupdensity γ and the number of contacts in a social group, q . To illustrate the main resultsmore concisely, we will first identify operating regimes on the throughput–delay trade-off asfollows: • Regime A (Low social group density regime): q = ω (1) or { q = Θ(1) and ≤ γ < } • Regime B (Medium social group density regime): q = Θ(1) and ≤ γ ≤ • Regime C (High social group density regime): q = Θ(1) and γ > .That is, the entire operating regimes are categorized by γ and q .Note that similarly as in other studies on the throughput scaling laws of ad hoc networks[25]–[42], [45], [46], it is possible to have no outage event by globally controlling interferencebased on the TDMA operation among square routing cells. Thus, we do not employ anyretransmission scheme in our case since we will make the outage probability approach zero.Next, we will briefly account for the MH protocol in ad hoc networks, which was originallyintroduced in [25] and then was generalized in [40]. The overall procedure of the MH protocolis described as follows: • The network is divided into square routing cells of area a ( n ) = Ω(log n/n ) in a densenetwork ( a ( n ) = Ω(log n ) in an extended network) ensuring that each cell includes atleast one node w.h.p. [25], [40]. • Draw a line connecting a source to its destination and perform MH routing horizontallyor vertically by using the adjacent routing cells passing through the line until its packetreaches the corresponding destination. • By virtue of the central processor, each routing cell operates the t -TDMA to avoid hugeinterference, where t > is some small constant independent of n .The network-decomposed HC protocol will be described in detail in Section IV-B. EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 9
A. Throughput–Delay Trade-off in Dense Networks
In this subsection, we show the throughput–delay trade-off achieved by the MH andnetwork-decomposed HC protocols in a dense network. The number of S–D lines passingthrough each cell in the MH protocol is first specified in the following lemma.
Lemma 3:
For a ( n ) = Ω(log n/n ) , the total number of S–D lines passing through each cellis O ( E [ d s,v ] n p a ( n )) , where E [ d s,v ] is the average distance of an S–D pair. Proof:
By generalizing the arguments in [40, Lemma 3] to the case of an arbitraryaverage distance of an S–D pair, one can easily show that the total number of S–D linespassing through each cell is given by O ( E [ d s,v ] n p a ( n )) . This completes the proof of thislemma.The main result achieved by the MH protocol in a dense network is shown in the followingtheorem. Theorem 1:
The throughput–delay trade-off for the MH protocol in a wireless dense networkadopting the social formation model is given by ( T ( n ) , D ( n )) = (cid:18) Θ (cid:18) √ a ( n ) (cid:19) , Θ (cid:18) √ a ( n ) (cid:19)(cid:19) for Regime A (cid:18) Θ (cid:18)(cid:16) n log n (cid:17) γ − √ a ( n ) (cid:19) , Θ (cid:18)(cid:0) log nn (cid:1) γ − √ a ( n ) (cid:19)(cid:19) for Regime B (cid:18) Θ (cid:18)q na ( n ) log n (cid:19) , Θ (cid:16)q log na ( n ) n (cid:17)(cid:19) for Regime C , (4)where the area of each square cell is given by a ( n ) = Ω(log n/n ) . Proof:
Using Lemmas 1 and 3, the number of S–D lines passing through each cell, u ( n ) ,is given by u ( n ) = O (cid:16) n p a ( n ) (cid:17) for Regime A O (cid:16)(cid:0) log nn (cid:1) γ − n p a ( n ) (cid:17) for Regime B O (cid:18)q log nn n p a ( n ) (cid:19) for Regime C . Due to the fact that the total traffic through each cell is the traffic generated by all the S–Dlines passing through the cell and the cell throughput is
Θ(1) under the TDMA operation,the aggregate throughput T ( n ) is lower-bounded by T ( n ) = Θ (cid:18) nu ( n ) (cid:19) = Θ (cid:18) √ a ( n ) (cid:19) for Regime A Θ (cid:18)(cid:16) n log n (cid:17) γ − √ a ( n ) (cid:19) for Regime B Θ (cid:18)q n log n √ a ( n ) (cid:19) for Regime C . (5)We now turn to computing the average packet delay. Since each hop covers a distance of Θ( p a ( n )) and the delay is at most a constant times the number of hops, from Lemma 1, the EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 10 delay D ( n ) is given by D ( n ) = Θ E [ d s,v ] p a ( n ) ! = Θ (cid:18) √ a ( n ) (cid:19) for Regime A Θ (cid:18)(cid:0) log nn (cid:1) γ − √ a ( n ) (cid:19) for Regime B Θ (cid:18)q log nn √ a ( n ) (cid:19) for Regime C . (6)Hence, using (5) and (6) finally leads to (4), which completes the proof of this theorem.From the above result, the following observations can be found according to each operatingregime. In Regime A (i.e., the low social group density regime), the throughput–delay trade-off for the MH protocol is the same as that in [40], in which no social group exists. This isbecause small values of γ correspond to the case where the social group members are widelydistributed over the whole network and hence the average distance of an S–D pair is notreduced compared to the network case with no social group. In Regime B, as γ increases, thethroughput is improved and the delay is reduced by virtue of the decreased average distanceof an S–D pair. In Regime C, the maximum throughput Θ( n ) can be achieved by only using Θ(1) hops. This is consistent with the throughput scaling result according to the social groupdensity in [21].
