Performance of Opportunistic Epidemic Routing on Edge-Markovian Dynamic Graphs

Transcription

1 IEEE TRANSACTIONS ON COMMUNICATIONS, TCOM-9-63 Performance of Opportunistic Epiemic Routing on Ege-Markovian Dynamic Graphs John Whitbeck, Vania Conan, an Marcelo Dias e Amorim arxiv:99.29v3 [cs.ni] 25 Nov 2 Abstract Connectivity patterns in intermittently-connecte mobile networks (ICMN) can be moele as ege-markovian ynamic graphs. We propose a new moel for epiemic propagation on such graphs an calculate a close-form expression that links the best achievable elivery ratio to common ICMN parameters such as message size, maximum tolerate elay, an link lifetime. These theoretical results are compare to those obtaine by replaying a real-life contact trace. Inex Terms Intermittently-connecte Mobile Networks, Network Moeling, Markovian Ranom Graphs, Epiemic Routing I. INTRODUCTION Intermittently connecte mobile networks (ICMN) emerge from the social processes that bring mobile evices into contact. Due to high noe mobility an frequent lack of en-to-en connectivity in such networks, message transport is usually hanle in a store-an-forwar fashion by elay/isruptivetolerant network (DTN) routing protocols []. The topology of a real-life network of mobile evices evolves over time as links come up an own. A network s connectivity graph is efine by associating each noe to a vertex an aing an ege between any pair of noes that are currently in contact (i.e., within transmission range of each other). Successive snapshots of the evolving connectivity graph yiel a ynamic graph, i.e., a time-inexe sequence of static connectivity graphs. Their theoretical stuy is therefore important for unerstaning the unerlying network ynamics. In this paper, we propose a new Markovian moel for flooing on ege-markovian ynamic graphs [2]. Unlike previous work on asymptotic behavior [2], our approach assumes source-estination pairs for messages, a finite number of noes, as well as finite link capacities an message sizes. Our main contribution is a close-form expression of the bunle elivery ratio as a function of bunle size, maximum tolerate elay, an the ynamics of the unerlying ege- Markovian ynamic graph. Using this moel, we show that A poster of this work was presente at the ACM SIGCOMM Workshop on Networking, Systems, Applications on Mobile Hanhels (Mobihel 29). This version is more etaile an contains many more results. This work has been partially supporte by the ANR project Crow uner contract ANR-8-VERS-6. John Whitbeck is with both UPMC Sorbonne Universités an Thalès Communications, France. john.whitbeck@lip6.fr. Vania Conan is with Thalès Communications, France. vania.conan@fr.thalesgroup.com. Marcelo Dias e Amorim is with LIP6/CNRS UPMC Sorbonne Universités, France. marcelo.amorim@lip6.fr. Bunles are message aggregates. They can contain anything from a message fragment to several messages []. /$. c 2 IEEE the message elivery ratio increases for smaller bunles, but that the achieve gain is boune an only significant when the constraints on message elivery elay are tight. Finally, we compare our moel s preictions to results from a real-life connectivity trace obtaine in a rollerblaing tour. In Section II, we briefly escribe the ege-markovian ynamic graph moel. We then calculate the elivery ratio for epiemic routing in Section III, before iscussing the the impact of bunle size on elivery ratio in Section IV. We then compare these theoretical insights to results from a real ata set, the Rollernet experiment, in Section V. II. EDGE-MARKOVIAN DYNAMIC GRAPHS In the rest of this paper, in both the theoretical an experimental parts, we represent the a hoc network forme by N mobile noes as a connectivity graph that evolves in iscrete time. Depening on the context, we will use the terms vertex (resp. ege) an noe (resp. link) interchangeably. The time step τ is equal to the shortest contact or inter-contact time. In a real-life trace, τ is equal to the neighborhoo scanning sampling perio. Eges come up or own at the beginning of each time step, but the topology then remains static until the next time step. Previous work on ynamic graphs focuse on graphs with increasing numbers of vertices or eges [3], but i not account for noe mobility an/or link instability. More recently, Chaintreau et al. use simple sequences of uniform ranom graphs for moeling ranom temporal graphs in orer to analyze the iameter of opportunistic mobile networks [4]. Pellegrini et al. explore the notion of connectivity over time but this approach loses all information about the orer in which contact opportunities appear [5]. Unfortunately, none of these moels capture the strong correlation between successive connectivity graphs. Ege-Markovian ynamic ranom graphs were first introuce by Clementi et al. as a generalization of time-inepenent ynamic ranom graphs to capture the strong epenence between the existence of an ege at a given time step an its existence at the previous time step [2]. While such a moel may be use to stuy a wie variety of ynamic graphs, in this paper, we will focus on its application to ICMNs. Dynamic ranom graph base moels, incluing the one in this paper, have an exponential (or geometric) inter-contact time istribution. In real-life atasets this may not always be the case. Inee, when the unerlying social ynamics are strong, the inter-contact istribution follows a power law [6]. However,

