A Graph-Based Model for Disconnected Ad Hoc Networks - CiteSeerX

A Graph-Based Model for Disconnected Ad Hoc Networks Francesco De Pellegrini, Daniele Miorandi, Iacopo Carreras and Imrich Chlamtac CREATE-NET International Research Center via Solteri 38/A, 38100 Trento, Italy email: [email protected]

Abstract— Recently, research on disconnected networks has been fostered by several studies on delay-tolerant networks, which are designed in order to sustain disconnected operations. We focus on the emerging notion of connectivity which exists in such networks, where the message exchange between nodes is enforced by leveraging storage capabilities at intermediate relays, with the aim of achieving connectivity over time. The problem, under the constraint of intermittent connectivity, is hence to devise efficient mechanisms for message delivery, and evaluate the performance thereof. In this paper, we introduce a graph-based model able to capture the evolution of the connectivity properties of such systems over time. We show that, for most networks of interest, such connectivity graphs can be modeled as Erd¨os-Renyi random graphs. Furthermore, we show that, under a uniformity assumption, the time taken for the connectivity graph to become connected scales n as Θ( n log ) with the number of nodes in the network. Hence λ we found that, using epidemic routing techniques, message delay log2 n ). The model is validated by numerical simulations is O( λnlog log n and by a comparison with the connectivity patterns emerging from real experiments. Index Terms— wireless networks, connectivity graph, delaytolerant networking, Erd¨os-Renyi graphs, message latency.

I. I NTRODUCTION The current Internet protocols are designed assuming that the underlying network complies to certain requirements, i.e., rather small end-to-end round-trip delays, the existence of a path between source and destination and small link error rates. In practice, these assumptions fail for a wide spectrum of communication systems, where connectivity is intermittent due to sudden and repeated changes of environmental conditions, such as nodes moving out of range, drop of link capacity or short end-node duty cycles, as in power-saved sensor networks. Further customary examples of application scenarios include underwater networks, satellite networks, tactical and emergency networks [1–3]. Networks operating in such conditions, nevertheless, can sustain a wide set of applications which are not compromised by significant delays, such as periodical massive data exchanges, database updates, file transfers, e-mailing and, in general, those applications where data are valid on a This work has been partially funded by the European Commission within the framework of the BIONETS project EU-IST-FET-SAC-FP6-027748, www.bionets.org.

much longer time scale than topology changes. Thus, such disconnected architectures could prove strategical where the deployment of a wired infrastructure is not devisable due to cost problems. All the aforementioned examples share the property of being disconnected most of the time. Recently, a network paradigm, namely Delay Tolerant Network (DTN), has been proposed to enforce resilience to repeated network partitions [1, 4, 5]. In order to overcome frequent network partitions, DTN nodes leverage their storage capabilities, so that the common notion of routing could be better rephrased as store-carry-andforward [6]. The opportunities for pairs of nodes to communicate are ephemeral and, to this respect, communication among nodes is conveniently described by the set of node contacts [1, 7]. When a contact between a pair of nodes occurs, they are able to exchange a certain amount of data, namely a message. Contacts describe the occurrence of several cases resulting from intermittent links, ranging from the case of LEO satellite networks, where contacts have a regular pattern, to contacts generated by mobile devices, coming by chance into mutual radio range. Due to their potential impact, most of research on DTNs focused on sparse mobile ad hoc networks [3, 5–7], a domain where intermittent connectivity is due to node mobility and to limited radio coverage. In such cases, traditional end-to-end communication techniques have to be enforced or replaced with some mechanisms to reconcile disrupted paths. To this respect, in particular, some authors denote DTN networks from a different perspective, calling them Disruption Tolerant Networks [5], stressing the resilience to unreliable network links and the lack of a robust connected topology. This paper is concerned with the scalability of delay tolerant networks, with respect to the increase of the number of nodes. In particular, we focus on the notion of connectivity over such networks. The contribution of this paper can be resumed as follows. First, we introduce a graph model able to describe the connectivity properties of a DTN over a finite time interval. Furthermore, we show that, for a uniform scenario, the resulting contact graph is an Erd¨os-Renyi (ER) graph. Then, we leverage the behavior of ER graphs in order to derive scaling laws in the number of nodes for message delay. Our starting point is that, since in the case of DTNs

connectivity is not guaranteed all the time, but over time, we can build a connectivity graph that adheres to the dynamics of the contacts occurring among nodes. When such graph gets connected, we leverage the length of the longest route in ER graphs, to obtain scaling laws in the number of nodes. To some extent our approach is the analogous of similar techniques used to bound the delay in IP networks [8]. The paper is organized as follows. In section II, we describe related works on DTNs, and focus on the notion of connectivity graphs given in previous works. Section III describes the system model we work with. In Section IV, we present the main results on the scaling law of message delay in disconnected mobile ad hoc networks, under epidemic routing techniques. Section V reports numerical results, where we tested our model against both synthetic mobility traces and real-world ones. Finally, the last section is devoted to concluding remarks.