Remark 1:
In our dense network, the throughput–delay trade-off for the MH protocol isillustrated in Fig. 2 when q = Θ(1) . The red arrows in Fig. 2 represent the throughput–delayscaling results achieved by the MH protocol according to different values of γ . More specifi-cally, when ≤ γ < (Regime A), the trade-off is given by ( T ( n ) , D ( n )) = (cid:18) √ a ( n ) , √ a ( n ) (cid:19) for a ( n ) = Ω (cid:0) log nn (cid:1) and a ( n ) = O (1) . In Regime B, as an exemplary value of γ , the trade-off for γ = is illustrated in the figure for a ( n ) = Ω (cid:0) log nn (cid:1) and a ( n ) = O (cid:16) √ n (cid:17) , where ( T ( n ) , D ( n )) = (cid:18) Θ (cid:18)(cid:16) n log n (cid:17) √ a ( n ) (cid:19) , Θ (cid:18)(cid:0) log nn (cid:1) √ a ( n ) (cid:19)(cid:19) . When γ > (Regime C),the trade-off is given by ( T ( n ) , D ( n )) = (cid:16) Θ (cid:16) n log n (cid:17) , Θ(1) (cid:17) for a ( n ) = Θ (cid:0) log nn (cid:1) , whichcorresponds to a single point in the figure. As seen in Fig. 2, the throughput–delay trade-offgets improved significantly as the social group density γ increases. Remark 2:
To better understand the above trade-off between throughput and delay for theMH protocol in the dense network, we express T ( n ) as a function of D ( n ) as follows: T ( n ) = Θ ( D ( n )) for Regime A Θ ( D ( n ) n γ − − ǫ ) for Regime B Θ ( D ( n ) n − ǫ ) for Regime C (7)for an arbitrarily small ǫ > . It is again observed that the throughput T ( n ) given the delay D ( n ) can be greatly improved as γ increases.We modify the HC protocol in [42] so that the network under our social formation modeloperates properly, which will be explained in detail in Section IV-B. We herein summarizeour main results for the proposed network-decomposed HC protocol when the notion of socialrelationships is incorporated into ad hoc networks. The main results for a dense network areshown as follows. EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 11 ( )
D n ( )
T n n n n n n n g > MH g = g£ < HC(Regime A)(Regime B) (Regime C)HCMHMH HC
Fig. 2. The throughput–delay trade-off in a dense network when q = Θ(1) . Red and blue arrows denote the trade-offs forthe MH and network-decomposed HC protocols, respectively. Black lines indicate the best trade-offs achieved by one ofthese two protocols. The factor of log n is omitted for simplicity. Theorem 2:
The throughput–delay trade-off for the network-decomposed HC protocol in awireless dense network adopting the social formation model is given by ( T ( n ) , D ( n )) = (cid:0) Θ (cid:0) n b − ǫ (cid:1) , Θ (cid:0) n b + ǫ (cid:1)(cid:1) for Regime A (cid:0) Θ (cid:0) n b − ( γ − b − − ǫ (cid:1) , Θ (cid:0) n (3 − γ ) b + ǫ (cid:1)(cid:1) for Regime B (Θ ( n − ǫ ) , Θ ( n ǫ )) for Regime C (8)for an arbitrarily small ǫ > , where ≤ b < . Proof:
See Appendix A.From Theorem 2, the following insightful observations are found according to each operat-ing regime. In Regime A, it is shown that the throughput–delay trade-off derived along withthe social formation model is the same as that with no social behavior. In other words, thenetwork behaves as if there is no social group when the social group density γ is small. InRegime B, as γ increases, the throughput is improved and the delay is reduced. In Regime C,the maximum throughput Θ( n ) is achieved with a very small delay. These observations aresimilar to those for the MH case. The throughput–delay trade-off for the network-decomposedHC protocol is illustrated in Fig. 2 (see blue arrows). Remark 3:
When < γ < (Regime A), the throughput–delay trade-off for the network-decomposed HC protocol is given by ( T ( n ) , D ( n )) = (Θ( n b − ǫ ) , Θ( n b + ǫ )) for ≤ b < . InRegime B, the trade-off achieved by the network-decomposed HC protocol for γ = (as anexemplary value of γ ) is given by ( T ( n ) , D ( n )) = (Θ( n ( b +1) / − ǫ ) , Θ( n b/ ǫ )) for ≤ b < .When γ > (Regime C), the trade-off for the network-decomposed HC protocol is given by ( T ( n ) , D ( n )) = (Θ( n − ǫ ) , Θ( n ǫ )) . From Fig. 2, it is seen that for ≤ b ≤ , performanceon the trade-off for the MH and network-decomposed HC protocols is the same, but for < b < , the network-decomposed HC protocol has a higher throughput than that of theMH protocol at the expense of an increased delay. In the dense network, the best trade-offscan thus be achieved by the network-decomposed HC protocol and are depicted by blacklines in Fig. 2. EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 12
