Not Always Sparse: Flooding Time in Partially Connected Mobile Ad Hoc Networks
aa r X i v : . [ c s . N I] N ov Not Always Sparse: Flooding Time inPartially Connected Mobile Ad Hoc Networks
Lorenzo Maggi and Francesco De PellegriniCREATE-NETvia alla Cascata, 56/D, 38123 Trento (Italy)E-mail: { lmaggi,fdepellegrini } @create-net.orgSeptember 10, 2018 Abstract
In this paper we study mobile ad hoc wireless networks using thenotion of evolving connectivity graphs. In such systems, the connec-tivity changes over time due to the intermittent contacts of mobileterminals. In particular, we are interested in studying the expectedflooding time when full connectivity cannot be ensured at each pointin time. Even in this case, due to finite contact times durations, con-nected components may appear in the connectivity graph. Hence,this represents the intermediate case between extreme cases of fullymobile ad hoc networks and fully static ad hoc networks. By using ageneralization of edge-Markovian graphs, we extend the existing mod-els based on sparse scenarios to this intermediate case and calculatethe expected flooding time. We also propose bounds that have re-duced computational complexity. Finally, numerical results validateour models.
Research in ad hoc networks has been focusing on several trade-offs emergingin such systems, and several models able to reproduce their properties havebeen proposed in literature. In particular, two milestone results in this areacaptured the properties of fully static ad hoc networks on the one hand [14],1nd the case of fully mobile ad hoc networks on the other hand [13]. Fromthe constructive proof in [13] we learn that there exists a class of routingprotocols for mobile ad hoc networks which are scalable in the number ofnodes, under the hypothesis of i.i.d. mobility . This fact motivated a largeeffort in the study of the so called store-carry-and-forward routing protocols.Such protocols are employed for a specific class of mobile ad hoc networks,namely mobile Delay Tolerant Networks (DTNs). DTNs are networks whereconnectivity is intermittent and the duration of end-to-end paths is not suffi-cient to deliver a message from source to destination using traditional routingtechniques. Instead, store-carry-and-forward is used in such systems, mean-ing that message copies stored at local nodes’ memory can be delivered topeer mobiles whenever they enter radio range [20][5][12][16]. To this respect,mobility compensates for the lack of connectivity: for instance, the so calledepidemic routing performs flooding by releasing a copy of the message to anynode without the message in radio range.Furthermore, in the highly-dynamic scenario depicted above, a customaryassumption is that nodes cannot acquire knowledge of the network topology.As a consequence, the process of data exchange in mobile DTNs is typicallyrepresented by means of a point process where events are contact times, i.e.,the time instants when mobile nodes are in radio range and can exchangemessage copies [16, 1].However, the effect of mobility is not always such to fully disconnectthe network at each point in time. Indeed, full i.i.d. mobility captureswell the extreme case of networks where the delay for routing a message isdominated by the effect of intermeeting times, i.e., times between successivemeeting of two mobile nodes. In practice, despite connectivity is evolvingand intermittent due to mobility, when contact times have finite durationthen connected components may exist. Thus, within those “connectivityislands”, the delay for message diffusion needs not to be dominated by theduration of intermeeting times.In this work, we are interested in the performance figures of the floodingtime in mobile ad hoc networks in such intermediate regime. The floodingtime is defined as the first time instant in which all nodes are informed, giventhat a message is generated by one source node at time t = 0. The proof in [13] was developed for the two hop routing protocol where relays canonly receive messages from a source and deliver to the destination. Our definition of intermeeting times refers to the residual time from the end of acontact until a novel contact occurs. evolving one [8][10], i.e., links may appear and disappear repeatedly. Hence, thenotion of diameter of a graph does not apply to this context. Rather, assuggested in [6], the flooding time appears to be the right metric to encodeconnectivity properties of such networks. In the rest of the paper, we willcharacterize this metric as a function of the number of nodes of the networkand of the mobility parameters of terminals.The paper is organized as follows. Sect. 2 describes relevant backgroundon evolving graphs and remarks the novelties introduced in our framework,compared to existing works. In Sect. 3 we describe our network model,whereas in Sect. 4 we provide an easy expression for the flooding time in thelimit case when intermeeting times are much larger than contact durations.In Sect. 5 we develop our main result in the general case. Also, in Sect.6we introduce approximations for the flooding time which require lower com-putational complexity. Finally, we deliver numerical results in Sect.7 and aconcluding section ends the paper. The main notation used in the rest of thepaper is reported in Tab. 1.