The literature reports several results of real experiments on delay tolerant networks [5, 16, 18]. In [5], the DieselNet network was deployed over a wide urban area, using buses as mobiles. Authors of [16] describe the use of human mobility to diffuse informations through portable devices. In such works, authors discuss several technical issues. Special attention is given to existing routing protocols for DTNs, where the major design target is to maximize the fraction of delivered packets and to minimize the latency between source and destination. It is quite apparent that one major difficulty encountered in DTNs is the derivation of suitable robust routing algorithms. We can either leverage an oracle and design message delivery by node contact sequences, or we can use epidemic routing techniques instead, but, message delivery ultimately relies on the underlying contact process and on a notion of connectivity over time.

II. R ELATED W ORKS

Even though the common notion of a connectivity graph is well established in ad hoc network literature, defining a similar, useful notion of a connectivity graph for DTNs is not immediate. Some proposals exist in literature. The authors of [1] propose the notion of DTN multigraph, where several edges may exist between pairs of nodes, each weighted with the capacity and delay functions. Also, space-time graphs have been defined in [19], under perfect information of connectivity at each point in time. Space time-graphs are constructed sampling the instantaneous connectivity over a set of consecutive disjoint time intervals, where topology and link capacity do not change. Such instances are then pasted and linked by directed edges in order to construct the final instance of the space-time graph, which ultimately is an oriented graph. The construction basically corresponds to the time-expanded graph construction proposed in [20]. There, the effects of dynamic changes of the network edge capacities are included in a transhipment problem where arcs posses both a capacity and a traversal time. The corresponding maximum dynamic network flow problem was formulated over a time-expanded graph. The works [21, 22] provide a general framework for the problem of dynamic network topology. They introduce the notion of an evolving graph, which is a sequence of graphs constructed from the presence schedule of all nodes and links over a set of intervals. They introduce the notion of journey which corresponds, over time, to the concept of a path. The authors proved that the problem of building a minimum cost spanning tree in such domain (evolving spanning tree) is N Phard [22]. Here, as in most DTN literature, we will implicitly refer to the definition of a journey, but, we do not assume any apriori knowledge on the dynamics of connectivity.

From the theoretical viewpoint, a reference work for mobile DTNs can be considered the pioneering work of Glossgauser and Tse [9]. The results therein focused on the (broader) problem of the capacity of ad hoc networks. The main finding was that mobility can be leveraged to increase capacity. The hint was a viable direction to overcome the severe limitations in static networks capacity [10]. Along this line, later research proved that such improvement comes at the price of an inherent trade-off with the delay required to deliver a message to destination [11]. Furthermore, the authors of [12] confirmed that throughput results on the reference two-hop protocol in [9] are tight, in the sense that redundant messages cannot increase the network capacity, but, nevertheless, redundancy is beneficial to improve delay. Most literature on DTNs addresses the problem of routing, where a key role is played by the knowledge on input variables such as contact times, traffic demands or memory occupation [2]. The assumptions for routing in DTNs may range from the case of perfect or partial information, when a so called oracle is assumed to trace input variables, to the case of routing with zero knowledge. In [6, 13], the authors describe a network architecture, where special nodes, named ferries, carry data among nodes. In [6], ferry mobility is designed to minimize the duration of the ferry tours and achieve high contact probabilities with nodes. Multiple ferries route design with minimum delay is studied in [13]. In this paper we exclude apriori to operate mobility control. A different research line adopts epidemic diffusion algorithms [14, 15], also named epidemic routing or controlled flooding. A leitmotiv of epidemic message diffusion is then how to fill the gap between pure flooding, on one hand, and the original two hop protocol in [9], on the other. The paper in [16], gives insight in the behavior of DTNs diffusion based on the opportunistic exchange of data between nodes in radio range. An in deep analysis of the two-hop relaying protocol is detailed in [17].

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III. S YSTEM M ODEL We restrict our attention to a special class of DTN networks, i.e., DTNs of mobile nodes, where intermittent connectivity is due to the high node mobility. In these networks, mobility is the engine that permits network-wide information diffusion. We remark that, on one hand, this popular DTN model refers

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at x ∈ R [24]. We define as intrameeting times the sequence {Yn }n∈Z , where Yn = Tn − Tn−1 . Notice that the notion of intrameeting times used in our framework is different from the notion of intermeeting times used in literature (see for example in [16]), which refers to the successive meetings of a specific pair of nodes. Finally, with standard notation, the diameter of graph G is the greatest distance between any two vertices in G. The average degree of nodes of G will be briefly denoted d.

