Capacity and Delay Scaling for Broadcast Transmission in Highly Mobile Wireless Networks
CCapacity and Delay Scaling for BroadcastTransmission in Highly Mobile Wireless Networks
Rajat Talak, Sertac Karaman, and Eytan Modiano
Abstract —We study broadcast capacity and minimum delayscaling laws for highly mobile wireless networks, in which eachnode has to disseminate or broadcast packets to all other nodes inthe network. In particular, we consider a cell partitioned networkunder the simplified independent and identically distributed (IID)mobility model, in which each node chooses a new cell at randomevery time slot. We derive scaling laws for broadcast capacityand minimum delay as a function of the cell size. We propose asimple first-come-first-serve (FCFS) flooding scheme that nearlyachieves both capacity and minimum delay scaling. Our resultsshow that high mobility does not improve broadcast capacity,and that both capacity and delay improve with increasing cellsizes. In contrast to what has been speculated in the literaturewe show that there is (nearly) no tradeoff between capacity anddelay. Our analysis makes use of the theory of Markov EvolvingGraphs (MEGs) and develops two new bounds on flooding timein MEGs by relaxing the previously required expander propertyassumption.
I. I
NTRODUCTION
We study all-to-all broadcast capacity and delay scalingbehavior in mobile wireless networks. Interest in mobilewireless networks has increased in recent years due to theemergence of autonomous aerial vehicle (UAV) networks.Dense networks of small UAVs are being used in a widerange of applications including product delivery, disaster andenvironmental monitoring, surveillance, and more [2]–[6]. Ourwork is motivated by the need to disseminate timely controlinformation in such networks [5]–[8]. An important commu-nication operation that needs to be performed in exchangingsafety critical information is that of all-to-all broadcast, whereeach vehicle or node broadcasts its current state or locationinformation to all other vehicles in its vicinity.We consider a cell partitioned network with N nodes, shownin Figure 1, in which a unit square is partitioned into C cells.Due to interference, only a single packet transmission cantake place in the cell at a given time, and all other nodesin the cell can correctly receive the packet. Different cells canhave simultaneous packet transmissions. This simple modelcaptures the essential features of interference and helps obtainkey insights into its impact on throughput and delay [9]–[11].We consider IID mobility, where, at the end of every slot, eachnode chooses a new cell uniformly at random. This mobilitymodel was used in [9], [12] to capture the impact of highmobility, and the resultant intermittent network connectivity,on throughput and delay. Moreover, this model serves as a The authors are with the Laboratory for Information and Decision Systems(LIDS) at the Massachusetts Institute of Technology (MIT), Cambridge, MA. { talak, sertac, modiano } @mit.edu Fig. 1. Network partitioned into C = a N cells. Each cell of area a N . good model for UAV networks where rapid mobility andintermittent connectivity are common [5]–[7].We study all-to-all broadcast capacity and delay scaling asa function of node density. Here, capacity is defined as themaximum rate at which each node can transmit packets to allother nodes in the system and delay as the average time takenby a packet to reach every node in the system. We say that anetwork is dense if the number of vehicles or nodes per cell isincreasing with N , and sparse otherwise. Thus, if the cell sizegrows as cN − α , for some c > , then the network is dense for < α < and sparse for α ≥ .We show that as the network gets more dense the all-to-allbroadcast capacity increases to reach a maximum scaling of /N . Interestingly, delay decreases as the network gets denser.In fact, both, capacity and delay attain their best scaling in N when the cell size is just smaller than order /N , i.e., when α = 1 − (cid:15) for a small positive (cid:15) . We further note that thebest per-node capacity scaling of /N is the same as thatcan be achieved in a static wireless network, thus, mobilitydoes not improve network capacity. This is in contrast to theunicast case where it was shown in [13] that mobility improvescapacity. Our scaling results are summarized in Table I.We propose a simple first-come-first-serve (FCFS) floodingscheme that achieves capacity scaling, up to a log N factorfrom the optimal when the network is sparse and up to a log log N factor from the optimal when the network is dense.The FCFS flooding scheme also achieves the minimum delayscaling when the network is sparse, and up to a factor of log log N from minimum delay when the network is dense. a r X i v : . [ c s . I T ] A ug ABLE IC
APACITY AND A VERAGE D ELAY
Capacity
Upper bound FCFS flooding(Theorem 1) (Eqn. (53))Sparse: α ≥ N α N α N Dense: < α < N N N Average Delay
Lower bound FCFS flooding(Theorem 2) (Eqns. (54) and (52))Sparse: α ≥ N α − log N N α − log N Dense: < α < N Thus, nearly optimal throughput and delay scaling is achievedsimultaneously.The IID mobility model was analyzed for unicast and mul-ticast operations in [9] and [12], respectively, using standardprobabilistic arguments. In contrast, we use the abstraction ofMarkov evolving graphs (MEG), and flooding time bounds forMEGs [14]. An MEG is a discrete time Markov chain withstate space being a collection of graphs with N nodes. AnMEG of the IID mobility model can be constructed by drawingan edge between two nodes in the same cell and viewing thenetwork as a graph at each time step. Flooding time, is then,the time it takes for a single packet to reach all nodes from asingle source node.A flooding time bound for MEGs was derived in [14]. Itrelied on an expander property which states that whenever m nodes have the packet then in the next slot at least km new nodes will receive the packet with high probability, forsome k > . However, this strong requirement does not alwayshold. For example, when the IID mobility model is sparse,this expander property cannot be guaranteed. We derive twonew bounds on flooding time in MEGs by relaxing the strongexpander property requirements imposed in [14]. These newbounds are of independent theoretical interest. This work firstappeared in MobiHoc 2017 [1]. A. Previous Work