Models for communications based on evolving graphs have been proposed inliterature since the seminal work [9]. In [3], the authors provide an asymptoticanalysis of the cover time of a graph where edges can be modified at eachtime step by an adversarial. Covering is shown to be feasible in a timewhich is polynomial in the size of the graph, despite examples when suchtime is exponential exist for standard random walkers. In this paper weare interested in studying the flooding time: a similar analysis to the oneprovided here appears in [7]. The authors there provide upper and lowerbounds for the flooding time in edge-Markovian dynamic graphs, under the3able 1: List of symbols used throughout the paper λ − : expected contact duration µ − : expected intermeeting time p : probability that an edge is in state ON at time 0 N : number of nodes F (1 , N −
1) : (also F (1)) exact flooding time; expected time for a messagegenerated by 1 node to reach the remaining N − F (1 , N −
1) : (also F (1)) flooding time in the case of infinitesimal contactduration ( p = 0 , µ ↑ ∞ ) F (1 , N −
1) : (also F (1)) upper bound for F (1 , N − F (1 , N −
1) : (also F (1)) lower bound for F (1 , N − S ( a ) i ( N ) : event that i nodes are informed, but only a of them are possiblyactive, i.e., connected with other nodes W ( c ) i ( N ) : event that i informed nodes are connected to other c nodes F ( a ) ( i, N − i ) : flooding time when i nodes are already informed, but only a ofthem are possibly active C N , C N , C N : computational complexity for F (1 , N − F (1 , N − F (1 , N −
1) respectively G t : connectivity graph at time t assumption that the stationary node distribution at every time step is almostuniform and the transmission radius r is over the connectivity threshold. Themodel proposed uses discretization in both time and space. In [6] the authorsprove that the flooding time is O (cid:16) log N log(1+ Np ) (cid:17) and Ω (cid:16) log NNp (cid:17) : those results arein line with our results for the continuous-time case.Flooding time is also a well investigated subject in graph theory. Authorsof [17] extend classical percolation results for the diameter of random graphs[4]. The relevant result is that the flooding time distribution can be expressedin closed form for the class of Erd¨os Renyi random graphs where link delaysare exponentially distributed. Along the same research line, the study in[2] derives analogous results for regular graphs. Results of [17] resemble theform that we obtain here. However, in our case, the link presence is governedby an on-off process and the topology is not static. In the context of networkcapacity analysis, indeed, the beneficial effect of mobility has been analysedin the original work [13] and in several follow-ups, including the case with apopulation of heterogeneous devices characterized by different intermeetingintensities [11]. 4he beneficial effect of mobility onto the capacity of ad hoc networksappeared in [13] under the assumption of i.i.d. mobility occurs in the unitarydisc. Capacity results were extended to the case of clustered networks in [18]:indeed, the presence of a clustered hierarchical structure, is showed to lowerthe minimum asymptotic degree required to maintain asymptotic connectedgraphs. The reduction per node obeys to a factor N − d , where d is the clustersize.Furthermore, several papers extended the analysis to the case of DTNs,where all nodes are disconnected with high probability. In this context, onemay model connectivity using evolving graphs and derive delay bounds [8].The authors of [19] identify edge-Markovian dynamic graphs as a promisingmodel to capture the main features of store-carry-and-forward routing pro-tocols. Novel contributions:
Customary models used in ad hoc networks considereither a fixed topology, i.e., the case of a static ad hoc network, or a fullydisconnected network, i.e., the DTN case. The analysis proposed in thispaper is able to explore the intermediate case when connectivity is still in-termittent as assumed in DTNs, but, whenever a connected component ispresent, messages can span several hops with very small delay. Moreover, ifcompared to the analysis in [6], the continuous time model we employ hasthe key advantage that the closed forms and bounds proposed in this papercan be specialized to span different conditions of the sub-critical regime, de-pending on the ratio between the contact time duration and the intermeetingtimes.
We consider a delay tolerant network (DTN) with a set N of N mobile nodes.Any pair of nodes can exchange a message when in radio range, i.e., when-ever their distance is smaller than a certain threshold r >
0. Since terminalsroam within a region of size larger than r , the connectivity between any twoterminals is intermittent.We model this situation by a connectivity graph where nodes represent theterminals and edges evolve over time. At each time t ≥
0, the undirectededge between any two nodes can be either in state ON, i.e., the two terminalscan communicate with each other, or in state OFF, i.e., they are off-range.5he ON–OFF process governing the presence of edge i is stochastic and fol-lows an alternating renewal process. Once an edge is ON, it remains activefor a random time M ( i )1 ; then it switches to the OFF state, where it remainsfor a time L ( i )1 . It then goes ON for a time M ( i )2 , and so on and so forth.We assume that the random variables A ( i ) = { M ( i ) j , L ( i ) j } j ≥ are i.i.d. . Theintermeeting time L ( i ) j , i.e., the time between the end of a contact and thenext one, is an exponential random variable with mean λ − >