IV. DTN C ONNECTIVITY G RAPH In this section we introduce some definitions aimed at capturing the properties over time of DTNs connectivity. The main definition introduced here is the DTN connectivity graph, in the following simply connectivity graph. Given a DTN and the associated marked point process {Zn }n , the connectivity graph G(t0 ,D) = (V, E) is a graph where the set of vertices is V = {1, 2, . . . , n} and the set of edges E is characterized as (i, j) ∈ E if and only if ∃ Tn such that t0 ≤ Tn ≤ to + D, σn = (i, j). (1)

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to a practical case of interest, but the results we are presenting in the following easily generalize to other DTN networks. In what follows, we assume that all the random processes of interest are defined on a suitable probability space {Ω, F, P}. In particular, we consider a set of n nodes over an area A, each of them denoted by a unique ID, for simplicity 1, 2, 3, . . . , n. In DTNs of mobile nodes, the notion of a contact interval denotes the period of time when two nodes are able to exchange a message. We let two nodes be able to communicate when they are within reciprocal radio range R > 0, and communications are bidirectional. We also introduce two further simplifying hypothesis: (a1) Contact intervals are sufficient by long to exchange all messages: this let us consider nodes meeting times only, i.e., time instants at which a pair of not connected nodes fall within reciprocal radio range; (a2) We assume that all nodes have infinite buffer; We observe that storage, in DTN literature, is a primary concern, especially when epidemic routing is employed. In sight of the previous assumptions, the bounds provided in this work should be considered a reference ideal case. In what follows, also, the underlying assumption is that the network is in the subcritical regime, meaning that it is not connected with high probability [23]. Furthermore, we assume that the size of each connected component is small and thus multi-hopping is not feasible. Given a DTN and a mobility pattern, we associate the DTN with a marked point process {Zn }n∈Z = {Tn , σn }n∈Z [24]. Sequence {Tn }n∈Z represents meeting times. The marks of the sequence {σn }n∈Z take the form σn = (i, j), where i, j ∈ S denotes the (unordered) pairs of IDs of the devices falling within communication range. We associate with Tn a counting process N , defined as P N (A) = δTn , for each A ⊂ R, δx being the Dirac measure

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Fig. 1. Example of the construction of a DTN connectivity graph. A realization for the sequence of the intrameeting times is reported along the vertical time axis on the left, marks are reported on the right of the axis.

It is clear that set of edges E is a function of time and, in the following, we will denote parameter D as the window length. The above definition describes also the simple procedure to construct the connectivity graph. We start with an empty graph with no edges and n nodes at time t0 , and we parse the time interval up to time t0 + D. Each time we register a contact between a pair of nodes (i, j), if such a pair of nodes meet for the first time, then we add the corresponding edge (i, j) to the set of edges of the connectivity graph. In Fig. 1, we reported an example of the connectivity graph which corresponds to the mobility pattern of 5 nodes. The connectivity graph is built over time: each time a new pair of nodes meet, an edge is drawn between them. The mobility pattern, in particular, is captured by the sequence of marks corresponding to intrameetings, stamped along the time axis. We remark two features of the connectivity graph that are clarified from the example. First, replicated edges are counted only once; we notice, that edge (3, 2) is counted once, even if the corresponding mark appears twice. The connectivity graph collects, to some extent, the presence information on the contact process, meaning that if an edge exists, then at least one contact between two nodes appeared in the given interval. Second, the information on the order of the sequence of contacts between nodes is not preserved. In practice, even if nodes 2 and 5 are connected in the resulting graph, this does not guarantee that a message can propagate along such path: for example, node 5 could exchange a message and relay it to node 2 using node 3 as intermediate relay, but the converse is not possible. Nevertheless, the cumulative information on the contact process retained by the connectivity graph let us obtain scaling laws for the delay in DTN networks, under uniformity assumptions on the mobility pattern.