2 IEEE TRANSACTIONS ON COMMUNICATIONS, TCOM in ifferent scenarios, it may follow an exponential law [7]. Interestingly, the inter-contact istribution of any mobility moel in a boune omain necessarily exhibits an exponential cutoff [8]. In an ege-markovian ynamic graph with N vertices, each ege is consiere inepenently an can be in one of two states: either or. Let p (resp. p ) be the probability of transitioning to the (resp. ) state. The transition matrix for each ege is therefore M = p p p p. () The contact (T ) an inter-contact (T ) times are istribute geometrically an their expecte values are E(T ) = τ p an E(T ) = τ p. Inee, the number of time steps require to leave the (resp. ) state is the number of trials neee to get one success in a Bernoulli process with probability p (resp. p ). Let π (resp. π ) be the stationary probability of being in state (resp. ). We have π = p p +p an π = p p +p. Finally, the average noe egree is (N )π. III. TUNING MESSAGE SIZE TO MEET DELAY CONSTRAINTS A. Preliminaries We assume that, when up, all links have equal capacity φ an thus can transport the same quantity φτ of information uring one time step. We refer to φτ as the link size. Small values of τ therefore mean that the network topology s characteristic evolution time is short an thus only small amounts of information may be transmitte over a link uring one time step. We efine the bunle size as numerically proportional to τ: αφτ. By abuse of language, however, we will simply refer to α as the bunle size. For example, a bunle of size 2 () is only able to traverse links that last for more than 2 time steps, whereas a bunle of size.5 is able to traverse two links uring each time step. Furthermore, each bunle can only tolerate a certain maximum elay. We note the maximum number of time steps, beyon which a elivery is consiere to have faile. By abuse of language, we will simply refer to as the maximum elay. Epiemic routing was one of the first methos propose for ealing with intermittent connectivity in mobile a hoc networks [9]. Each message is flooe into the entire network. Upon meeting, two noes first exchange message vectors escribing which messages they currently hol, before requesting from one another copies of the messages they o not yet have. Following the epiemic analogy, a noe is sai to be infecte by a message upon receiving a copy of it []. Epiemic routing is particularly useful for theoretical purposes, since its elivery ratio is also that of the optimal single-copy time-space routing protocol. Our goal is to calculate the elivery ratio of a bunle using epiemic routing. To be successful, elivery has to occur without exceeing the maximum allowe elay. To this en, we introuce a new Markovian moel for epiemic propagation on the ege-markovian ynamic graph of the previous Section. For the sake of simplicity, the moel will first be escribe for α =. In Sections III-C an III-D, we will respectively escribe how to aapt the previous moel when the bunles are smaller (α < ) an larger (α > ) than the link size. B. Bunles fit in a time slot (α = ) Source a wishes to transmit a bunle to estination b using epiemic routing. Eges change states at the beginning of each time step. During one time step, an infecte noe infects all of its irect uninfecte neighbors, an only those, since the bunle size is an bunles can therefore only perform one hop per time step. Let V be the set of the noes in the network. After k time steps, noes other than b fall in one of three isjoint sets: Those that have just been infecte: J k. Those that have been infecte at time step k or before: I k. Those that have not yet been infecte: S k = V \ (I k J k {b}). This istinction is necessary to etermine who can be infecte at time step k +. Inee, if a noe belongs to I k, then all its neighbors at the en of time step k are in I k J k. It can only infect new noes if an ege to a clean noe in S k comes up at time step k +. However, a noe in J k may have eges to some clean neighbors in S k which may become infecte at time step k + if the ege remains up. In this paper, we are only intereste in the probability that b receives a copy in at most time steps. In this case, the only information necessary to characterize the state of the epiemic is the number of noes i an j in I k an J k, respectively. The elivery ratio can be obtaine as the absorption probability of the Markov chain escribe hereafter. States. The epiemic can be escribe as a Markov chain on the following 2 + N(N ) 2 states: Init: The initial state in which only the origin a is infecte. This state is transient. Succ: The estination b has been infecte. This state is absorbing. States (i, j) for i N an j N i. These are also transient. Primitives. The transition probabilities are functions of the following primitives. Given two sets of noes U an W, if each noe of U can infect each other noe in W with probability p, we efine the probability that m noes in W will be infecte: ) P inf (m, p, U, W ) = pf B (m, ( p) U, W, (2) where pf B (m, p, n) is the probability ensity function of a binomial istribution of n inepenent events with probability p. A noe that has just been infecte (i.e., J k ) can contaminate the estination the following roun with probability π, while noes that have been infecte for two or more time steps (i.e., I k ) can o so with probability p. If I k = i an J k = j, then the probability of infecting the estination b uring the next time step is: P succ (i, j) = π j ( p ) i. (3) Transition Probabilities. The state Succ is absorbing. Any transitions from the Init state, can be calculate as transitions