0. Conversely,we do not make further assumptions on the duration of a contact M ( i ) j : it isa generic random variable with mean µ − ≥ { A ( i ) } Ni =1 are i.i.d. . Thisassumption allows our problem of computing the flooding time to be analyt-ically tractable.We note that no assumption on the independence between L ( i ) j and M ( i ) j isrequired. Hence, our model is still valid if the duration of the contact betweenany two terminals somehow influences the following intermeeting time.Let P ( t ) be the probability that an edge is ON at time t . From classicresults in renewal theory (e.g. [15], Thm 3.4.4),lim t →∞ P ( t ) = E [ M ( i ) j ] E [ M ( i ) j ] + E [ L ( i ) j ] = µ − µ − + λ − := p. (1)We point out that our graph model is a generalization of a continuous timeedge-Markovian graph [6], in which the contact duration M ( i ) j between anytwo terminals is also an exponential random variable, independent of L ( i ) j .In that case, the edge process follows a continuous-time Markov chain withtransition rate matrix Q = (cid:16) − λ λµ − µ (cid:17) .In this paper we are interested in studying the expected flooding time F (1 , N −
1) := E [ f (1 , N − N − informed at time t if, at time t , it has a copy of the message. The described relay protocol iscalled unrestricted multi-copy protocol in [12] or epidemic routing [20].In order to develop our analysis, we consider the intermeeting dominated case, i.e., the transmission time is negligible on the scale of the ON-OFF6rocess governing the link presence: hence, any node instantaneously copiesthe carried message to all the nodes it is connected to.More specifically, we provide the exact expression of F (1 , N − F (1 , N − F (1 , N − at time 0. Thus, if we let E t be the setof undirected edges in state ON at time t ≥ G t the corresponding con-nectivity graph, then G = ( E , N ) is an Erd¨os-R´enyi graph with parameter p . A notation remark: for compactness’ sake, we will drop the dependenceof our variables on N whenever possible. For instance, the flooding time F (1 , N −
1) will be F (1), and same simplification holds for its lower andupper bounds F and F , respectively. Before tackling the computation of the flooding time in the general case with λ − > µ − ≥
0, it is useful to develop a simpler and more restrictivecase in which the intermeeting process between any two terminals is a pointprocess, i.e., the contacts’ average duration is null (or µ ↑ ∞ ) .In the general model introduced later, more than one link may be activeat the same time with positive probability: therefore, in the intermeetingdominated case, the message originated by the source node can be spread tomore than one node simultaneously.Conversely, in the point-like contact case developed in this section, theevent that two or more links are in state ON at the same time occurs withnull probability. In other words, the set of edges E t is almost always emptyfor t ≥
0. Moreover, if we suppose that E t contains one edge for some t ≥ E t contains another edge with null probability. It is then apparent that This is justified if we assume that the network, at time 0, has not been observed fora time long enough (see Eq. 1) We will precise later in Thm. 2 the notion of the sparse regime case as the limit forthe general case.
7n this section we study the sparse regime of DTNs: a message created bythe source can only be transmitted via separate successive hops between pairof terminals.Let us call F (1 , N −
1) the flooding time in the point-like case. Whenever itsdependence on N is clear, we will call it F (1). Clearly, F (1) constitutes anupper bound for the flooding in the general case for finite µ , i.e., F (1 , N − ≥ F (1 , N − , ∀ λ, p, N. In fact, the probability of having connected components in an Erd¨os-R´enyigraph increases with the probability p that an edge is in state ON. In Sect.6.2 we provide a second upper bound for F (1), which is tighter than F (1)for large N .To proceed further, let F ( i ) the flooding time with point-like contactsunder the hypothesis that i nodes are informed, i.e., they have the messageand can forward it to other nodes. Therefore, we can write F ( N ) = 0 , F ( N −
1) = 1 λ ( N − F ( i ) = 1 λi ( N − i ) + F ( i + 1) , ≤ i ≤ N − . Using the recursive expression above, it follows that F (1) = N − X i =1 λi ( N − i ) = 2 λN N − X i =1 i = 2 λN H N − , where H n is the n -th harmonic number. By the classic relation Z N x dx ≤ H n ≤ Z N x − dx, we easily conclude: 2 ln NλN ≤ F (1) ≤ N − λN .Therefore, F (1 , N − ∈ Θ( N − ln N ) and, since F (1) is an upper boundfor the flooding time F (1) when µ − >
0, the asymptotic bound for F (1)writes F (1 , N − ∈ O ( N − ln N ) . Flooding time
After studying in Sect. 4 the flooding time F (1) in the specific case of point-like contact ( µ − ↓ F (1) in the moregeneral non-sparse case with contact average duration µ − ≥
0. Hence, F (1)equals F (1) calculated at p = 0.We recall here two technical assumptions made in Sect. 3, namely in-stantaneous transmission during contacts and exponential intermeeting timewith mean λ − .Further modelling is required with respect to the point-like contact caseof Sect. 4. In fact, at any time t , in the connectivity graph G t there existstrongly connected components with more than one edge with positive prob-ability. Hence, if at least one node n belonging to a connected component isinformed, then the message can be spread instantaneously to all the nodesconnected with n . The model introduced in this paper is meant precisely tocapture such topological feature characterizing mobile ad hoc networks, inorder to go beyond the sparse regime.In particular, in this case, we need to distinguish among informed nodes. Ateach time t , in fact, there are two classes of nodes: active and not active . Anode is active if it possibly has some link in state ON (each with probability p ): we hence define the auxiliary notation F ( a ) ( i ), which represents the ex-pected flooding time when i nodes are informed and 0 ≤ a ≤ i of them areactive.Moreover, due to the symmetry of the problem, F (1) does not depend onthe identity of the source. The exact expression of the flooding time F (1) ispresented in iterative form in the following Theorem, whose proof is in theAppendix. Theorem 1.