In [8], we considered the impact of wireless interferenceconstraints on the ability to exchange timely control infor-mation in UAV networks. We showed that, in guaranteeinglocation awareness of other vehicles in the networks, wirelessinterference constraints can limit mobility of aerial vehiclesin such networks. This result motivates us to study the delayand capacity scalings of all-to-all broadcast in mobile wirelessnetworks.Broadcast has been studied before in the contexts of dis-seminating data packets in wireless ad-hoc networks [15],[16], sensor information in sensor networks, and in exchangingintermediate variables in distributed computing [17]. Scalinglaws for capacity and delay in wireless networks have receivedsignificant attention in the literature. Capacity scaling forunicast traffic, in which each node sends packets to only oneother destination node, was analyzed in [18], [19]. It wasshown that the capacity scales as / √ N log N with increasing N . Minimum delay scaling for the static unicast network wasanalyzed in [10], where it was also shown that it is not possibleto simultaneously achieve minimum delay and capacity. Thisimplied a tradeoff between capacity and delay. In [13], it wasshown that if the nodes were mobile, then a constant per nodecapacity that does not diminish with N can be achieved. Theseminal works of [18] and [13] led to the analysis of capacityand delay scaling under various mobility models includingIID [9], Markov [10], Brownian motion [20], and RandomWaypoint [21]. Capacity-delay tradeoffs were observed in eachof these settings.Broadcast has been studied in static wireless networksin [15], [16], [22], [23]. It was shown that the per-node broad-cast capacity scales as /N in static wireless networks [16].However, to the best of our knowledge, optimal delay scalingsfor static broadcast has not been analyzed. In [12], the authorsconjectured a capacity-delay tradeoff for multicast, and byimplication for broadcast as a special case, under IID mobility.However, in this paper, we show that there is nearly nocapacity-delay tradeoff for broadcast. In particular, we proposea scheme that (nearly) achieves both capacity and minimumdelay, which is up to a log log N factor when the networkis dense and up a log N factor when the network is sparse.Moreover, we show that the capacity scaling does not improvewith mobility, unlike in the unicast case [13].Although, throughput and delay scalings have been inves-tigated under various communication operations and mobilitymodels for the past 15 years, the same problem under broad-cast has not been thoroughly analyzed even for the simplestIID mobility model. In [12], delay bounds were obtained formulticast, however, these bounds are very weak when appliedto the all-to-all broadcast operation. By using and extendingthe theory of MEGs developed in [14] we are able to obtaintight bounds on delay.Flooding time bounds on MEG have been used for variousnetwork models in [14], [24], [25]. To the best of our knowl-edge, this is the first time that these techniques are being usedin the mobility setting. Moreover, the new bounds derived inSection IV could be of independent interests and can also beapplied to models considered in [14], [24], [25]. B. Organization
The paper is organized as follows. In Section II we describethe system model, and in Section III we derive bounds oncapacity and minimum delay. In Section IV, we summarizethe flooding time upper bound result of [14], and derive twonew upper bounds on flooding time for MEGs. In Section V,we apply these results to our setting and, in Section VI, weuse it to analyse the FCFS flooding scheme. We propose asingle-hop scheme in Section VII that achieves capacity for asparse network. We conclude in Section VIII.II. S
YSTEM M ODEL
Consider the network of Figure 1 with N nodes that areuniformly distributed over a unit square. The size of each cellis a N = C = cN − α , for some α > and c > .We consider alotted time system, with the duration of each slot normalizedto unity. The duration of each slot is sufficient to complete thetransmission of a single packet. We use the IID mobility modelof [9] in which each node, at the end of every slot, choosesa new cell/location uniformly at random, and independent ofother node’s locations.Packets arrive at each node according to a Poisson process,at rate λ . Note that the arrivals happen over continuous time,and therefore, two or more packets can arrive during a slot.In this paper we make extensive use of order notation. Forinfinite sequences { a N } and { b N } , a N = O ( b N ) implies lim N →∞ a N b N ≤ c for some c > and a N = Θ ( b N ) implies a N = O ( b N ) and b N = O ( a N ) . We write a N ≤ N b N if thereexists a N ≥ such that for all N ≥ N we have a N ≤ b N .Positive constants are denoted by c , c . . . .III. F UNDAMENTAL L IMITS : C
APACITY AND M INIMUM D ELAY
We now obtain upper-bound on rate λ and a lower-boundon achievable delay. A. Capacity
Each node receives an inflow of packets at rate λ , and eachof these packets have to be broadcast to all other nodes in thenetwork. A communication scheme is said to achieve a rateof λ if at this arrival rate the average number of backloggedpackets in the network does not increase to infinity. Thecapacity of the network is the maximum achievable rate. Westart with a simple upper-bound on the capacity. Theorem The achievable rate λ is bounded by λ ≤ N − (cid:16) − (1 − a N ) N − (cid:17) (1) = (cid:26) Θ (cid:0) N α (cid:1) if α ≥ (sparse) Θ (cid:0) N (cid:1) if < α < (dense) . (2) Proof:
For an intuitive argument, consider a scheme thatachieves a rate of λ . Then the average number of packetreceptions per slot must be at least N ( N − λ under thisscheme, because there are ( N − destinations for each ofthe N sources. However, the total number of receptions perslot cannot be more than the average number of nodes in eachcell, across all cells. Thus, N ( N − λ ≤ average no. receptions in each slot (3) ≈ C N (cid:88) k =2 k P [ k nodes in a cell ] (4) = 1 a N N (cid:88) k =2 k (cid:18) Nk (cid:19) a kN (1 − a N ) N − k (5) = N (cid:110) − (1 − a N ) N − (cid:111) . (6)In (4), the summation starts from k = 2 as there must be atleast two nodes in a cell to have a transmission. The above intuition turns out to be true. Scaling law of the upper boundis then obtained by substituting a N = cN − α . The completeproof is given in Appendix A.This capacity upper bound is in fact achievable. The single-hop scheme in Section VII achieves capacity when the networkis sparse and the FCFS flooding scheme in Section VI achievescapacity, up to a log log N factor, when the network is dense.Typically, one expects to have larger broadcast capacity withincreasing cell sizes, i.e., with decreasing α . A larger cell sizeimplies more nodes in a given cell, and hence, more receptionsper slot can occur by exploiting the broadcast nature of thewireless medium. Theorem 1, however, shows that the capacityremains constant at Θ (cid:0) N (cid:1) for < α < . This is because,larger cell sizes also result in fewer transmission opportunitiesin every slot due to interference. As a result capacity remainsconstant when < α < . B. Minimum Delay