A. Mobility model Even if the connectivity graph can be defined for any DTN, to simplify our analysis, we restrict to mobile DTNs which can be modeled through a particular class of mobility patterns. In particular, we will refer to the class of mobility patterns, named the Marks-Memoryless (M 2 ), introduced in [25]. The M 2 class collects all the mobility patterns where the mobility pattern is such that (1) marks {σn } are i.i.d; (2) intrameeting times {Yn }n∈Z are nonnegative i.i.d. The statistical tests performed in [25], proved that the M 2 shows a very good statistical fit with mobility models used in ad hoc networking, such as Random Waypoint Mobility (RWM), Random Direction Model (RDM) together with their generalizations [26]. Notice that such a class of mobility models is “memoryless” in the sense that (i) there is no correlation between the IDs of the nodes that meet in successive meeting times (ii) the process {Tn }n∈Z is a renewal process [27]. Nevertheless, the term memoryless does not imply that the mobility pattern is memoryless. On the contrary, it somewhat implies some form of regularity in the mobility patterns, such that the sequences {σn }n∈Z and {Yn }n∈Z are i.i.d. Furthermore, we denote by π the stationary probability distribution of the marks σn , so that we have: π(i, j) = P [σ0 = (i, j)], i, j ∈ S. (2) B. Diffusion and Bounds

As the DTN connectivity graph carries the cumulative information on the meetings occurring up to a given instant, its evolution depends on the window length D considered. Furthermore, under the hypothesis that π(i, j) > 0 for ∀i, j, it is trivial that the connectivity graph will converge to the complete graph as D → ∞. Therefore, edge (i, j) will belong to the connectivity graph with a probability which in general is a continuous, non-decreasing function of the window size ∂ pij (D) ≥ 0. D, pij = pij (D), ∂D We introduce a further uniformity assumption, meaning that all nodes move according to the same mobility model. Hence, by symmetry, the marks will have a uniform distribution,
which means π(i, j) = 1/C, where C = n2 is the maximum number of possible edges. If this is the case, the growth of the connectivity graph described by (1), will be performed adding incrementally edges at each intrameeting, but, with no preferential attachment. Hence, we are basically excluding the occurrence of power laws [28], which might appear in the case of non-uniform marks distribution. Thus, pij (D) = p is independent of the considered edge. This means that the connectivity graph will be an Erd¨os-Renyi random graph [29]. In the following we will use two features of ER graphs. The first property states that connectivity occurs abruptly above a corresponding threshold value for p(n). The scaling of p(n) for ER graphs is such that almost all ER graphs will be connected when the average degree of a node, p(n − 1), scales faster than log(n), and disconnected below such threshold.

Second, we will employ a result on the diameter of ER random graphs, stating that above the threshold, the diameter of random graphs is almost surely concentrated on a few log n values around log np [30]. For general properties of almost all graphs see [29, 31]. In the rest of the paper, we show that it is possible to leverage general properties of ER graphs to characterize how the typical interval of observation should scale in order to make the DTN connectivity graph a connected one. In particular, we want to determine the corresponding scaling law of the window length, at the increase of the number of nodes. It turns out that the time needed in order for all the nodes to be connected in the connectivity graph grows as a mild superlinear function of n. In what follows, we will briefly refer to D ∗ as the critical window (length). Note that a major assumption, in order to derive next results, is that the intensity of the meeting process is finite, 0 < λ < ∞, and does not depend on the number of nodes. In order to simplify our analysis, we will further ask for stationarity on the mobility process, which let us drop the dependence from the initial instant. Hence, given window length D, we let L(D) the number of edges of the connectivity graph, and L(D) = E [L(D)]. We also denote p∗ = p(D∗ ) the ER edge probability at the threshold, so that p∗ (n − 1) = log(n). (3) Also, the condition on the average degree of a graph brings n(n − 1)p∗ = 2L(D ∗ ), (4) and, finally, we obtain L(D∗ ) = n log(n)/2. (5) 1) Critical window: Equation (5) relates the number of nodes and the critical window D ∗ , through the average number of edges of the connectivity graph. It turns out that D∗ can be expressed directly as a function of n, as shown in the next sections. Then, the following result holds:1 Theorem 1 In a DTN of n mobile nodes, satisfying a Marks Memoryless mobility model, it holds   n log(n) . (6) D∗ = Θ λ The proof of the above statement is derived in the Appendix for the discrete model. 2) Fluid approximation: Here, we will gain direct insight into the dynamics of connectivity introducing a fluid approximation, under the assumption that n  1. Under the fluid approximation, the number of novel edges added at each intrameeting can be expressed in terms of the arrival counting process. In fact, under the uniformity assumptions introduced before, the following rate equation regulates the variation in 1 Given

two functions f (n) and g(n), we use the following notation: (i) f (n) f (n) f (n) = o(g(n)) if g(n) → 0 as n → +∞, (ii) f (n) = O(g(n)) if g(n) is upper bounded for n large enough, (iii) f (n) = ω(g(n)) if g(n) = o(f (n)), (iv) f (n) = Ω(g(n)) if g(n) = O(f (n)) and (v) f (n) = Θ(g(n)) if f (n) = O(g(n)) and f (n) = Ω(g(n)).