The flooding time F (1) can be expressed as F (1) = (1 − p ) N − (cid:18) λ ( N −
1) + F (1) (2) (cid:19) + N − X c =1 (cid:18) N − c (cid:19) p c (1 − p ) N − − c F ( c ) (1 + c ) , (2)9 here, for ≤ i ≤ N − , ≤ a ≤ i − , F ( a ) ( i ) = (1 − p ) a ( N − i ) (cid:18) λi ( N − i ) + F (1) ( i + 1) (cid:19) ++ N − i − X c =1 (cid:18) N − ic (cid:19)h − (1 − p ) a i c (1 − p ) a ( N − i − c ) F ( c ) ( i + c ) (3) and, for ≤ a ≤ N − : F ( a ) ( N −
1) = (1 − p ) a λ ( N − . (cid:3) p small In this section we wish to find the relation between the flooding time withpoint-like contacts, F (1), and the general flooding time F (1) for small valuesof the stationary probability p . More specifically, we wish to find an expres-sion of the kind F (1) = F (1) − χp + o ( p ), where χ is some positive constant.We then obtain the following result. Lemma 1.
The flooding time F (1 , N − can be written for values of p sufficiently close to 0 as F (1 , N −
1) = F (1) − λ − H N − p + o ( p ) , where H n = P ni =1 /i is the n -th harmonic number. (cid:3) The proof of Lemma 1 is deferred to the Appendix. We conclude that,for p small, the flooding time decreases approximately linearly in p withcoefficient λ − H N − , which behaves like log( N ) + γ for large values of N ,where γ is the Euler-Mascheroni constant. After providing the exact expression of the flooding time F (1) in Thm. 1, inthis section we focus on issues related to its computation. First we providethe matricial form of the linear system of equations in (2) and (3), thenwe show the complexity of the computation of F (1), in terms of number ofrequired additions and multiplications. From expressions (2) and (3), wenotice that ( N − N − / F (1), namely F ( a ) ( i ), for i = 2 , . . . , N − = 1 , . . . , i −
1. Hence, the equations in (2) and (3) can be rewritten as TF = d , where F is the column vector: F = (cid:2) F (1: N − ( N − , F (1: N − ( N − , . . . , F (1) (2) , F (1) (cid:3) T and F ( a : a + k ) is defined as [ F ( a ) , F ( a +1) , . . . , F ( a + k ) ]. T is a square, lower tri-angular matrix of dimension ( N − N − / T and d , let us define the index mapping:Ψ( i, a ) = ( N − N − − i ( i − a, with 2 ≤ i ≤ N − , ≤ a ≤ i − ∪ ( i = 1 , a = 1)Then, T i,i = 1 , ≤ i ≤ ( N − N − / T Ψ( i,a ) , Ψ( i +1 , = − (1 − p ) a ( N − i ) + − ( N − i ) [1 − (1 − p ) a ] (1 − p ) a ( N − i − T Ψ( i,a ) , Ψ( i + c,c ) = − (cid:18) N − ic (cid:19)h − (1 − p ) a i c (1 − p ) a ( N − i − c ) , ≤ c ≤ N − i − d Ψ( i,a ) = (1 − p ) a ( N − i ) λi ( N − i ) . For example, when N = 5, the linearsystem has the following structure: • • • • • • • F (1) (4) F (2) (4) F (3) (4) F (1) (3) F (2) (3) F (1) (2) F (1) = d where the symbol • stands for the generic nonnull element.Now we are ready to compute the complexity of the flooding time: theproof requires the enumeration of the operations performed in order to solvethe linear system TF = d and it is omitted for the sake of space.11 roposition 5.1. The number C N of operations (i.e. multiplications or ad-ditions) required to compute F (1 , N − both equal the number of nonnullelements of the matrix T below the main diagonal, i.e. C N = ( N − N +17 N − / . (cid:3) With our model we can bound the expected flooding time as a function ofthe network size N . To this respect we referred to the point-like bound F (1)as the limit case when µ diverges. However, it is interesting to compute thescaling law for the network when such bound is actually the limit case forlarge N ’s. In fact, our analysis addresses the case when the network is in thesub-critical regime, i.e., with a terminology of random graphs analysis, thegraph is disconnected almost surely.In particular, with respect to the connectivity properties of the system,the dynamic graph parameters λ = λ ( N ) and µ = µ ( N ) may depend onthe underlying mobility model in non trivial fashion. The question in turnis under which scaling regime the point-like bound is guaranteed to be stillthe limit of a scaling law which is consistent with the sub-critical regime(when the graph is connected indeed we expect the bound to be zero). Inother words: as N diverges, how should the contact time duration and theintermeeting time scale in order to approach the point-wise limit while stillin the sub-critical regime for every N ? In practice, consistent scaling lawsare ensured by the following result. Theorem 2.
Let µλ = Ω (cid:0) N log N (cid:1) , then F (1 , N −
1) = O ( N − log N ) . (cid:3) Proof.