Remark 4:
To better understand the above trade-off between throughput and delay for thenetwork-decomposed HC protocol in the dense network, we express T ( n ) as a function of D ( n ) as follows: T ( n ) = Θ ( D ( n ) n − ǫ ) for Regime A Θ ( D ( n ) n γ − − ǫ ) for Regime B Θ ( D ( n ) n − ǫ ) for Regime C (9)for an arbitrarily small ǫ > . It is observed that the throughput–delay trade-off in (9) for thenetwork-decomposed HC protocol is the same as (7) for the MH protocol within a factor of n ǫ as long as the delay D ( n ) scales up to its maximum achieved by MH (refer to Fig. 2). Itmeans that the inherent relation between the two protocols with respect to the throughput–delay trade-off remains the same, but a general trade-off can be achieved by incorporatingthe results for the network-decomposed HC protocol. B. Throughput–Delay Trade-off in Extended Networks
We next show the throughput–delay trade-off of an extended network. The trade-off achievedby the MH protocol is first presented in the following theorem.
Theorem 3:
The throughput–delay trade-off for the MH protocol in a wireless extended network adopting the social formation model is given by ( T ( n ) , D ( n )) = (cid:18) Θ (cid:18)q na ( n ) α +1 (cid:19) , Θ (cid:18)q na ( n ) (cid:19)(cid:19) for Regime A (cid:18) Θ (cid:18)(cid:16) n log n (cid:17) γ − q na ( n ) α +1 (cid:19) , Θ (cid:18)(cid:0) log nn (cid:1) γ − q na ( n ) (cid:19)(cid:19) for Regime B (cid:18) Θ (cid:18) n √ a ( n ) α +1 log n (cid:19) , Θ (cid:16)q log na ( n ) (cid:17)(cid:19) for Regime C(10)where the area of each square cell is given by a ( n ) = Ω(log n ) . Proof:
Similarly as in the proof of Theorem 1, the throughput–delay trade-off for theMH protocol in an extended network can easily be proved by computing the received signalpower that is expressed in a different manner due to the power limitation as well as usingthe fact that the area of each cell, a ( n ) , is increased by a factor of n compared to a densenetwork. Since the received signal-to-noise ratio (SNR) for the signal transmitted from thenearest-neighbor cell scales as (cid:18) √ a ( n ) (cid:19) α , we obtain the throughput result in (10) by using(4), which completes the proof of this theorem.In the extended network, the throughput–delay trade-off for the MH protocol is illustrated inFig. 3 when q = Θ(1) and α ∈ { , } . The red arrows in the figure represent the throughput–delay scaling results achieved by the MH protocol according to different values of γ . Remark 5:
To better understand the above trade-off between throughput and delay for theMH protocol in the extended network, we express T ( n ) as a function of D ( n ) as follows: T ( n ) = Θ (cid:0) D ( n ) a ( n ) − α/ (cid:1) for Regime A Θ (cid:0) D ( n ) n γ − − ǫ a ( n ) − α/ (cid:1) for Regime B Θ (cid:0) D ( n ) n − ǫ a ( n ) − α/ (cid:1) for Regime Cfor an arbitrarily small ǫ > . Compared to the dense network case (see Remark 2), thethroughput T ( n ) given the delay D ( n ) is reduced by a factor of a ( n ) − α/ , which correspondsto the amount of SNR loss. EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 13 ( )
D n ( )
T n n n n n n n MH HC n MH HC HCMH(Regime A) (Regime C)(Regime B) g = g£ < g > n (a) α = 2 ( ) D n ( )
T n n n n n n n MHHC n MH HC HCMH (Regime C) g > (Regime B) g = (Regime A) g£ < n n (b) α = 3 Fig. 3. The throughput–delay trade-off in an extended network when q = Θ(1) . Red and blue arrows denote the trade-offsfor the MH and network-decomposed HC protocols, respectively. Black lines indicate the best trade-offs achieved by oneof these two protocols. The factor of n ǫ is omitted for simplicity. Now, let us turn to summarizing our main results for an extended network when theproposed network-decomposed HC protocol is employed.