the number of edges of the connectivity graph ∂L L =1− , (7) ∂N C where C is the number of all possible edges; the interpretation of (7) is immediate: the probability that a new edge is added at intrameeting times is given by the fraction of pending edges. From (7), we have ∂(L − C) 1 = − ∂N , (8) L−C C which solves for L = C(1 − e−N /C ). (9) Replacing (9) into (5), we obtain   log(n) . (10) N /C = − log 1 − n−1 Using the Newton-Mercator series for log(1 + x) = ∞ X (−1)k+1 xk /k, −1 < x ≤ 1, we can write (10) as

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n 1 log(n) + n(n − 1)o(log(n)/n), (11) 2 2 and, since from renewal theory N /D converges to λ, it follows D∗ = O(n log(n)/λ). Conversely, we can use the inequality x < − log(1 − x), x < 1, x 6= 0, and from (10) we can write also 1 (12) N > (n − 1) log(n). 2 ∗ ∗ so that D = Ω((n − 1) log(n)/λ). Finally, since D = Θ(n log(n)/λ), we obtain again the result found in Thm. 1. Notice that, in the above calculations, we assumed λ = Θ(1). But, this may require, in turn, a suitable scaling law for the nodes’ speed. To see this, we consider the dense network case [32], where n nodes move in the unit square. There, each node encounters on the order of nv(n) nodes per second, so that the number of intrameetigs per second is on the order of n2 v(n): this rationale shows that λ = Θ(1) holds if the nodes’s speed scales as v = Θ(n−2 ). In the extended case, conversely, nodes move in the squared box of area √ n: here the typical distance between nodes increases as n, so that a rescaling argument from the dense case suggests that the 3 nodes’ speed needs to scale as v = Θ(n− 2 ). 3) Delay bound: Let us assume that a message is generated by node i and intended for node j: since the network we consider is subcritically connected, with unitary probability, the message cannot be delivered using any multihopping technique, the reason simply being that almost surely a path between any two nodes does not exist. Nevertheless, using opportunistic forwarding it is possible to deliver the message, storing it at intermediate nodes. We remark, that under a Marks Memoryless model, using either an epidemic diffusion mechanism [14, 15] or the two-hop routing mechanism [9], the message will be delivered to the destination node with unitary probability over an infinite time interval. As we noticed before, an important issue for DTNs is the mean delay for a message originating at node i to reach node j, which we denote Di,j . Finally, we find the following scaling law for the message delay, whose proof is deferred to the Appendix. N =

Theorem 2 In a DTN of n mobile nodes, satisfying a Marks Memoryless mobility model, it holds   n log2 (n) . (13) Dij = O λ log(log(n)) 4) Discussion of the results: As we described before, the existence of a path between two nodes in the connectivity graph does not guarantee that a message can be delivered between the end nodes. Nevertheless, the scaling law of D ∗ gives insight on how to dimension certain routing parameters in DTNs. One of such parameters, which is quite critical in DTN epidemic routing, is the duration of timeouts after which a message can be safely discarded. In the case of epidemic routing, in fact, this becomes a major concern, due to the replication of multiple copies of the same message. Under disconnected operations, in practice, a common paradigm is the use of k-relaying protocols [16], where every node replicates a copy of a message k times to nodes not already receiving the message. The indications from Thm. 1 is that message timeouts should scale as Ω(n log(n)/λ). Also, if messages are acknowledged at the receiver side, the result  of 2Thm. 2 n log (n) . suggests that the timeout scaling should be O λ log(log(n))

We notice that the result in Thm. 2 provides also a loose lower bound on the storage requirements of DTNs networks. If the message generation rate at each node is 0 < µ < ∞, a straightforward application of Little’s law suggests that the number of messages stored will increase at least as µDij . Thus, neglecting the storage for relaying, in a DTN memory n log2 (n) requirements are Ω λ log(log(n)) . V. N UMERICAL R ESULTS

As we described before, one main rationale underlying our model is that, under uniformity assumptions on mobility, the DTN connectivity graph matches an ER graph. We validated our model comparing the typical parameters [31] of the connectivity graph of a DTN with the corresponding values of a ER random graph. The tests were performed as follows. Given a trace of the contact process, and a value of D, we generated the connectivity graphs over consecutive disjoint intervals. For each realization of a connectivity graph, we calculated the corresponding ER graph parameter p = d/(n − 1). Then we averaged and compared the degree distribution, the clustering coefficient and the graph diameter.
The clustering coefficient of a node i is the ratio C`i = Ti / k2i , where Ti is the number of edges among the ki neighbors of node i, and the size of a complete graph with the same number of nodes; the clustering coefficient of graph G is the average of C`i over the set of nodes. The clustering coefficient describes the probability of a node to belong to a complete cluster of nodes [31]. For a ER random graph of parameter p, C` = p, since there exist no preferred attachment rule. Finally, we validated the fluid model approximation for the number of edges of the connectivity graph.