Consider the class of graphs G ( N, p ) where p = p ( N ) = λµ + λ : accord-ing to the classic result [4][Thm 7.3 pp. 164], we can fix a c ∈ R , and if p = { log N + c + o (1) } /N , then lim N →∞ P ( G p is connected) → e e − c . Here,under our assumptions, indeed, there exists N such that λµ > b N log N , where b >
0, which implies for N ≥ N p ( N ) = λ ( N ) λ ( N ) + µ ( N ) = 11 + µλ < λµ < b log NN so that p ( N ) = o (cid:0) log NN (cid:1) . Since the point-like case is an upper bound, thestatement follows. 12hm. 2 explains to which extent the point process bound is the limitfor our model: this is the case when the intermeeting time grows muchfaster than the contact duration. Under this law, the network approachesthe conditions of a sparse network which is typically used in literature forDTN models [1, 12]. In Thm. 1 we provided the exact expression of the flooding time F (1). It isthe solution of a linear system of equations, which can be solved via itera-tive substitution. It can be calculated at a cost of ∼ N / F (1) with lower complexity, by providinga lower bound F (1) and an upper bound F (1) for F (1) in Sects. 6 and 6.2,respectively. In this section we intend to provide a lower bound F (1) for the floodingtime F (1), whose computation complexity is bounded asymptotically by thecomplexity of F (1) (see Prop. 5.1). To this aim, we need to simplify theexpression in (2) by formulating some convenient approximations. Beforegoing deeper into the analysis, we still need to define two concepts. Definition 1. S ( a ) i ( N ) is defined as the event that, among i i nformed ter-minals, only a ≤ i of them are possibly a ctive, i.e. connected with some ofthe N − i uninformed nodes. In details, let I ⊂ N be the set of informedterminals ( |I| = i ) and let A ⊆ I ( |A| = a ). Then, i ) all the a ( N − i ) edgesbetween A and N \ I and ii ) all the edges within N \ I are independently instate ON with probability p . Also, iii ) all edges between I \ A and
N \ I areOFF with probability . Definition 2. W ( c ) i ( N ) is defined as the event that c nodes are c onnectedto the i informed nodes. More formally, let I ⊂ N be the set of informedterminals ( |I| = i ) and let C ⊆ N \ I ( |C| = c ). Then, for any terminal I p pp pp I S (2)3 (5) W (1)3 (5) C Figure 1: Illustrations of an instance of the event S (2)3 (5) (on the left side)and of W (1)3 (5) (on the right side). The dashed edges are ON with probability p . n ∈ C there exists a link in state ON between n and an informed node n ′ ∈ I .Moreover, all the edges between I and N \ ( I ∪ C ) are in state OFF withprobability 1. Hence, the message is forwarded instantaneously to all thenodes in C . It is straightforward to see that the following relation holds: (cid:16) W ( c ) i ( N ) , S ( a ) i ( N ) (cid:17) = S ( c ) i + c ( N ) . (4)In other words, the message is transmitted at time t + to the c connectednodes, which are the only ones which can possibly forward the message tosome of the remaining N − i − c uninformed nodes at time t + . In fact, theevent ( W ( c ) i , S ( a ) i ) rules out the possibility that the a nodes possibly active attime t are capable to retransmit the message at time t + .Instead, in order to compute a lower bound for F (1) we assume that allthe i + c informed nodes are still able to spread the message to the othernodes with probability p . More formally, we replace the correct equation onthe left-hand side of the following expression: (cid:16) W ( c ) i ( N ) , S ( a ) i ( N ) (cid:17) = S ( c ) i + c ( N ) replace −→ S ( i + c ) i + c ( N ) . (5)with the right-hand side one (compare with Eq. 4). In other words, by (5)we overestimate the activity of the edges, by assigning a probability of beingactive p also to those edges that are known being in state OFF. Under thisapproximation, the flooding process speeds up, and we can lower bound thecorrect expression of F (1) in (2) with F (1), that we define in the following.14 roposition 6.1. The flooding time under the approximation in (5) is F (1) ,where F (1) = (1 − p ) N − (cid:20) λ ( N −
1) + F (2) (cid:21) ++ N − X i =1 (cid:18) N − i (cid:19) p i (1 − p ) N − − i F ( i + 1) , and where F ( i ) = (1 − p ) i ( N − i ) (cid:20) λi ( N − i ) + F ( i + 1) (cid:21) ++ N − i − X c =1 (cid:18) N − ic (cid:19) (cid:2) − (1 − p ) i (cid:3) c (1 − p ) i ( N − i − c ) F ( i + c ) . for ≤ i ≤ N − , and F ( N −
1) = (1 − p ) N − / ( λ ( N − . Moreover, F (1) is a lower bound for the flooding time F (1) . (cid:3) We notice that the expression of F (1) can be computed iteratively, start-ing from F ( N − F ( N − , . . . , F ( i +1)to compute F ( i ), for i = N − , . . . , p small Under Assumption (5), we actually consider the OFF edges as having aprobability p of being ON. Intuitively, if the stationary probability p tendsto 0, F (1) should approximate the actual flooding time F (1) with increasingaccuracy. We now prove this formally and we also provide an expression ofthe lower bound F (1) for p small, in a similar fashion as we did for the exactflooding time F (1) in Sect. 5.1. Lemma 2.
The lower bound F (1) of the flooding time can be written forvalues of p sufficiently close to 0 as F (1) = F (1) − λ − ( N − p + o ( p ) . (cid:3) Lemma 2 confirms the intuition that when p tends to 0, then F (1) → F (1). Moreover, in a right neighbourhood of p = 0, F (1) decreases linearlyin p , and proportionally to the number of nodes N . The reader should com-pare this result with Lemma 1, claiming that the exact flooding time F (1)decreases as λ − H N − p for p small, where H N diverges as log( N ).15 .1.2 Computational complexity In order to compute F (1) we need to solve an ( N − N −
1) linearsystem of equations of the form
A F = c (see proof of Lemma 2), where theunknowns are F ( i ) for i = 1 , . . . , N −
1. The number of operations requiredto solve the linear system equals the number of elements of the matrix A above the main diagonal. The following result follows. We leave its proof tothe reader. Proposition 6.2.