Theorem 4:
The throughput–delay trade-off for the network-decomposed HC protocol in awireless extended network adopting the social formation model is given by ( T ( n ) , D ( n )) = (cid:0) Θ (cid:0) n b − α/ − ǫ (cid:1) , Θ (cid:0) n b + ǫ (cid:1)(cid:1) for Regime A (cid:0) Θ (cid:0) n ( b − α/ − γ )+1 − ǫ (cid:1) , Θ (cid:0) n (3 − γ ) b + ǫ (cid:1)(cid:1) for Regime B (Θ ( n − ǫ ) , Θ ( n ǫ )) for Regime C (11)for an arbitrarily small ǫ > , where ≤ b < . Proof:
See Appendix B.In contrast to the results for the dense network in (8), the throughput scaling achieved bythe network-decomposed HC protocol in the extended network depends highly on the path-
EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 14 loss exponent α in Regimes A and B (i.e., the low and medium social group density regimes,respectively), while the delay scaling is the same for both dense and extended networks. Thisis because in the above two regimes, a bursty transmission whose duration depends on α isneeded due to the power limitation, which will be explained in detail in Section IV-C. Remark 6:
From Theorem 4, T ( n ) for the network-decomposed HC ptorocol in an extendednetwork is expressed as a function of D ( n ) as follows: T ( n ) = Θ (cid:0) D ( n ) n − α/ − ǫ (cid:1) for Regime A Θ (cid:0) D ( n ) n α ( γ − / − ǫ (cid:1) for Regime B Θ ( D ( n ) n − ǫ ) for Regime Cfor an arbitrarily small ǫ > .In the extended network, the throughput–delay trade-off for the network-decomposed HCprotocol is also illustrated in Fig. 3 when q = Θ(1) and α ∈ { , } (see blue arrows), andthe following interesting observations in comparison with the results for the MH protocol aremade according to each operating regime. Remark 7:
First, it is shown in Fig. 3 that the trade-offs for the MH and network-decomposed HC protocols are improved as the social group density γ increases as in thedense network configuration. We now recall that in the extended network, the throughput fora given delay depends on the path-loss exponent α , unlike the dense network case. When α = 2 , the trade-off achieved by the network-decomposed HC protocol is always superiorto or equal to that of the MH protocol for all the operating regimes. On the other hand,as α increases, there exists operating regimes such that the MH protocol is dominant. Moreprecisely, in Regime A, it is not difficult to show that if α > √ , then the MH protocoloutperforms the network-decomposed HC protocol, and there is a crossover between twolines achieved by both protocols otherwise. In Regime B, the MH protocol is superior to thenetwork-decomposed HC protocol if γ ≤ α − α − α − . Otherwise, there is a crossover between two lines achieved by both protocols, where thenetwork-decomposed HC protocol has better trade-off performance in the low throughputregime as depicted in Fig. 3(b). In Regime C, both MH and network-decomposed HC protocolsachieve the best throughput of Θ( n ) with a very small delay of Θ(1) in which the social groupdensity γ becomes high, as in the dense network case.IV. T HROUGHPUT –D ELAY T RADE - OFF FOR N ETWORK -D ECOMPOSED
HCIn this section, we shall first introduce the network-decomposed HC protocol so that thenetwork under our social formation model operates suitably, and then derive its throughput–delay trade-off in a dense network. The trade-off achieved by the network-decomposed HCprotocol in an extended network is also shown through network transformation.