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Synthetic mobility traces We first considered the contact traces recorded in a simulation of a set of N = 60 mobile nodes moving according to a Random Waypoint mobility model, over a square area of 10000 m2 with a speed of 5 m/s and zero pause intervals; the initial set-up of simulations adhered to the stationarity conditions indicated in [33]. The simulations were performed using the Omnet++ simulation platform [34]. The first quantity under consideration is the probability mass distribution of nodes’ degree. As depicted in Fig. 2, connectivity graphs match the binomial degree distribution of a ER graph. Nevertheless, certain structural properties of graphs are not caught by the plain degree distribution. Thus, we further tested the clustering coefficient and, as depicted in Fig. 3, we found a very good match. Notice that for small values of the window length, the clustering coefficient is over-estimated: this happens because a large fraction of isolated nodes exists, thus lowering the average value. Finally, we tested the behavior of the diameter of connectivity graphs at the increase of the window length, reporting also the reference trend for the ER model, when samples are generated with the same parameter p.2 As expected, DTN connectivity graphs show a decreasing diameter at the increase of the window length. We notice that the diameter of the DTN connectivity graphs resemble closely the same trend shown by the ER graphs. In the case 2 We

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Fig. 5. Fluid model approximation of the number of edges of the connectivity graph, RWP mobility model.

of the RWP mobility, the very good match with our model is not surprising, since the underlying mobility process provides a fairly uniform distribution in the marks. Also, we tested the fluid model against the actual growth of the connectivity graph, and, as reported in Fig. 5, a close match exists with the outcomes of synthetic mobility traces. Real-world traces. We performed further tests on the contact traces generated by the DieselNet networks, and available online [5, 35]. Such contact traces were collected on a daily base, and the core network is represented by 20 devices, using a IEEE802.11 interface and mounted on buses. A preliminary analysis showed that the mark distribution, in this case, is not uniform, and, moreover, certain contacts do not appear in the trace, meaning that, as expected, certain nodes do not meet at all (this is quite intuitive since, usually, not all bus routes are supposed to intersect). We notice that, in the case of the DieselNet, the contact traces are clearly not stationary, and to this extent, p represents only a benchmark for our model; one major consequence is that p is not guaranteed to be monotonically non decreasing, depending both on the window length and the initial epoch of observation t0 . In order to have a significant number of contact events, we did not considered nodes which generated less than 20 contacts over the day. Thus, the network is restricted to a subnetwork of 17 nodes, for a run of 76686 s. In the overall trace, the maximum number of possible contact pairs was C = 136,

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Fig. 7. Clustering coefficient versus window length, DieselNet trace 3022005.

but 56 pairs were not present. The mean intrameeting time is 156.629 s with a maximum intrameeting time of 2262 s. The degree distribution shows that the connectivity graph has a heavier tail for the real-world trace, meaning that there exist a few nodes which connect more than others (preferential connectivity), for example, as reported in Fig. 6, for D = 30000 s there is a larger frequency of nodes of degree 10 and 11 than expected, and, meanwhile, a non-negligible number of nodes which experienced only one contact. The tests on the clustering, see Fig. 7, confirm that, as we expected, realworld traces have clustering coefficients higher than in the corresponding ER model, as it appears at higher values of the window length. Furthermore, the diameter is higher than in the case of the ER model, as shown in Fig. 8. This confirms the results on clustering, indicating a certain locality in the contact sequence between nodes. Finally, as reported in Fig. 9, the growth of the connectivity graph, is much slower than predicted by our model. This is in line with the fact that, under preferential attachment, many of the contacts will be repeated compared to the case of the ER model, which assumes uniform mark probability. VI. C ONCLUSIONS In this paper, we presented a graph based model which gives insight in the connectivity properties of disconnected mobile ad hoc networks, which are emerging recently as a novel research field for delay tolerant applications. Under uniformity assumptions in the mobility pattern, we derived a scaling law

Fig. 9. Fluid model approximation of the number of edges of the connectivity graph, DieselNet trace 3022005.