Another important performance measure is the delay. Thedelay of a packet is defined as the time from the arrival of thepacket to the time the packet reaches all its N − destinationnodes. The delay of a communication scheme is the averagedelay, averaged over all packets in the network. To obtain alower-bound on the network’s delay performance we define a single packet flooding scheme that transmits a single packet toall other nodes in the network. As we show later, this lower-bound provides a fundamental limit on delay.Single packet flooding scheme: At the beginning of thefirst slot, only a single node has the packet.1) In every cell, randomly select one packet carryingnode to be the transmitter in that slot. If no suchnode exists in a cell no transmission occurs in thatparticular cell.2) In each cell, the transmitter node (if present) trans-mits the packet to all other nodes in the cell.3) If all nodes have the packet then terminate theprocess, otherwise repeat from step 1.The single packet flooding scheme is clearly the fastest wayto disseminate a packet to all nodes in the network. Hence,a lower-bounded is given by the time it takes for a singlepacket to reach all other nodes under the single packet floodingscheme.The analysis of the single packet flooding scheme relies onthe following observation: if h nodes have the packet at agiven time slot then the number of nodes that will receive thepacket in the next slot, N ( h ) , is a binomial random variableBin ( N − h, − (1 − a N ) h ) .To see this, let H = { , , . . . h } and H = { h + 1 , h +2 , . . . N } denote the set of nodes that have and do not havethe packet at a given time slot, respectively. For the node i that has not received the packet, i.e. i ∈ H , let X i be abinary valued random variable that is if node i receivesthe packet in the next slot and otherwise. The probabilityhat the node i does not receive the packet in the nextslot is the probability that no node of H lies in the samecell as node i . This happens with probability (1 − a N ) h aslocations of node’s are independent and identically distributed(i.i.d.). Hence, P [ X i = 0] = (1 − a N ) h . Also, the X i s areindependent across i ∈ H as, again, the node locations arei.i.d. and uniform. Since N ( h ) = (cid:80) i ∈ H X i the result follows.We use this to obtain a lower-bound on delay. Theorem Any achievable average delay D is lower-bounded by D ≥ (cid:26) Θ (cid:0) N α − log N (cid:1) if α ≥ (sparse) Θ (1) if < α < (dense) . (7) Proof:
As a lower-bound we compute the time it takes forthe single packet flooding scheme to terminate. Let K t denotethe number of nodes that have the packet after t slots; where K = 1 . Let T N be the flooding time, i.e., the first time when K t = N . Let A i , for ≤ i ≤ K t , be the number of newnodes to which node i transmits the packet in slot t + 1 . Wethen have K t +1 = K t + K t (cid:88) i =1 A i . (8)Since E [ A i | K t ] ≤ ( N − a N , we have E [ K t +1 | K t ] = E (cid:34) K t + K t (cid:88) i =1 A i | K t (cid:35) , (9) ≤ K t (1 + ( N − a N ) , (10)for all t ≥ . Applying this recursively, we obtain E [ K t ] ≤ (1 + ( N − a N ) t . (11)Now, using Markov inequality we have E [ T N ] ≥ t P [ T N > t ] . (12)The event { T N > t } is same as { K t < N } . Hence, we have E [ T N ] ≥ t P [ K t < N ] , (13) = t (1 − P [ K t ≥ N ]) , (14) ≥ t (cid:18) − E [ K t ] N (cid:19) , (15)where the last inequality follows from Markov inequality.Using (11), we obtain E [ T N ] ≥ t (cid:18) − N (1 + ( N − a N ) t (cid:19) , (16)for all t ≥ . Since (16) is a valid lower-bound for all valuesof t ≥ , setting t = / N log(1+( N − a N ) for α ≥ and t = / N α log(1+( N − a N ) for < α < yields the result.In Figure 2, we plot the lower-bound on average delay D asa function of α . We observe that as the network gets sparser Fig. 2. Lower bound on achievable average delay D as a function of α . the number of nodes receiving the flooded packet per celldecreases, thereby, increasing the broadcast delay. Thus, thelower-bound is a non-decreasing function of α . However, for < α < the delay bound is a constant O (1) , and remainsunchanged. Clearly, if C = 1 , i.e. if the entire network is asingle cell, then the broadcast delay will be as the packetcan reach all other nodes in a single transmission. In the nexttwo sections we show that this lower-bound on average delayis in fact achievable, up to log log N factor.IV. F LOODING T IME IN M ARKOV E VOLVING G RAPHS
In order to gains further insights into the flooding time of thepacket flooding scheme we use the theory of Markov evolvinggraphs (MEG), to help us derive the necessary upper bound onthe flooding time. We start with a brief introduction to MEGand a review of pertinent results.Let G be a family of graphs with node set [ N ] = { , , . . . N } . The Markov chain M = ( G t ) t ∈ N , where G t ∈G , with state space G is called a MEG. Note that G is afinite set. For our network model of Figure 1, if we drawedge between i and j whenever both nodes i and j lie inthe same cell, the resulting time evolving graph is an MEG.When the MEG has a unique stationary distribution we call ita stationary MEG. In this work, we assume that a stationaryMEG starts from it’s stationary distribution. The IID mobilitymodel results in one such stationary MEG, as every graphformation can follow any other in G . We now describe thesingle packet flooding scheme in MEG.Single packet flooding for a MEG: In the first slotonly a single node s has the packet, i.e. I = { s } . Here, I t ⊂ [ N ] denotes the set of nodes that have the packet attime t . In every slot t ≥ :1) Identify the neighbors of I t that are not in I t : N ( I t ) = { neighbours of I t in G t \ I t } . (17)2) Transmit the packet to each node in N ( I t ) . We,thus, have I t +1 = I t (cid:91) N ( I t ) . (18) Since the state space G is finite, it always has at least one stationarydistribution. ) If I t = [ N ] then stop, else start again from Step 1.Let T N be the flooding time , i.e., the time it takes forthis process to terminate. Note that, this scheme reduces tothe single packet flooding scheme of Section III for ournetwork model. An upper bound on flooding time was derivedin [14]. This bound depended on the MEG satisfying certainexpander properties. We summarize this result in Theorem 3,and provide two new bounds on flooding time in Theorem 4and Theorem 5.The expander property of MEG is defined in terms of theexpander property of a static graph [14]. Definition 1:
A graph G = ([ N ] , E ) is said to be ([ h , h ] , k ) -expander if for every I ⊂ [ N ] such that h < | I | ≤ h we have | N ( I ) | ≥ k | I | , (19)where N ( I ) is the set of all neighbours of nodes in I thatare not already in I .We now use this to define the expander property of MEG. Definition 2:
Stationary MEG M = ( G t ) t ∈ N is ([ h , h ] , k ) -expander with probability p if P [ G is ([ h , h ] , k ) -expander ] ≥ p. (20)If the graph is ([ h − , h ] , k ) -expander then for notationalsimplicity we say that it is ( h, k ) -expander. To show that astationary MEG is ( h, k ) -expander we have to evaluate theprobability P (cid:92) | I | = h {| N ( I ) | ≥ k | I |} . (21)The following upper bound on flooding time was derivedin [14]. Theorem [14] For a stationary MEG, if P (cid:34) s (cid:92) i =1 { G is an ([ h i − , h i ] , k i ) -expander } (cid:35) ≥ N − c N (22)for some c > , h ≤ h < h < · · · < h s = N ,a non-increasing sequence k ≥ k ≥ · · · ≥ k s > , and s ∈ { , , . . . N } then the flooding time T N = O (cid:32) s (cid:88) i =1 log ( h i /h i − )log(1 + k i ) (cid:33) , (23)with probability at least − c N for some c > . A stationary MEG may not always satisfy the expanderproperty required by (22). In such a case, we provide thefollowing two bounds for flooding time for a stationary MEG. Theorem If for every h ∈ , , . . . N − and for all I ⊂ [ N ] with | I | = h , there exists a function p ( h ) suchthat P [ N ( I ) = 1] ≥ N p ( h ) > then the flooding time T N = O (cid:32) N − (cid:88) h =1 p ( h ) (cid:33) , (24)with probability at least − e − c N for some c > . Proof:
We denote X ∼ Geo ( p ) when X is a ge-ometrically distributed random variable with parameter p ,that is, P [ X = k ] = p (1 − p ) k − for all k ≥ . Let X h ∼ Geo ( P [ N ( h ) = 1]) and Z h ∼ Geo ( p ( h )) for all h ∈ { , , . . . N − } . It is clear that X h ≤ N Z h a.s. for all ≤ h ≤ N − . If the packet transmissions were to take placeonly at the occurrences of the events { N ( h ) = 1 } , the floodingtime would be much larger, and would equal (cid:80) N − h =1 X h . Thisimplies T N ≤ N − (cid:88) h =1 X h (25)Further, since P [ N ( h ) = 1] ≥ N p ( h ) we have X h ≤ N Z h a.s. for all h . This implies T N ≤ N − (cid:88) h =1 X h ≤ N N − (cid:88) h =1 Z h . (26)Now, using the concentration bound given in Lemma 6 ofAppendix E on { Z , . . . Z N − } and substituting t = µ = (cid:80) N − h =1 1 p ( h ) we obtain P (cid:34) N − (cid:88) h =1 Z h > c µ (cid:35) ≤ (1 − p ∗ ) µ exp (cid:26) − c −
34 ( N − (cid:27) , (27)for some c ≥ , where p ∗ = min h ∈{ , ,...N − } p ( h ) . Notethat (1 − p ∗ ) µ ≤ . We, thus, have P (cid:34) N − (cid:88) h =1 Z h > c µ (cid:35) ≤ exp (cid:26) − c −
34 ( N − (cid:27) (28) = Θ (exp {− c N } ) , (29)for some positive constant c . From (26) and (29) we have P (cid:34) T N ≤ c N − (cid:88) h =1 p ( h ) (cid:35) ≥ N − exp {− c N } . (30)Notice that instead of P [ N ( I ) = 1] ≥ N p ( h ) > if wehave the condition P [ N ( I ) ≥ ≥ N p ( h ) > the same resultholds, using an identical proof.Theorem 4, does not use any expander properties of theMEG. It can happen that a stationary MEG satisfies thexpander property for some subsets I ⊂ [ N ] but not all. Inthis case Theorem 4 may not give a very tight bound. Wecan combine the ideas of Theorem 3 and 4 to establish thefollowing result. Theorem For a stationary MEG if1) there exists a s ∈ { , , . . . N } , strictly increasingsequence < h < h < · · · < h s = N , and anon-increasing sequence k ≥ k ≥ · · · ≥ k s > such that P (cid:34) s (cid:92) i =2 { G is ([ h i − , h i ] , k i ) -expander } (cid:35) ≥ N − c N , (31)for some c > ,2) for ≤ h ≤ h , for all I ⊂ [ N ] such that | I | = h we have P [ N ( I ) = 1] ≥ N p ( h ) > , (32)and3) h ≥ c log N is such that lim N →∞ h log N = ∞ , (33)then T N = O (cid:32) h (cid:88) h =1 p ( h ) + s (cid:88) i =2 log ( h i /h i − )log (1 + k i ) (cid:33) , (34)with probability at least − c /N for some c > . Proof: I t ⊂ [ N ] denotes the number of nodes that havethe packet at time t ≥ . Let T be the first time at which atleast h nodes get the packet, i.e., T = min { t ≥ || I t | ≥ h and | I | = 1 } , (35)and T N = T N − T . Clearly, T N will be less than the timeit takes for the packet to reach all nodes if the system wereto start with exactly h nodes carrying the packet, i.e., T N ≤ T (cid:48) N = min { t ≥ || I t | = N and | I | = h } . (36)Following the same arguments listed in [14] for the proof ofTheorem 3, while using the expander property (31), we have T (cid:48) N = O (cid:32) s (cid:88) i =2 log ( h i /h i − )log (1 + k i ) (cid:33) , (37)with probability at least − c /N for some c > .Following the same arguments in the proof of Theorem 4,while using (32), yields T = O (cid:32) h (cid:88) h =1 p ( h ) (cid:33) , (38) with probability at least − exp {− c h } for some c > .From (33), it is clear that h > γ log N for any γ > . Thisimplies − exp {− c h } ≥ − exp {− c γ log N } , (39) ≥ − N c γ , (40)for any γ > . Choosing any γ ≥ /c yields T = O (cid:32) h (cid:88) h =1 p ( h ) (cid:33) , (41)with probability at least − c /N for some c > . We knowthat T N ≤ T + T (cid:48) N . Using (37) and (41) we obtain thedesired result.The results also hold if we replace the condition P [ N ( I ) = 1] ≥ N p ( h ) > with P [ N ( I ) ≥ ≥ N p ( h ) > . (42)Theorems 3, 4, and 5 give a high probability upper boundon flooding time, and not an upper bound on average floodingtime. In the next section we apply these results to obtain ahigh probability upper bound on flooding time for our networkmodel, and show that it nearly scales as the lower bound onaverage flooding time obtained in Theorem 2 of Section III.In Section VI, we use this fact to propose a FCFS floodingscheme that achieves the high probability upper bound as itsaverage delay.V. F LOODING T IME FOR THE
IID M
OBILITY M ODEL
We now apply the high probability upper bounds on floodingtime from Theorems 3, 4, and 5 of Section IV to our networkmodel. As to which of the three results we use depends onwhether the network is sparse or dense. Let M denote thestationary MEG for our network model of Figure 1, and let G be it’s stationary distribution. Theorem The flooding time is T N = (cid:26) O (cid:0) N α − log N (cid:1) if α ≥ (sparse) O (log log N ) if < α < (dense) , (43)with probability at least − c N for some c > . Proof:
We derive this by showing the expander propertiesof the network M . We split the proof into three cases: <α < , ≤ α < , and α ≥ .1) < α < : In this case, the expander properties ofTheorem 3 hold. Note that E [ N ( h )] = ( N − h ) (cid:104) − (1 − c/N α ) h (cid:105) . (44)It is also easy to see that − (1 − c/N α ) h = Θ ( h/N α ) if h/N α → , and − (1 − c/N α ) h = Θ(1) if h/N α → . When h/N α = Θ(1) , both are true. We, therefore,have E [ N ( h )] = (cid:26) Θ (
N h/N α ) for ≤ h ≤ N α Θ( N ) for N α + 1 ≤ h ≤ N/ . (45)Since, in both cases we have E [ N ( h )] → ∞ , we can useLemma 5, the concentration bound on the binomial dis-tribution, to show that the event { N ( h ) ≥ c E [ N ( h )] } occurs with high probability for some < c < .This proves that the graph is ( h, k ( h )) -expander where k ( h ) = c E [ N ( h )] h , i.e., P N/ (cid:92) h =2 { G is ( h, k ( h )) -expander } ≥ N − c N , (46)for some c > where k ( h ) = (cid:26) c N − α for ≤ h ≤ N α c Nh for N α + 1 ≤ h ≤ N/ , (47)for some c , c > . See Appendix B for a detailedproof. This satisfies the expander property requirementsof Theorem 3. Applying Theorem 3, we obtain T N = O (log log N ) , (48)with probability at least − c N for some c > . Weprove this in Appendix B.2) ≤ α < : In this case, the expander properties ofTheorem 5 hold. Note that hN α → for all ≤ h ≤ N/ . We, thus, have (cid:16) − (1 − c/N α ) h (cid:17) = Θ ( h/N α ) .Using the expression for E [ N ( h )] in (44) we have N ( h ) = Θ ( N h/N α ) = Θ (cid:0) h/N α − (cid:1) .Here, E [ N ( h )] does not always go infinity in N . How-ever, we observe that, for all βN α − log N + 1 ≤ h ≤ N/ and for any β > , E [ N ( h )] → ∞ as N → ∞ .We can then use Lemma 5, the concentration bounds forbinomial distribution, to derive the following expanderproperty for βN α − log N + 1 ≤ h ≤ N/ : P N/ (cid:92) h>βN α − log N (cid:110) G is (cid:16) h, c N α − (cid:17) -expander (cid:111) ≥ N − c N , (49)for some c , c > and provided β > c for some c > .For ≤ h ≤ βN α − log N , E [ N ( h )] need not alwaysgo to infinity, and can in fact go to zero. Due tothis, the network M does not satisfy any expanderproperty for all ≤ h ≤ βN α − log N . Therefore, wederive a lower-bound on the probability P [ N ( h ) ≥ .In particular, there exists c > such that P [ N ( h ) ≥ ≥ N c (cid:0) − exp (cid:8) − h/N α − (cid:9)(cid:1) , (50) Fig. 3. High probability upper bound and the average lower-bound on floodingtime T N as a function of α . for all h ∈ { , , . . . βN α − log N } . See Appendix Cfor a detailed proof. This satisfies the conditions ofTheorem 5. From this, one can obtain T N = O (cid:0) N α − log N (cid:1) , with probability at least − c N for some c > . Weprove this in Appendix C.3) α ≥ : In this case, the conditions of Theorem 4 hold.Since α ≥ , we have h/N α → for all ≤ h ≤ N/ . This implies − (1 − c/N α ) h = Θ ( h/N α ) . Thus,using (44), we have E [ N ( h )] = Θ ( N h/N α ) → forall ≤ h ≤ N/ . This shows that the network M doesnot satisfy any expander property. We, therefore, derivea lower-bound on P [ N ( h ) = 1] . There exists a c > such that P [ N ( h ) = 1] ≥ N c ( N − h ) hN α , (51)for all ≤ h ≤ N − . See Appendix D for a detailedproof. This satisfies the condition of Theorem 4, usingwhich one can obtain T N = O (cid:0) N α − log N (cid:1) , with probability at least − c N for some c > . Weprove this in Appendix D.Figure 3 compares the high probability upper bound with theaverage lower-bound on flooding time T N from Theorem 2.We observe a gap of at most O (log log N ) when < α < .For all other values of α the upper and lower-bounds are of thesame order. The lower-bound on flooding time was derived inTheorem 2, which was also the lower-bound on the achievableaverage delay. In the next section, we show that a simple FCFSflooding scheme achieves the high probability upper bound onflooding time as its achievable average delay.VI. FCFS F LOODING S CHEME
We propose a scheme that is based on the idea of singlepacket flooding described in Section III. In this scheme, onlya single packet is transmitted over the entire network at anygiven time. Packets are served sequentially by the network ona FCFS basis. Each packet gets served for a fixed durationof U N . The packet is dropped if within this duration it is noteceived by all the other ( N − nodes. We call this the FCFSpacket flooding scheme.FCFS Packet Flooding: Packets arrive at each of the N nodes at rate λ .1) Among all the packets that have arrived, select theone that had arrived the earliest. At this time onlyone node, i.e. the source node, has this packet.2) In every cell, randomly select one packet carryingnode (if it exists) as a transmitter.3) Selected nodes transmit in each cell during the slotwhile all other nodes in the corresponding cellsreceive the packet.4) Repeat Steps 2 and 3 for U N time slots.5) After U N slots, remove the current packet from thetransmission queue and go to Step 1.Since we abruptly terminate the process in Step 5 after U N slots, it can happen that the packet has not reached all the ( N − destination nodes. To ensure that this happens rarelylet U N = (cid:26) c N α − log N if α ≥ (sparse) c log log N if < α < (dense) , (52)for some positive constants c and c such that T N < U N with probability − N . Such constants exists by Theorem 6.This leads to a vanishingly small packet drop rates. We nowobtain the capacity and delay performance of this FCFS packetflooding scheme. Theorem The FCFS packet flooding schemeachieves a capacity of λ = Θ (cid:16) N α log N (cid:17) if α ≥ (sparse) Θ (cid:16) N log log N (cid:17) if < α < (dense) . (53)Furthermore, the delay achieved at this rate is D =Θ ( U N ) . Proof:
The packets arrive at each node according to aPoisson process, at rate λ . Thus, the sum packets arrivals inthe networks is also a Poisson process of rate N λ . The servicetime for each packet under the FCFS packet flooding schemeis nothing but U N . Thus, the system can be thought of as aM/D/1 queue, with an arrival rate of N λ and service time of U N . The waiting time for such a system is given by [26] ˜ W = U N + U N ρ − ρ ) , (54)for any arrival rate N λ < U N , where ρ = N U N λ < is thequeue utilization. Selecting any ρ < , we obtain ˜ W = Θ( U N ) and λ = Θ (cid:16) NU N (cid:17) . Substituting U N from (52), we obtain theresult. This implies that the delay lower-bound of Theorem 2 isachieved, up to a gap of O (log log N ) , when the network isdense, i.e. < α < . We also see that the achieved throughput λ is less than the capacity upper bound of Theorem 1 by afactor of log log N when < α < , and by a factor of log N ,when α ≥ . The log log N gap appears due to the exact samegap between the flooding time upper and lower bounds when < α < . The log N factor gap for α ≥ occurs even thoughthe flooding time upper and lower bounds are asymptoticallytight. This, we conjuncture, is because the FCFS floodingscheme does not allow simultaneous transmissions of differentpackets, which leads to inefficient utilization of availabletransmission opportunities.We summarize these results in Table I. Unlike the unicastcase, where a capacity-delay tradeoff has been observed [9],[10], [21], nearly no such tradeoff exists for the broadcastproblem, and both capacity and minimum delay can be nearlyachieved simultaneously.VII. S INGLE H OP S CHEME
We now propose a single-hop scheme that achieves thecapacity upper-bound of Theorem 1 when the network issparse, i.e. α ≥ . In this scheme, a packet reaches it’sdestination from a source in a single hop, i.e. by direct sourceto destination transmission. This scheme only allows for asingle receiver in each cell, thus, ignores the broadcast natureof the wireless medium. The scheme still achieves the upper-bound capacity as the number of nodes in a cell tends to bevery small in the sparse case.Single-Hop Scheme: Each node makes ( N − copiesof an arrival packet, one for each receiving node. Figure 4illustrates this for node , where a copy of an arrivingpacket at node is transferred to each of the queues Q ,j for all ≤ j ≤ N .1) In each cell, select a pair of nodes at random. If acell contains fewer than nodes no transmissionsoccur in that cell.2) For the selected pair in every cell, assign, uniformlyand randomly, one node as a transmitter and theother as receiver.3) For each transmitter-receiver pair, if the transmitternode has a packet for the receiver node, transmit it,else remain idle.4) Wait for the next slot to begin, and restart theprocess from Step 1.The scheme is opaque to which node pairs are chosen as thesource-destination pairs. Thus, every queue Q i,j is activatedat the same rate. This implies that all the queues Q i,j haveidentical service rates. Hence, (cid:88) i (cid:54) = j r i,j = N ( N − r , . (55) ig. 4. Node 1 makes ( N − copies of every arriving packet, one for eachqueue Q ,j for ≤ j ≤ N . Service rate of Q ,j is denoted by r ,j . The left hand side of (55) corresponds to the total tate ofservice opportunities across the network, which is given by Cp , where p is the probability that there are at least two nodesin a cell: p = 1 − (1 − a N ) N − N a N (1 − a N ) N − . Thus, N ( N − r , = Cp , which gives, r , = CpN ( N − . (56)Hence, any arrival rate λ < r , will yield a stable networkunder the single-hop scheme. The delay achieved by thisscheme is lower-bounded by the delay in the single queue.Since each queue is Bernoulli arrival and Bernoulli service,the waiting time in each queue is given by ¯ W = − λr , − λ .Setting λ = r , we obtain ¯ W = Θ (1 /r , ) . We summarizethis in the following result. Theorem The single hop scheme achieves a capacityof λ SH = (cid:26) Θ (cid:0) N α (cid:1) if α ≥ (sparse) Θ (cid:0) N − α (cid:1) if < α < (dense) , (57)Furthermore, the delay achieved at this rate is D SH ≥ (cid:26) Θ ( N α ) if α ≥ (sparse) Θ (cid:0) N − α (cid:1) if < α < (dense) . (58)Hence, the single hop scheme achieves the capacity upper-bound for α ≥ . Thus, the capacity upper bound in Theorem 1is indeed achievable.VIII. C ONCLUSION
We considered the problem of all-to-all broadcast transmis-sions, in a networks of highly mobile nodes. We derived thebroadcast capacity and minimum delay scaling, in the numberof vehicles N , and showed that the capacity cannot scale betterthan /N . This, in conjunction with earlier known results forstatic network [16], proves that the broadcast capacity does notimprove with high mobility. This is in contrast with the unicastcase for which mobility improves network capacity [13].We further showed that both, the capacity and minimumdelay scalings, can be nearly achieved, simultaneously. We proposed a simple FCFS flooding scheme, that nearly achievesthis both capacity and minimum delay scaling. The floodingtime bound for Markov evolving graphs (MEG), proposedin [14], was used to analyze the proposed scheme. We derivedtwo new bounds on flooding time for MEG, which may be ofindependent theoretical interest.A PPENDIX
A. Proof of Theorem 1
Let λ be the rate achieved by a scheme. If X h ( T ) is thenumber of packets delivered to the destination in exactly h hops by time T then for an (cid:15) > we have T (cid:88) h ≥ X h ( T ) > N ( N − λ − (cid:15) (59)for all T > T (cid:15) , for some T (cid:15) > .If Z ki ( t ) is a binary random variable which equals if thereare k nodes in cell i in slot t then the total number of packet re-ceptions by time T is at most (cid:80) Ci =1 (cid:80) Nk =2 (cid:80) Tt =1 ( k − Z ki ( t ) .Hence, (cid:88) h ≥ hX h ( T ) ≤ C (cid:88) i =1 N (cid:88) k =2 T (cid:88) t =1 ( k − Z ki ( t ) . (60)Combining (59) and (60) we obtain C (cid:88) i =1 N (cid:88) k =2 T T (cid:88) t =1 ( k − Z ki ( t ) ≥ T (cid:88) h ≥ hX h ( T ) , = 1 T X ( T ) + 1 T (cid:88) h ≥ hX h ( T ) , ≥ T X ( T ) + 2 T (cid:88) h ≥ X h ( T ) . Using (59) we obtain C (cid:88) i =1 N (cid:88) k =2 T T (cid:88) t =1 ( k − Z ki ( t ) ≥ T X ( T )+ 2 (cid:18) N ( N − λ − (cid:15) − T X ( T ) (cid:19) . Taking T → + ∞ we have C (cid:88) i =1 N (cid:88) k =2 ( k − p ( k ) ≥ Cp + 2 ( N ( N − λ − (cid:15) − Cp ) , = 2 N ( N − − (cid:15) − Cp, (61)where p ( k ) is the probability that there are k nodes in a celland p is the probability that there are at least two nodes in acell; we use the fact that lim sup T → + ∞ X ( T ) T ≤ Cp . Taking (cid:15) → , we obtain N ( N − λ ≤ Cp + C N (cid:88) k =2 ( k − p ( k ) . (62)ubstituting p ( k ) = (cid:0) nk (cid:1) a kN (1 − a N ) N − k and computing thebinomial sum we obtain N ( N − λ = N (cid:16) − (1 − a N ) N − (cid:17) . (63)Therefore, ( N − λ ≤ (cid:16) − (1 − a N ) N − (cid:17) , (64) = 12 (cid:18) − (cid:16) − cN α (cid:17) N − (cid:19) . (65)When < α < , we have N/N α → ∞ . In which case, ( N − λ ≤ (cid:18) − (cid:16) − cN α (cid:17) N − (cid:19) = Θ(1) . (66)Hence, λ = O (1 /N ) . When α ≥ , either N/N α → or N/N α → c for some c > . This implies ( N − λ ≤ (cid:18) − (cid:16) − cN α (cid:17) N − (cid:19) = Θ ( N/N α ) . (67)Hence, λ = O (1 /N α ) . B. Proof of Expander Property and Flooding Time when <α < Lemma For ≤ h ≤ N α E [ N ( h )] = Θ (cid:0) hN − α (cid:1) , (68)and for all N α + 1 ≤ h ≤ N/ E [ N ( h )] = Θ( N ) . (69) Proof:
We know that N ( h ) ∼ Bin (cid:18) N − h, − (cid:16) − cN α (cid:17) h (cid:19) . Therefore, E [ N ( h )] = ( N − h ) (cid:20) − (cid:16) − cN α (cid:17) h (cid:21) . If h/N α → then − (1 − c/N α ) h ch/N α → , (70)and if h/N α → c , for some c > , then − (1 − c/N α ) h ch/N α → − exp {− cc } cc . (71)Since f ( x ) = − exp {− cx } cx is a decreasing function in x ,from (70) and (71) − e − c c ≤ lim N →∞ − (1 − c/N α ) h ch/N α ≤ , for all ≤ h ≤ N α . This implies − e − c c ≤ lim N →∞ E [ N ( h )] cN h/N α ≤ , for all ≤ h ≤ N α . This proves (68).If h/N α → ∞ then lim N →∞ − (1 − c/N α ) h = 1 , (72)and if h/N α → c , for some c > , then lim N →∞ − (1 − c/N α ) h = 1 − e − cc . (73)Since f ( x ) = 1 − e − cx is an increasing function of x , from (72)and (73) we have − e − c ≤ lim N →∞ − (1 − c/N α ) h ≤ , for all N α +1 ≤ h ≤ N/ . This implies − e − c ≤ lim N →∞ E [ N ( h )] N ≤ , for all N α +1 ≤ h ≤ N/ . This proves (69).Lemma 1 implies that for all h , E [ N ( h )] → ∞ as N → ∞ .Using Lemma 5 of Appendix E, we have P (cid:2) N ( h ) < η c N − α h (cid:3) ≤ N exp (cid:8) − c N − α h (cid:9) , for some η ∈ (0 , , c > , and for all ≤ h ≤ N α . Usingthis and union bound, we obtain P (cid:34) N α (cid:91) h =1 (cid:8) N ( h ) < η c N − α h (cid:9)(cid:35) ≤ N α (cid:88) h =1 P (cid:2) N ( h ) < η c N − α h (cid:3) , ≤ N α (cid:88) h =1 exp (cid:8) − c N − α h (cid:9) , ≤ N α (cid:88) h =1 exp (cid:8) − c N − α (cid:9) , = N α exp (cid:8) − c N − α (cid:9) ≤ N c N , (74)for some c > . This implies P (cid:34) N α (cid:92) h =1 (cid:8) N ( h ) ≥ η c N − α h (cid:9)(cid:35) ≥ N − c N . (75)This proves the expander property for ≤ h ≤ N α . Similarly,we obtain P (cid:34) N α (cid:92) h =1 { N ( h ) ≥ η c N } (cid:35) ≥ N − c N , (76)for some η ∈ (0 , and c > , which is the expanderproperty for N α +1 ≤ h ≤ N/ . To prove (46) we observe thatif P [ A ] ≥ N − c /N and P [ B ] ≥ N − c /N , for some pos-itive constants c and c , we have P [ A ∩ B ] ≥ N − c /N for some positive constant c . Note that the constant c > does not depend on h ; see Lemma 5 inAppendix E. ) Computing Flooding Time: We now apply Theorem 3to obtain an upper bound on flooding time. N/ − (cid:88) h =1 log (cid:0) h +1 h (cid:1) log (1 + k ( h )) = N α − (cid:88) h =1 log (cid:0) h +1 h (cid:1) log (1 + c N − α )+ N/ − (cid:88) h = N α log (cid:0) h +1 h (cid:1) log (1 + c N/h ) , (77)for some c > and c > . The first term in the expressioncan be simplified as N α − (cid:88) h =1 log (cid:0) h +1 h (cid:1) log (1 + c N − α ) = log (cid:16)(cid:81) N α − h =1 h +1 h (cid:17) log (1 + c N − α ) , (78) = log N α log (1 + c N − α ) , (79) = Θ (cid:18) log N α log N − α (cid:19) = Θ(1) . (80)The second term in (77) can be simplified as N/ − (cid:88) h = N α log (cid:0) h +1 h (cid:1) log (1 + c N/h ) ≤ N/ − (cid:88) h = N α h log (1 + c N/h ) , = Θ (cid:32)(cid:90) N/ N α dhh log (1 + c N/h ) (cid:33) , where the first inequality is because log (cid:0) h (cid:1) ≤ h . Toevaluate the integral, substitute y = c N/h to obtain (cid:90) N/ N α dhh log (1 + c N/h )= (cid:90) c N − α c dyy log(1 + y ) , = (cid:90) c N − α c y ) dy log(1 + y ) + (cid:90) c N − α c y (1 + y ) dy log(1 + y ) , ≤ (cid:18) c (cid:19) (cid:90) c N − α c y ) dy log(1 + y ) , = (cid:18) c (cid:19) (cid:2) log log (cid:0) c N − α (cid:1) − log log(2 c ) (cid:3) , = Θ (log log N ) . (81)This implies N/ − (cid:88) h = N α log (cid:0) h +1 h (cid:1) log (1 + c N/h ) = Θ (log log N ) . (82)Hence, from (77), (80), and (82), the flooding time is upperbounded by O (log log N ) . C. Proof of Expander Property and Flooding Time when ≤ α < Let β > . We show that the network has expander propertyfor βN α − log N + 1 ≤ h ≤ N/ for some β > , andprove a lower bound on probability P [ N ( h ) ≥ for ≤ h ≤ βN α − log N . Lemma For every (cid:15) > we have P [ N ( h ) ≥ ≥ N − (1 + (cid:15) ) exp (cid:8) − h/N α − (cid:9) , (83)for all ≤ h ≤ βN α − log N . Proof:
Since P [ N ( h ) ≥
1] = 1 − P [ N ( h ) = 0] , weevaluate P [ N ( h ) = 0] . We know that N ( h ) ∼ Bin (cid:16) N − h, − (1 − c/N α ) h (cid:17) . We thus have P [ N ( h ) = 0] = (1 − c/N α ) h ( N − h ) . (84)This implies lim N →∞ P [ N ( h ) = 0]exp (cid:110) − c h ( N − h ) N α (cid:111) = 1 . (85)Note that h ( N − h ) N α = hN α − − h N α , and the first term inthe expression dominates the scaling with N for ≤ h ≤ βN α − log N . Hence, lim N →∞ P [ N ( h ) = 0]exp {− cN h/N α } = 1 , (86)for all ≤ h ≤ βN α − log N . This implies that for every (cid:15) > P [ N ( h ) = 0] ≤ N (1 + (cid:15) ) exp (cid:8) − ch/N α − (cid:9) , (87)all ≤ h ≤ βN α − log N . This proves that for every (cid:15) > P [ N ( h ) ≥
1] = 1 − P [ N ( h ) = 0] ≥ N − (1 + (cid:15) ) exp (cid:8) − h/N α − (cid:9) , (88)all h ∈ { , , . . . βN α − log N } . Lemma For every (cid:15) > we have (1 − (cid:15) ) hN α − ≤ N E [ N ( h )] ≤ N (1 + (cid:15) ) hN α − , (89)for all βN α − log N + 1 ≤ h ≤ N/ . Proof:
We know that N ( h ) ∼ Bin (cid:18) N − h, − (cid:16) − cN α (cid:17) h (cid:19) . Therefore, E [ N ( h )] = ( N − h ) (cid:20) − (cid:16) − cN α (cid:17) h (cid:21) . (90)Note that if h/N α → then − (1 − c/N α ) h ch/N α → , (91)and h/N α → for all βN α − log N + 1 ≤ h ≤ N/ . Thisimplies lim N →∞ E [ N ( h )] cN h/N α = 1 , (92)or all βN α − log N + 1 ≤ h ≤ N/ . This proves the result.From Lemma 3, we note that E [ N ( h )] → ∞ as N → ∞ forall βN α − log N ≤ h ≤ N/ . Using Lemma 5 of Appendix E,we obtain for a given (cid:15) > P (cid:20) N ( h ) < η (1 − (cid:15) ) chN α − (cid:21) ≤ N exp (cid:26) − c chN α − (cid:27) , (93)for some η ∈ (0 , , c > , and all h ∈ { βN α − log N +1 , . . . N } . This, with union bound, implies P N/ (cid:91) h = βN α − log N +1 (cid:26) N ( h ) < η (1 − (cid:15) ) chN α − (cid:27) ≤ N/ (cid:88) h = βN α − log N +1 P (cid:20) N ( h ) < η (1 − (cid:15) ) chN α − (cid:21) , ≤ N N/ (cid:88) h = βN α − log N +1 exp (cid:26) − c chN α − (cid:27) , ≤ N exp (cid:26) − c c βN α − log N + 1 N α − (cid:27) , = Θ ( N exp {− c β log N } ) , = Θ (cid:18) N c β − (cid:19) , (94)for some c > . Choosing β > /c we have P N/ (cid:91) h = βN α − log N +1 (cid:26) N ( h ) < η (1 − (cid:15) ) chN α − (cid:27) ≤ N c N , for some c > . This implies P N/ (cid:92) h = βN α − log N +1 (cid:26) N ( h ) ≥ η (1 − (cid:15) ) chN α − (cid:27) ≥ N − c N , which proves the expander properties of (49).