A. Conventional HC Protocol
We will first briefly explain the original HC protocol [30] and its modified one [42] in anad hoc network. It will be shown how the HC protocol is further modified to work properlyfor our network with the distance-based social formation model in the subsequent subsections.In the original HC protocol [30], packets are transmitted through the following three phases: • The network is divided into multiple clusters, each having M nodes, where M = O ( n ) . EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 15 • During the first phase, each source distributes its M bits to other nodes in the samecluster, one bit for each node. • During the second phase, a long-range multiple-input multiple-out (MIMO) transmissionis then performed between two clusters having a source and its destination, one at atime. • During the last phase, each node in a cluster quantizes its received observations anddelivers the quantized data to the rest of nodes in the same cluster. Each destination candecode its packets by collecting all quantized observations.When each node transmits data within its cluster, another smaller-scaled cooperation canbe applied in the same manner by dividing each cluster into smaller ones. By recursivelyapplying this procedure, it is possible to establish the hierarchical strategy in the network.One drawback of the original HC protocol [30] is that it requires an extremely large bulk-size Θ( n h ) , i.e., the minimum number of bits that should be transmitted between each S–D pair islarge, where h > is the hierarchy level. Due to a large bulk-size and inefficient scheduling,the delay scaling of the HC protocol can be much worse than that of the MH protocol. Inorder to improve the delay performance, the original HC protocol was modified in [42] byreducing the bulk-size and enhancing scheduling. Using the modified HC protocol [42], thethroughput–delay trade-off is given by ( T ( n ) , D ( n )) = Θ( n b / log n, n b log n ) where ≤ b < . The parameter b depends on the size of a cluster with ≤ b ≤ hh +1 . B. Network-Decomposed HC Protocol
Both the original HC protocol [30] and its modified one [42] were designed assuming thatthe geographic distance of between a source node and its destination node scales as O (1) ina dense network due to the random S–D pairings. Since the distance of an S–D pair in ournetwork may scale at a lower rate than that of the network size depending on the parametersof the social formation model, the HC protocols in [30], [42] may not work effectively. Thismotivates us to further modify the HC protocol in [42] by taking into account the notion of network decomposition as in the following.
1) Network Decomposition:
To fully exploit the characteristics of our distance-based socialformation model, we modify the HC protocol so that the reduced distance of an S–D paircan be carefully incorporated into the protocol design. Owing to the central processor thatleverages geo-information of nodes, it is possible to divide a network into multiple non-overlapping subnetworks, each of which operates using the existing HC protocol in [42] witha reduced network size accordingly. When we denote d s,v as the distance between a source s and its destination v , we choose the side length l ( n ) of a subnetwork as E [ d s,v ] n ǫ for anarbitrarily small ǫ > so that s and v can coexist inside one subnetwork w.h.p., since thedistance between an S–D pair does not scale at a faster rate than E [ d s,v ] within a factor of n ǫ w.h.p. (refer to Lemma 2). The subnetworks of constant side length l ( n ) = E [ d s,v ] n ǫ (12)having S–D pairs are illustrated in Fig. 4. There are m := nl ( n ) nodes on average in eachsubnetwork, where the HC protocol in [42] is employed within each subnetwork.
2) Edge Node Problem and Protocol Refinement:
When the whole network is decomposedinto multiple subnetworks as in Section IV-B1, there may exist some sources in a subnetwork,whose destinations are outside of the subnetwork, as depicted by red circles in Fig. 4. Packets
EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 16 l ( n ) l ( n ) Fig. 4. The illustration of subnetworks for the network-decomposed HC protocol, where a subnetwork is depicted as abox, a source or destination node is denoted by a circle, and the line between two circles indicates the relationship of S–Dpairings. The red circle represents the node whose transmission is performed across the adjacent subnetwork. l ( n ) l ( n ) l ( n )/2 (a) Shifted to the right by l ( n ) / l ( n ) l ( n ) l ( n )/2 (b) Shifted up by l ( n ) / l ( n ) l ( n ) (c) Shifted diagonally by √ l ( n ) / Fig. 5. The 4-TDMA operation of our network-decomposed HC protocol. of these sources in the subnetwork cannot be delivered to the corresponding destinationsusing the existing HC framework that should be employed only inside each subnetwork. Inorder to solve this edge node problem, we apply a 4-TDMA operation with which all thesubnetworks are shifted to the right, shifted up by l ( n ) / , and shifted diagonally by √ l ( n ) / in consecutive time slots (see Fig. 5). By doing so, the packets of the vast majority of S–Dpairs can be successfully delivered in one of time slots. Note that this subnetwork shift basedon the 4-TDMA strategy does not fundamentally change the throughput and delay scaling. Ineach time slot, the subnetworks operate in parallel using the HC protocol with Θ( m ) nodesin each subnetwork.