The number of operations required to compute the floodingtime lower bound F (1 , N, is C N = (cid:0) N − (cid:1) . (cid:3) By comparing the computational complexity required to compute theexact flooding time F (1) and its lower bound F (1) (see Prop. 5.1 and 6.2,respectively), we see that C N < C N for N ≥
5, and C N = C N for N ≤ C N ∼ N , C N ∼ N . Hence, approximating the exact expression of the flooding time F (1) via thelower bound F (1) becomes more and more convenient when the number ofterminals N increases. After providing a lower bound for the flooding time F (1) with low com-plexity, we now devise an upper bound, called F (1 , N − N . Hence, the computationof F (1 , N −
1) already yields F (1 , K −
1) for all
K < N . As such, the complex-ity required to calculate the first N values of the bound is O ( N ), whereas O ( N ) operations are needed for the exact value of the flooding time.We observed in Sect. 4 that the exact flooding time with point-like contacts, F (1 , N − F (1 , N − F . Let I be the set of informed nodes at time t . As soonas a connection between I and some C ⊆ N is established at time t ′ ≥ t ,the message is copied instantaneously to C . Then, we assume that all theterminals in I deactivate and no longer participate in the flooding process.16his corresponds to progressively remove informed nodes from the originalsystem and to consider the flooding process on the remaining subgraph.More specifically, in order to derive the upper bound we perform thefollowing substitution: (cid:16) W ( c ) i ( N ) , S ( a ) i ( N ) (cid:17) = S ( c ) i + c ( N ) replace −→ S ( c ) c ( N − i ) . (6)Clearly, the spreading process under the approximation (6) is slower than inthe exact case because the number of nodes capable of retransmitting themessage decreases over time.Thus, the flooding time calculated under assumption (6) provides an upperbound for F (1), namely F (1). Proposition 6.3.
The flooding time under the approximation in (6) is F (1) ,where F (1) = (1 − p ) N − (cid:20) λ ( N −
1) + F (1 , N − (cid:21) ++ N − X c =1 (cid:18) N − c (cid:19) p c (1 − p ) N − − c F ( c, N − c − , where, for ≤ i ≤ n − , F ( i, n − i ) = (1 − p ) i ( n − i ) (cid:20) λi ( n − i ) + F (1 , n − i − (cid:21) + n − i − X c =1 (cid:18) n − ic (cid:19)h − (1 − p ) i i c (1 − p ) i ( n − i − c ) F ( c, n − i − c ) ,F ( n − ,
1) = (1 − p ) n − λ ( n − . Moreover, F (1) is an upper bound for the exact flooding time F (1) . (cid:3) We defer the proof of Proposition 6.3 to the Appendix.
We now investigate the computational complexity C N of the upper bound F (1 , N − F (1 , N − (1 , N − F (1 , N − F (2 , N − F ( N − , . . . F (2 , N − . . . F ( N − , F (2 , N − F (1 , N − Figure 2: The quantities in black are available when F (1 , N −
2) has alreadybeen computed. The quantities in red need to be calculated in order to derive F (1 , N − Proposition 6.4.
Let C N be the number of operations to compute F (1 , N − .Then, C N = C N for all integers N . (cid:3) Interestingly, the complexity of F (1 , N −
1) equals the complexity of theexact flooding time F (1 , N − F as F iscomputationally profitable when we need to evaluate F (1 , N −
1) for severalvalues of N . Once F (1 , N −
2) has already been computed indeed, in orderto compute the F (1 , N −
1) we only need to perform an incremental numberof operations equal to (compare with Eq. 10 and Fig. 2)( N −
2) + ( N −
4) + ( N −
5) + · · · + 1 = ( N − N − − N + 3 . We begin our numerical investigations by displaying in Fig. 3 the exactflooding time F (1) when the size of the network N varies. We also illustratethe bounds F (1), F (1), and the flooding time in sparse regime F (1). Inthe next sections we go deeper in the analysis by showing numerically therelations among these quantities for different values of p and N . The main objective of this paper is to account for the existence, at eachinstant, of connected components in the connectivity graph of intermittently18
10 20 30 40 5000.511.52 N F (1 , N –1) F (1 , N –1) F (1 , N –1) F (1 , N –1) Figure 3: Exact flooding time F (1 , N −
1) ( λ = 1, p = 0 . F (1 , N − F (1 , N − F (1 , N − p = 0). When this is not thecase, i.e., p >
0, one may be tempted to use still the same model, whichis certainly appealing for the simplicity of the expression of flooding time F in that regime (see Section 4). Nevertheless, we now show numericallythat, assuming that the regime is non-sparse ( p > p = 0) may betremendously high even for reasonable values of the stationary probability p . In Figure 5(a) we illustrate the ratio between the flooding time in thesparse regime, F (1 , N − F (1 , N − p and N . As expected, the approximationerror is negligible for p small. Nevertheless, it becomes indeed unacceptablefor p > . N ≤ We now report some numerical results on the tightness of the lower andupper bounds F (1) and F (1), proposed in Sections 6.1 and 6.2 respectively.19imilarly to Fig. 5(a), in Fig. 5(b) we illustrate the ratio between the upperbound F (1) and the exact expression of the flooding time F (1), for differentvalues of the stationary probability p and the number of terminals N . Fig.5(c) shows the same analysis for the lower bound F (1). We notice that theupper bound F (1) is a good approximation of F (1) for values sufficientlylarge of p , i.e. p > .