1) Computing the Flooding Time:
Set p ( h ) = 1 − c exp (cid:8) − ch/N α − (cid:9) , (95)for all h ∈ { , , . . . βN α − log N } and some c > . Weknow from Theorem 5 that the flooding time is upper boundedby βN α − log N (cid:88) h =1 p ( h ) + βN α − log N − (cid:88) h =1 log (cid:0) h +1 h (cid:1) log (1 + c /N α − ) , (96) Note that c does not depend on h ; see Lemma 5 in Appendix E. where c = η (1 − (cid:15) ) c . Computing the first term we get βN α − log N (cid:88) h =1 p ( h ) = βN α − log N (cid:88) h =1 − c exp {− ch/N α − } , = βN α − log N (cid:88) h =1 exp (cid:8) ch/N α − (cid:9) exp { ch/N α − } − c , = Θ (cid:32)(cid:90) βN α − log N exp (cid:8) ch/N α − (cid:9) exp { ch/N α − } − c dh (cid:33) . The integral equals (cid:90) exp (cid:8) ch/N α − (cid:9) exp { ch/N α − } − c dh = 1 c N α − log (cid:0) exp (cid:8) ch/N α − (cid:9) − c (cid:1) . We, thus, have βN α − log N (cid:88) h =1 p ( h ) = Θ (cid:0) N α − log (exp { β log N } − c ) (cid:1) , = Θ (cid:0) N α − log (cid:0) N β − c (cid:1)(cid:1) , = Θ (cid:0) N α − log N (cid:1) . (97)Computing the second term in the expression (96) we have βN α − log N − (cid:88) h =1 log (cid:0) h +1 h (cid:1) log (1 + c /N α − )= log (cid:16)(cid:81) βN α − log N − h =1 h +1 h (cid:17) log (1 + c /N α − ) , = log (cid:0) βN α − log N (cid:1) log (1 + c /N α − ) , = Θ (cid:18) log N log (1 + c /N α − ) (cid:19) , = Θ (cid:0) N α − log N (cid:1) , (98)where the last equality follows because log (cid:0) c /N α − (cid:1) =Θ (cid:0) /N α − (cid:1) . Therefore, from (97), (98), and (96) the flood-ing time is T N = O (cid:0) N α − log N (cid:1) with probability at least − c /N for some c > . D. Proof of Expander Property and Flooding Time when α ≥ In this case, distribution of N ( h ) is concentrated at N ( h ) =0 . We, therefore, seek a lower-bound on P [ N ( h ) = 1] in orderto apply Theorem 4. Since N ( h ) ∼ Bin (cid:16) N − h, − (1 − c/N α ) h (cid:17) , we have P [ N ( h ) = 1] = ( N − h ) (cid:20) − (cid:16) − cN α (cid:17) h (cid:21) × (cid:16) − cN α (cid:17) h ( N − h − , . (99)ote that hN α → for all h ∈ { , , . . . , N − } since α ≥ .This implies lim N →∞ − (cid:0) − cN α (cid:1) h ch/N α = 1 , (100)for all h ∈ { , , . . . N/ } . Also, since max h ∈{ , ,...N − } h ( N − h − ≤ N , and min h ∈{ , ,...N − } h ( N − h − ≥ N , we have e − c/ ≤ lim N →∞ (cid:18) − N α (cid:19) h ( N − h − ≤ . (101)Then, (99), (100), and (101) imply e − c/ ≤ lim N →∞ P [ N ( h ) = 1]( N − h ) h/N α ≤ . for all ≤ h ≤ N − . Thus, there exists a positive constant c such that P [ N ( h ) = 1] ≥ N c ( N − h ) hN α , for all ≤ h ≤ N − . This proves the property of (51) for p ( h ) = c ( N − h ) hN α , for all ≤ h ≤ N − .
1) Computing the Flooding Time:
Then the upper boundon flooding time given in Theorem 4 equals N − (cid:88) h =1 p ( h ) = N − (cid:88) h =1 N α /c ( N − h ) h , = 1 c N α N N − (cid:88) h =1 (cid:20) h + 1 N − h (cid:21) , = Θ (cid:0) N α − log N (cid:1) . (102) E. Concentration Bounds
We list here some concentration bounds that we use in ourproofs. The following Lemma is from Chap. 1 in [27].
Lemma If X ∼ Bin ( n, p ) for some p ∈ (0 , and µ = np then for all k ≥ µ P [ X ≥ k ] ≤ exp (cid:26) − µH (cid:18) kµ (cid:19)(cid:27) , (103)and for all k ≤ µ P [ X ≤ k ] ≤ exp (cid:26) − µH (cid:18) kµ (cid:19)(cid:27) , (104)where H ( a ) = 1 − a + a log a for all a > .We now extend this result to the following Lemma If X , X , . . . X g ( n ) are binomial randomvariables such that c f ( n ) ≤ N E [ X h ] ≤ N c f ( n ) , (105)for some positive constants c and c , where g ( n ) and f ( n ) are increasing functions of n . Then there exists an η ∈ (0 , and a positive constant c such that P [ X h < ηc f ( n )] ≤ N e − c f ( n ) , (106)for all h ∈ { , , . . . g ( n ) } . Proof:
For every h ∈ { , , . . . g ( n ) } , X h is a binomialrandom variable. Lemma 4 gives P [ X h < ηc f ( n )] ≤ exp (cid:26) − E [ X h ] H (cid:18) ηc f ( n ) E [ X h ] (cid:19)(cid:27) . (107)Evaluating the exponent of the right hand side, we get E [ X h ] H (cid:18) ηc f ( n ) E [ X h ] (cid:19) = E [ X h ] − ηc f ( n ) + ηc f ( n ) log (cid:18) ηc f ( n ) E [ X h ] (cid:19) , ≥ N c f ( n ) − ηc f ( n ) + ηc f ( n ) log ( c /c ) , = (cid:20) − ηη − log ( c /c ) (cid:21) ηc f ( n ) . (108)where the second inequality follows from the fact that c f ( n ) ≤ n E [ X h ] ≤ n c f ( n ) . Now, since − ηη can take anypositive real values for η ∈ (0 , , we have E [ X h ] H (cid:18) ηc f ( n ) E [ X h ] (cid:19) ≥ c f ( n ) , (109)for some η ∈ (0 , and c = (cid:104) − ηη − log ( c /c ) (cid:105) ηc > for the corresponding η . Notice that c does not dependon h , and hence, (109) holds for all h ∈ { , , . . . g ( n ) } .Combining (107) and (109) we obtain P [ X h < ηc f ( n )] ≤ n exp {− c f ( n ) } , (110)for all h ∈ { , , . . . g ( n ) } . Lemma Let X , X , . . . X n be independent geo-metrically distributed random variables with parameters < p ≤ p ≤ · · · ≤ p n , i.e., P [ X i = t ] = p i (1 − p i ) t − for all t ≥ . Let S n = (cid:80) ni =1 X i and µ = E [ S n ] = 1 p + 1 p + · · · + 1 p n . (111)Then, for some c ≥ , P [ S n > c ( µ + t )] ≤ (1 − p ) t exp {− (2 c − n/ } . (112) Proof:
The proof is given in [28].
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