3) Network-Decomposed HC Protocol:
We assume that packets are conveyed by the network-decomposed HC protocol only when the geographic distance between a source node and itsdestination node is shorter than Θ( l ( n )) without affecting the overall throughput–delay trade-off. By recalling Lemma 2, we note that the average number of S–D pairs whose distanceis less than Θ( l ( n )) is given by Θ( n ) (refer to Remark 9 for more details). According toSections IV-B1 and IV-B2, the proposed network-decomposed HC protocol is described asfollows: • Step 1) Our ad hoc network is divided into multiple non-overlapping subnetworks, each
EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 17 of which has size l ( n ) × l ( n ) . • Step 2) The HC protocol in [42] is employed in each subnetwork. • Step 3) All the subnetworks are shifted to the right by l ( n ) / compared to their originalpositions, and the HC protocol is employed in each shifted subnetwork. • Step 4) All the subnetworks are shifted up by l ( n ) / compared to the original positions,and the HC protocol is employed in each shifted subnetwork. • Step 5) All the subnetworks are shifted diagonally by √ l ( n ) / compared to the originalpositions, and the HC protocol is employed in each shifted subnetwork. • Steps 2), 3), 4), and 5) are repeated in a TDMA manner.Under the network-decomposed HC protocol, the throughput–delay trade-off of a wireless dense network adopting the social formation model is given by the expression in (8) (refer toTheorem 2). It is shown that the throughput–delay trade-off for the network-decomposed HCprotocol is improved as the social group density γ increases, since the number of subnetworksthat are activated in parallel becomes large with increasing γ . Remark 8:
In the original HC protocol, the average per-node transmit power required torun the HC protocol in dense networks is Pn but not P . This is because when the Θ( n ) × Θ( n ) long-range MIMO transmission is performed at the top level of the hierarchy, the averagedistance between two clusters (i.e., the transmitting nodes and the receiving nodes) is O (1) and an array gain of n can be obtained. On the other hand, when the network-decomposedHC protocol is employed in our dense network, the average per-node transmit power becomes Pm l ( n ) α = Pn l ( n ) α − (13)due to the fact that the network-decomposed HC protocol operates within a subnetwork of m nodes and the average distance between two clusters is given by O ( l ( n )) . Remark 9:
Our network-decomposed HC protocol operates within each subnetwork inparallel. However, from Lemma 2, there exists a non-zero probability that a relatively smallfraction of S–D pairs are further apart than the size of the subnetwork. In our work, we assumethat the packets of such S–D pairs are conveyed by the MH protocol. It is not difficult toshow that the resulting throughput–delay trade-off does not change at all by performing MHfor such pairs.
C. Extended Network Configuration
In the previous subsection, we have focused on the analysis for a dense network of unitarea. We now turn to analyzing an extended network of unit node density, whose size is √ n × √ n . Since the distance between a source and its destination is increased by a factorof √ n in the extended network, the received signal power will be decreased by a factor of n α/ . Hence, the extended network of size √ n × √ n can be treated as a dense network ofsize × with the reduced average per-node transmit power constraint of Pn α/ . Thus, if theaverage per-node transmit power required to perform the network-decomposed HC protocolin the dense network is bounded by Pn α/ , then the same throughput–delay trade-off for thenetwork-decomposed HC protocol can be achieved as in the dense network case. Since theaverage per-node transmit power depends on side length l ( n ) in (13), which varies accordingto the operating regimes, we compare the average per-node transmit power with Pn α/ in eachregime as follows. Note that this network transformation along with scaling parameters γ and α is not a straightforward extension of [30].In Regime A (i.e., the low social group density regime), using Lemma 1, (12), and (13), theaverage per-node transmit power for the network-decomposed HC protocol becomes Pn − ǫ in EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 18 our dense network. In order to satisfy the equivalent power constraint Pn α/ , we need a burstytransmission strategy similarly as in [30]. Specifically, the network-decomposed HC protocolis performed during a fraction n α/ − ǫ of the time with per-node power Pn − ǫ and remainssilent for the rest of the time. Then, the throughput is reduced to T ( n ) = Θ (cid:0) n b − α/ − ǫ (cid:1) whilethe delay is the same as that in the dense network. In Regime B (i.e., the medium social groupdensity regime), the average per-node transmit power required to run the network-decomposedHC protocol becomes Pn γ/ − α − − ǫ in the dense network due to the fact that l ( n ) dependson the social group density γ . In the regime, we need to use a bursty transmission that runsthe network-decomposed HC protocol during a fraction n (1 − α/ γ − ǫ of the time, resultingin T ( n ) = Θ (cid:0) n b − ( γ − b − − (1 − α/ γ − − ǫ (cid:1) with the same delay as in the dense network. InRegime C (i.e., the high social group density regime), the average per-node transmit power is Pn α/ − ǫ in the dense network due to l ( n ) = Θ( n − / ǫ ) . Hence, the network-decomposed HCprotocol runs during a fraction n − ǫ . The throughput is then given by T ( n ) = Θ( n − ǫ ) withthe same delay as in the dense network.Based on this bursty modification according to each operating regime, we can establish thethroughput–delay trade-off in (11) for the network-decomposed HC protocol in the extendednetwork with our social formation model (see Theorem 4). Compared to the dense networkcase, the throughput–delay trade-offs in Regimes A and B are degraded as the path-lossexponent α increases since the extended network is power-limited in these two regimes. Onthe other hand, in Regime C, the network is not power-limited due to the high social groupdensity, where the same throughput–delay trade-off can be achieved as in the dense networkcase. V. N UMERICAL E VALUATION