for message delay. We validated our model with synthetic mobility contact traces, and we found a fairly good agreement even with real-world traces. The model, anyhow, is not aimed at capturing the presence of preferential attachment rules, occurring when contacts are not evenly distributed among mobile nodes. Nevertheless, we showed that it can provide a first order description for such systems. An interesting direction for future research work is to model locality and preferential attachment rules in DTNs connectivity. Also, we believe that a central issue, as emerged from real-world analysis, is to trace non-stationary patterns of contacts among nodes. R EFERENCES [1] K. Fall, “A delay-tolerant network architecture for challenged internets,” in Proc. of ACM SIGCOMM, Karlsruhe, Germany, March 25–29, 2003. [2] S. Jain, K. Fall, and R. Patra, “Routing in a delay tolerant network,” SIGCOMM Comp. Comm. Rev., vol. 34, no. 4, pp. 145–158, Oct. 2004. [3] U. Lee, E. Magistretti, B. Zhou, M. Gerla, P. Bellavista, and A. Corradi, “MobEyes: smart mobs for urban monitoring with vehicular sensor networks,” UCLA CSD, Tech. Rep. 060015, 2006. [Online]. Available: http://netlab.cs.ucla.edu/wiki/files/mobeyestr06.pdf [4] S. Burleigh, L. Torgerson, K. Fall, V. Cerf, B. Durst, K. Scott, and H. Weiss, “Delay-tolerant networking: an approach to interplanetary Internet,” IEEE Comm. Mag., vol. 41, no. 6, pp. 128–136, June 2003. [5] J. Burgess, B. Gallagher, D. Jensen, and B. N. Levine, “Maxprop: Routing for vehicle-based disruption-tolerant networking,” in Proc. of IEEE INFOCOM, Barcelona, Spain, April 23–29, 2006. [6] M. M. B. Tariq, M. Ammar, and E. Zegura, “Message ferry route design for sparse ad hoc networks with mobile nodes,” in Proc. of ACM MobiHoc, Florence, Italy, May 22–25, 2006, pp. 37–48. [7] J. Leguay, A. Lindgren, J. Scott, T. Friedman, and J. Crowcroft, “Opportunistic content distribution in a urban setting,” in Proc. of ACM Chants, Florence, IT, September 15, 2006.

1−1/C

1 0

1

2

1/C

2/C

Fig. 10.










1/C

2/C

3/C

1−2/C

C−2

C−1

1−2/C

1−1/C

C

1

The embedded Markov-Chain for the semi-Markov process L(t).

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A PPENDIX Growth of the Connectivity Graph In order to express D ∗ as a function of n, we observe that L(t) can be described by a corresponding semi-Markov process [27]. In particular, the number of states x ∈ X is finite and corresponds to the number of edges L(t), i.e., X ∈ 1, 2, . . . , C. Also, as customary in the analysis of semi-Markov processes, sampling L(t) at the time when state transitions occur, each state i is associated to a transition probability to any other state j, Pi,j = P { x(Tn+1 ) = j| x(Tn ) = i }. Finally, the sojourn on every state x is regulated by the intrameeting times and the marks probability. In the following we denote τi is the sojourn time in state i. From the previous discussion, it follows that: (i) the transition probabilities correspond to the addition of one edge at the time i Pi,i+1 = P [x(Tn+1 ) = i + 1|x(Tn ) = i] = 1 − , C Pi,i = 1 − Pi,i+1 (14) Pn k (ii) the sojourn times are given by τk = h=1 Yh , where nk is a modified geometric random variable (r.v.) of parameter Pk,k+1 , meaning that P [nk = h] = Pk,k+1 (1 − Pk,k+1 )h−1 , independent of {Yh }. The embedded Markov Chain corresponding to L(t) is represented in Fig. 10. In what follows we employ the Laplace-Stieltjies transform (LST) of the density of a r.v. x with assigned probability density function, fx (t), Hx (s) = E [e−sx ] = LST [fx (t)| s]. Let also Fx (t) = P [ x ≤ t ] be the distribution function of x. The time θk taken for reaching the state k P is simply given k−1 by the summation of k sojourn times, θk = h=0 τh . If we let fθk (t) the probability density function of the time taken to reach state k, and Hθk (s) = LST [fθk (t)| s] the corresponding LST, then k−1 Y (15) Hθk (s) = Hτh (s). h=1

where Hτh (s) is the LST of the density function of τh . With standard notation, Gk (z) = E [z nk ] is the moment generating function of the discrete r. v. nk , ∞ X zPk,k+1 Gk (z) = Pk,k+1 z h (1−Pk,k+1 )h−1 = 1 − (1 − Pk,k+1 )z h=1 (16) so that Hτk (s) = Gk (HY (s)) boils down to (1 − Ck )HY (s) Pk,k+1 HY (s) = . (17) Hτk (s) = 1 − (1 − Pk,k+1 )HY (s) 1 − Ck HY (s) We can then write k Y (1 − Ch ) Hθk (s) = HYk (s) (18) 1 − Ch HY (s) h=1 Let us denote pk (t) = P [L(t) = k], so that pC (t) = P [θC ≤ t] = FθC (t)

pk (t) = P [ θk ≤ t] − P [ θk+1 ≤ t] = Fθk (t) − Fθk+1 (t), k = 1, . . . , C − 1

(19) p0 (t) = P [ θ1 > t] = 1 − Fθ1 (t). Now, we can then determine a closed form expression for the moment generation function of L(t), which writes C i X h z k pk (t) GL(t) (z) = E z L(t) = k=1