3. On the other hand, Fig. 5(c) confirms our analyticresults in Sect. 6.1.1: the lower bound F (1) well approximates the floodingtime F (1) for small values of p (also, see Lemma 2). In this paper we provided the expression of two different upper bounds forthe flooding time F (1 , N − F (1 , N −
1) (Sect. 4) and F (1 , N − F (1 , N −
1) in the sparse regime. Fig. 4 illustrates the comparisonbetween these two bounds. Numerical experiments showed that the bound F (1 , N −
1) is tighter than F (1 , N −
1) under two different regimes, namely i )for all N , if p is sufficiently large, or ii ) for N large enough, if p is sufficientlysmall. In other words, if we fix λ and p , then there exists b N such that F (1 , N − < F (1 , N − , ∀ N > b N . Hence, the case b N = 0 describesregime ii ). In this paper we focused on the flooding time for mobile ad hoc networks. Inthis context, the notion of diameter for static networks has little meaning.In our case, in fact, the network is disconnected almost surely at each pointin time, so that the diameter is infinite. Yet, mobility of terminals overcomesinstantaneous lack of connectivity and flooding time is still well defined. Weapplied a generalization of continuous time Markov-edge evolving graphs tothis context. The advantage of such models is that they allow to encode theimpact of finite contact durations into the flooding time calculation, wherethe event of possibly large connected components cannot be neglected. Insuch cases, the classic sparse models used in DTNs fail to capture such eventsthus providing conservative estimations. We provided the exact expressionof the flooding time F (1) in the non-sparse regime. Our continuous timemodel encompasses the sparse model as limit case. We computed a lower20
10 20 30 40 5000.511.52
N F (1 , N − F (1 , N − p Figure 4: Upper bounds F and F compared for λ = 1. F (1 , N − 1) either i ) for all N , if p > . ii ) for N sufficiently large for p < . p F (1 , /F (1 , F (1 , /F (1 , F (1 , /F (1 , F (1 , /F (1 , F (1 , /F (1 , −2 −1 p F (1 , /F (1 , F (1 , /F (1 , F (1 , /F (1 , F (1 , /F (1 , F (1 , /F (1 , p F (1 , /F (1 , F (1 , /F (1 , F (1 , /F (1 , F (1 , /F (1 , F (1 , /F (1 , (a) (b) (c) Figure 5: In (a) we show the ratio F (1) /F (1) for different values of p and N = 10 , . . . , 50: the approximation error becomes larger than 100% for p > . N ≥ 10. At p = 0 . 1, when N ≥ F (1) > F (1).Similarly, (b) and (c) illustrate the ratios F (1) /F (1) and F (1) /F (1), respec-tively. We see that F (1) well approximates F (1) for small values of p , while F (1) approaches the exact value F (1) for p sufficiently large, i.e., p > . F (1) and an upper bound F (1), both having a reduced computationalcomplexity. We studied analytically the behaviour of F (1) , F (1) , F (1) forsmall values of the stationary probability p and we investigated numericallythe bounds tightness with respect to the exact value F (1). We further showedthe error committed by approximating F (1) with the equivalent expression21n the sparse regime, F (1).There are two key directions that we have not explored in this work.First, the heterogeneity of the link ON-OFF processes has been neglectedfor tractability’s sake. An extension of the model in that direction wouldinclude the case when terminals have different mobility patterns, and thiswould permit tighter estimations. Also, in real mobility models, it is possibleto measure a certain degree of correlation among intermeeting events. Weplan to include such correlation effects into our model in our future researchwork. Proof of Theorem 1 Proof. For the sake of notation simplicity, we drop the dependence of theevents S ( a ) i and W ( c ) i on N . We observe that the joint event ( W ( c ) i , S ( a ) i ),with a ≤ i , is equivalent to the event S ( c ) i + c . In fact, when the set I ofinformed nodes transmit instantaneously the message to C , then the numberof informed nodes becomes I ∪ C , but only the nodes in C can possibly copythe message to N \ ( I ∪ C ) instantaneously. By the total probability rule, wewrite F (1) := E [ f (1) | S (1)1 ] as F (1) = N − X c =0 Pr (cid:16) W ( c )1 | S (1)1 (cid:17) E h f (1) | W ( c )1 , S (1)1 i = N − X c =0 Pr (cid:16) W ( c )1 | S (1)1 (cid:17) E h f (1) | S ( c ) c +1 i , (7) E h f (1) | S ( a ) i i = N − i X c =0 Pr (cid:16) W ( c ) i | S ( a ) i (cid:17) E h f (1) | W ( c ) i , S ( a ) i i = N − i − X c =0 Pr (cid:16) W ( c ) i | S ( a ) i (cid:17) E h f (1) | S ( c ) i + c i , (8)with a < i . We remark that E [ f (1) | S N ] = 0 because the transmission isinstantaneous. Due to this, we could truncate the sums in (7) and (8) up to c = N − i − 1. Now, we find an explicit expression for (7). When i nodesare informed and all the i ( N − i ) edges between I and N \ I are OFF, an22xponential time with mean 1 / ( λi ( N − i )) needs to be waited before theflooding process resumes. At that instant, there are i + 1 informed nodes butonly one of them, i.e. the newly informed one, can possibly instantaneouslycopy the message to the uninformed nodes. Therefore, we can writ E h f (1) | S (0) i i = 1 λi ( N − i ) + E h f (1) | S (1) i +1 i . Now we derive an explicit expression for the probability terms in (7) and (8).The event that c nodes (set C ) are connected to i informed nodes (set I ), ofwhich only a ≤ i are possibly active (set A ), is the intersection of the events i ) for each n ∈ C , there exists n ′ ∈ A such that the edge ( n, n ′ ) is ON, and ii ) all the edges between A and N \ ( I ∪ C ) are in state OFF. Thus, we canwrite Pr (cid:16) W ( c ) i | S ( a ) i (cid:17) = (cid:18) N − ic (cid:19)h − (1 − p ) a i c (1 − p ) a ( N − i − c ) . (9)Let us define F ( a ) ( i ) := E [ f (1) | S ( a ) i ]. We can interpret F (1) as F (1) (1). Then,the thesis easily follows by inspection. Proof of Proposition 6.1 Proof. By the substitution in (5), we modify (7) as F ′ (1) = P N − c =0 Pr (cid:16) W ( c )1 | S (1)1 (cid:17) E h f (1) | S (1+ c )1+ c i .By defining F ′ ( i ) := E [ f (1) | S ( i ) i ], under assumption (5) we modify (8) as F ′ ( i ) = P N − i − c =0 Pr( W ( c ) i | S ( i ) i ) F ′ ( i + c ) . Let us define F ( i ) := F ′ ( i ), 1 ≤ i ≤ N − 1. We compute Pr( W ( c ) i | S ( i ) i ) as in (9). The thesis follows byinspection. Proof of Lemma 2 Proof. Let us rewrite (6.1) as A F = c , where F = [ F (1) . . . F ( N − F , A is an upper triangular ( N − N − 1) matrix with diagonal elements23 i,i = 1 and A i,i +1 = − (1 − p ) i ( N − i ) − ( N − i ) (cid:2) − (1 − p ) i (cid:3) ×× (1 − p ) i ( N − i − = − o ( p ) A i,i + c = − (cid:18) N − ic (cid:19) (cid:2) − (1 − p ) i (cid:3) c (1 − p ) i ( N − i − c ) = o ( p ) ,c i ( p ) = (1 − p ) i ( N − i ) λi ( N − i ) = 1 − i ( N − i ) pλi ( N − i ) + o ( p )It is not difficult to show that F ( N − i, i ) = P N − i = N − i (1 − i ( N − i ) p ) / ( λi ( N − i )) + o ( p ), and in particular F (1) = N − X i =1 (1 − p ) i ( N − i ) i ( N − i ) − λ − ( N − p + o ( p )= F (1) − λ − ( N − p + o ( p ) Proof of Proposition 6.3 Proof. By performing the substitution in (6), we modify (7) as F ′′ (1 , N − 1) = P N − c =0 Pr (cid:16) W ( c )1 ( N ) | S (1)1 ( N ) (cid:17) E h f (1 , N − | S ( c ) c ( N − i .We define F ′′ ( i, n − i ) := E [ f (1) | S ( i ) i ( n )], n < N . Under assumption (6) wemodify (8) as F ′′ ( i, n − i ) = P N − i − c =0 Pr (cid:16) W ( c ) i ( n ) | S ( i ) i ( n ) (cid:17) F ′′ ( c, n − i − c ) . Letus define F ( i, n − i ) := F ′′ ( i, n − i ). The thesis follows by inspection. Proof of Lemma 1 Proof. We consider the matricial expression TF = d (see Sect. 5.2) andwrite the elements of d , T for p small: d Ψ( i,a ) = 1 λi ( N − i ) − aλi p + o ( p ):= α Ψ( i,a ) − β Ψ( i,a ) p + o ( p ) T Ψ( i,a ) , Ψ( i +1 , = − o ( p ) T Ψ( i,a ) , Ψ( i + c,c ) = o ( p ) , c = 2 , . . . , N − i − , T i,i = 1. We can write for 2 ≤ i ≤ N − F (1) ( i ) = α Ψ( i, − β Ψ( i, p + F (1) ( i + 1) + o ( p ) ,F (1) ( N − 1) = α Ψ( N − , − β Ψ( N − , p + o ( p ) F (1) = α Ψ(1 , − β Ψ(1 , p + F (1) (2) + o ( p ) . Hence, we find that F (1 , N − 1) = N − X i =1 d Ψ( i, = N − X i =1 λi ( N − i ) − N − X i =1 λi p + o ( p )= F (1) − λ − H N − p + o ( p ) Proof of Proposition 6.4 Proof. Let us prove the thesis by induction on N . 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