In this section, we perform computer simulations according to finite values of n to validatethat our analytical results show trends consistent with numerical results. We evaluate theperformance of the network-decomposed HC protocol in terms of the total throughput T ( n ) ,which can be computed as the sum of the transmission rates in parallel across all subnetworks.The capacity of each subnetwork is bounded by the rate of long-range MIMO transmissionbetween two clusters having a source and its destination in a subnetwork. A sufficient numberof nodes ( n = 256 ) are deployed so that a large-scale network is suitably modeled in practice.It is assumed that the network size is given by × (m ). All the transmit power is setto the same value as the noise variance.Now, we turn to describing how to generate multiple network configurations, each of whichhas different numbers of subnetworks, depending on values of the social group density γ . Let b γ denote the parameter b in Theorems 2 and 4 for given γ . In our simulations, we set b = .Then, from the results of the two theorems, we have b γ = b − γ = − γ ) to ensure that thedelays for all values of γ are the same in order sense. From the fact that the side length of asubnetwork in (12) is given by E [ d s,v ] in (3) for given γ , it follows that four values , . , log , and . for γ are chosen to generate such network configurations that have 1, 4,9, and 16 subnetworks, respectively. Note that γ = 2 corresponds to the baseline employingthe conventional HC protocol since there exists a single subnetwork.In Fig. 6, the total throughput versus the social group density γ is illustrated, where α ∈ { , . , } . From the figure, the following observations are made under our simulationenvironments: the throughput performance is degraded with increasing path-loss exponent α due to more severe path-loss attenuation; the total throughput is gradually enhanced as γ increases up to 2.4; and the total throughput is greatly improved when γ = 2 . since EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 19
The social group density γ T he t o t a l t h r oughpu t ( bp s / H z ) α =2.0 α =2.5 α =3.0 Fig. 6. The total throughput according to the social group density γ , where α ∈ { . , . , . } . the number of subnetworks is largely increased as the distance between a source and itsdestination, E [ d s,v ] , decreases with respect to γ . This result implies that the social behavioramong nodes in ad hoc networks has a significant impact on the throughput performance.VI. C ONCLUDING R EMARKS
By introducing a new HC protocol, we completely characterized the general throughput–delay trade-off of dense and extended ad hoc networks with the distance-based social forma-tion model parameterized by the social group density γ and the number of social contactsper node, q , where a source selects one of its social contacts as its destination uniformlyat random. More precisely, we proposed the network-decomposed HC protocol so that thenetworks operate properly under our social formation model in terms of maximizing thethroughput–delay trade-off. To more concisely show our main results, we also identified threeoperating regimes on the throughput–delay trade-off with respect to γ and q . In the densenetwork, we showed that when γ is small, the throughput–delay trade-off is the same asthe non-social behavior scenario; on the other hand, when γ increases, the throughput–delaytrade-off is significantly improved; and when γ becomes large, the maximum throughput Θ( n ) can be achieved via a single-hop transmission, leading to the delay of Θ(1) . In addition, weanalyzed the corresponding throughput–delay trade-off in the extended network through thenontrivial network transformation strategy, and investigated the operating regimes such thatthe network-decomposed HC protocol outperforms the MH protocol according to parameters γ and α . Suggestions for further research include analyzing performance on the throughputand delay when the size of social groups follows a well-known Zipf’s distribution instead ofassuming the same size of all social groups. Another interesting direction is to investigate theeffect of queuing delay on the throughput–delay trade-off for the network-decomposed HCprotocol in ad hoc networks with social relationships. EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 20 A PPENDIX AP ROOF OF T HEOREM m = nl ( n ) nodes is given by [42] ( T ( n ) , D ( n )) = Θ( m b / log m, m b log m ) , where ≤ b < . Since there are nm subnetworks over the whole network, the aggregatethroughput is nm times the throughput of each subnetwork while the delay of the networkremains the same due to the fact that all the subnetworks operate in parallel. Hence, thethroughput–delay trade-off of the network is given by ( T ( n ) , D ( n )) = Θ( m b − n/ log m, m b log m ) , (A.1)where ≤ b < . Substituting m = n E [ d s,v ] n ǫ into (A.1), where d s,v denotes the distancebetween a source s and its destination v , we have ( T ( n ) , D ( n )) = Θ( n b ( E [ d s,v ] n ǫ ) b − / log( n ( E [ d s,v ] n ǫ ) ) ,n b ( E [ d s,v ] n ǫ ) b log( n ( E [ d s,v ] n ǫ ) )) . In what follows, we derive the throughput and the delay according to each operating regime.