= (1 − Fθ1 (t)) + z(Fθ1 (t) − Fθ2 (t))

+ z 2 (Fθ2 (t) − Fθ3 (t)) + z 3 (Fθ3 (t) − Fθ4 (t)) . . . + z (C−1) (FθC−1 (t) − FθC (t)) + z C FθC (t) = 1 + (z − 1)

C X

z k−1 Fθk (t)

(20)

k=1

and the average edge number of the connectivity graph is C X d L(t) = GL(t) (z) = (21) Fθk (t) dz z=1

k=1

so that, in the Laplace domain, it holds  k (C−1)! HY (s) C X (k−1)! C sL(s) = . (22) Qk h h=1 1 − C HY (s) k=1 In order to proceed further with the results, we need to introduce some mild regularity assumptions on the intrameeting times, i.e., we ask that HY (s) exists over Re[s] > 0, as in the case of most nonnegative random variables, such as exponentials, Pareto, and all distributions with finite support. We can leverage the following

Lemma 1 If HY (s) exists in Re[s] > 0, then L(s) has a double pole in the origin and exists in the same region. Proof: The only possible poles are the zeros of the factors appearing on the denominator of (22), i.e., the solutions of HY (s) = Ck , for 2 ≤ k ≤ C. For k = C, a pole exists for s = 0, thus s = 0 having multiplicity 2. Now, if any other pole of L(s) exists, then it should be a real-valued pole and

satisfy, for some 2 ≤ k < C, and some sk > 0, Z +∞ C = fy (a)e−sk a da ≤ 1. k 0 But, this is not possible, since C/k > 1.

(23)

Proof of theorem 1 Proof: We prove first the asymptotic lower bound. From Lemma 1, L(s) is analytical for any Re[s] > 0. We notice that since L0 (t) ≥ 0, the Tauberian properties of the LST [36], let us write, for any s within the convergence region lim L(t)e−st = 0. (24) t→∞

Thus, let us choose sequence 0 < sn = λ/n, and fix a certain δ > 0: then, there exist a sequence D(n) so that, for every n, if t > D(n), then L(D(n)) < L(t) ≤ δeλD(n)/n . (25) Now, in order for the window length to be larger than the critical window length, from (5), we need to obtain L(D(n)) > n log(n)/2. Hence, from (25) a lower bound the critical ∗ window can be imposed as n log(n) < 2δeλD (n)/n . Then, n n n ∗ D (n) > λ log(2δn log(n)) = λ log(n) + λ log(2δ log(n)), which brings D ∗ (n) = Ω( nλ log(n)). Now, we have to prove the asymptotic upper bound. We notice that it holds FθC (t) ≤ FθC−1 (t) ≤ . . . ≤ Fθ0 (t), and from (21) (26) L(t) ≥ CFθC (t) Also, we consider a sequence of time instants t = nλ log n, and verify that, for large n, L( nλ log n) ≥ n log(n)/2. In fact, we can consider the stronger condition   n log n CFθC ≥ n log(n)/2 (27) λ which can be written  n log n log n . (28) ≥ Fθ C λ n−1 We observe that (28) is verified indeed for n large enough, since the left hand term is monotonically non decreasing, and the right hand term is monotonically decreasing. Hence, it follows that D ∗ (n) = O( nλ log(n)). Proof of theorem 2 Proof: Let us consider a sequence of consecutive time intervals and the corresponding sequence of connectivity graphs, associated with the Marks Memoryless model. Let V the average diameter of such graphs. We can choose window length D = (1 + )D ∗ for  > 0, such that the connectivity graphs will be connected with high probability. Hence, for message propagation, we can use Wald’s inequality and bound the mean delivery time as Di,j ≤ DV, (29) where the right hand term represents the worst case when at each interval only one hop is made towards the destination. Hence, we can plug (6) into (29), and write   n log2 (n) Di,j = O , (30) λ log(np(D)) where above the threshold V = log(n)/ log(np). The result follows since above the critical window np(D) ≥ log(n).