A. Regime A
For q ( n ) = ω (1) , since E [ d s,v ] = Θ(1) from Lemma 1, the throughput T ( n ) and the delay D ( n ) are given by ( T ( n ) , D ( n )) = Θ( n b + ǫ (2 b − / log n ǫ , n b +2 ǫb log n ǫ )= Θ( n b − ǫ , n b + ǫ ) . For q = Θ(1) and ≤ γ < , we have the same result as above due to E [ d s,v ] = Θ(1) . B. Regime B
For q = Θ(1) and ≤ γ ≤ , it follows that E [ d s,v ] = Θ (cid:16)(cid:0) log nn (cid:1) γ − (cid:17) from Lemma 1.Thus, we have T ( n ) = Θ n b (cid:18) log nn (cid:19) γ − n ǫ ! b − (cid:18) n (cid:16)(cid:0) log nn (cid:1) γ − n ǫ (cid:17) (cid:19) ,D ( n ) = Θ n b (cid:18) log nn (cid:19) γ − n ǫ ! b log n (cid:18) log nn (cid:19) γ − n ǫ ! , which can be simplified to ( T ( n ) , D ( n )) = (Θ (cid:0) n b − ( γ − b − − ǫ (cid:1) , Θ (cid:0) n (3 − γ ) b + ǫ (cid:1) ) . EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 21
C. Regime C
For q = Θ(1) and γ > , it follows that E [ d s,v ] = Θ (cid:18)q log nn (cid:19) from Lemma 1. Thus, wehave T ( n ) = Θ n b r log nn n ǫ ! b − n (cid:18)q log nn n ǫ (cid:19) ! D ( n ) = Θ n b r log nn n ǫ ! b log n r log nn n ǫ ! , which can be rewritten as ( T ( n ) , D ( n )) = (Θ ( n − ǫ ) , Θ ( n ǫ ) . In consequence, the throughput–delay trade-off for the network-decomposed HC protocol in the dense network is given bythe expression in (8), which completes the proof of this theorem.A PPENDIX BP ROOF OF T HEOREM Pn α/ . Note that from (13),the average per-node transmit power required for the network-decomposed HC protocol inthe dense network is Pm l ( n ) α = Pn l ( n ) α − (but not P ). We will compare this required powerwith the average per-node power constraint of Pn α/ in each regime. A. Regime A
For q ( n ) = ω (1) or { q = Θ(1) and ≤ γ < } , the network-decomposed HC protocol isperformed with the average per-node transmit power constraint of Pn l ( n ) α − = Pn ( n ǫ ) α − . Weuse a bursty transmission strategy similarly as in [30] in order to satisfy the correspondingpower constraint Pn α/ , where the network-decomposed HC protocol is performed during afraction n α/ − ǫ of the time with per-node power Pn − ǫ and remains silent for the rest of thetime. The throughput T ( n ) and the delay D ( n ) are then given by T ( n ) = Θ (cid:0) n b − α/ − ǫ (cid:1) and D ( n ) = Θ (cid:0) n b + ǫ (cid:1) , respectively. B. Regime B
For q = Θ(1) and ≤ γ ≤ , we use a bursty transmission that runs the network-decomposed HC protocol during a fraction n (1 − α/ γ − ǫ of the time. In this case, we have ( T ( n ) , D ( n )) = (cid:0) Θ (cid:0) n ( b − α/ − γ )+1 − ǫ (cid:1) , Θ (cid:0) n (3 − γ ) b + ǫ (cid:1)(cid:1) . C. Regime C
For q = Θ(1) and γ > , we use a bursty transmission that runs the network-decomposedHC protocol during a fraction n − ǫ , resulting in ( T ( n ) , D ( n )) = (Θ ( n − ǫ ) , Θ ( n ǫ )) .In consequence, the throughput–delay trade-off for the network-decomposed HC protocolin the extended network is given by the expression in (11), which completes the proof of thistheorem. EEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 22 R EFERENCES [1] B. Latan´e, J. H. Liu, A. Nowak, M. Bonevento, and L. Zheng, “Distance matters: Physical space and social impact,”
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