3 IEEE TRANSACTIONS ON COMMUNICATIONS, TCOM T = Init (,) (,) (2,) Succ Init π 2 π π π (,) ( p ) 2 ( p )p p (,) π ( p ) π ( p ) (2,) ( p ) 2 ( p ) 2 Succ (5) from a (, ) state. A state (i, j) can transition to either state Succ with probability P succ (i, j) or to another state (i + j, j ) with probability: ( P succ (i, j)) j m= { P inf (m, π, j, N i j) } P inf (j m, p, i, N i j m). (4) Delivery Ratio. Let T be the Markov chain s matrix of transition probabilities, i the initial state vector an s the state vector with coefficient for state Succ an for all others. Therefore, the elivery ratio (i.e., the probability of being in state Succ) after time steps is P eliv (, α = ) = it s. For example, let us consier a network with 3 mobile noes. The Markov chain escribing an epiemic propagation on its associate ege-markovian ynamic graph has 5 states: Init = (, ), (, ), (, ), (2, ), an Succ. Here s = [ ] T an its initial vector state vector is i = [ ]. Its matrix of transition probabilities, T, is etaile in Eq. (5). C. Bunles smaller than link size (α < ) When the bunle size is smaller than the link size, bunles may perform up to α hops uring one time step. Recall that the network topology instantly changes at the beginning of each time step, before the first hop. After that, the remaining hops happen on the same static network topology. To take this into account, we efine a static propagation matrix R using the same states as previously but tweaking the transition probabilities. In a static topology no new links can come up, hence Psucc static (i, j) = π j an the transition ( from state (i, j) to (i + j, j ) happens with probability P static succ (i, j) ) P inf (j, π, j, N i j). Finally, the elivery ratio (i.e., the probability ( of being in ) Succ after time steps) is P eliv (, α < ) = i T R α s. D. Bunles larger than link size (α > ) Bunles larger than the link size can only use links that last longer than α time steps. Computing the exact elivery ratio in this case requires one to keep track of the number of noes that will complete reception of the bunle in, 2,..., α time steps. This quickly becomes intractable. Instea one can easily calculate upper an lower bouns on the elivery ratio by consiering successive, non overlapping, intervals of α time steps an only the links that last longer than α time steps. The latter will hereafter be referre to as sufficiently long links. The lower boun is obtaine by taking into account only the sufficiently long links that either exist or come up at the.8.2 = = Bunle size (α) N = 2 p = /2 p = /2 Fig.. Influence of bunle size on elivery ratio for ifferent values of maximum elay (). Each value of correspons to two lines: its upper an lower bouns. beginning of an interval. This ignores links that come up later in the interval an hence unerestimates the propagation of the epiemic. More precisely, we replace π by π ( p ) α an p by p p α in Eqs. 3 an 4. The upper boun is obtaine by consiering that any sufficiently long link that comes up uring one interval will allow the full bunle to be transmitte over it by the en of the interval. For example, if an a sufficiently long link appears after one time step within the two-time-step interval, then we consier that the whole bunle can be transferre over that link before the next interval. This obviously overestimates the number of infecte noes at each time step. More precisely, we replace π by ( π + π ( ( p ) α ) ) ( p ) α an p by ( ( p ) α ) ( p ) α in Eqs. 3 an 4. If T l (resp. T u ) is the transition matrix obtaine for the lower (resp. upper) boun, then the elivery ratio after time steps is boune by it α l s Peliv (, α > ) it α u s. A. Influence of bunle size IV. DISCUSSION Fig. plots the elivery ratio as a function of the bunle size for ifferent values of maximum elay. Bunles larger than the link size see their elivery ratio severely egrae, though this is somewhat mitigate by longer maximum elays. On the other han, bunles smaller than the link size can make several hops in a single time step. This is a great avantage when the time constraints are particularly tight ( = 4 in Fig. ), but barely has any effect when the time constraints are looser. This also highlights the influence of noe mobility. Inee, since the actual bunle size is proportional to τ (see Section III-A), high noe mobility (i.e., small τ) makes the actual link size smaller an thus further constrains possible bunle size.

4 IEEE TRANSACTIONS ON COMMUNICATIONS, TCOM α = α = / N (a) Number of noes.8.2 α = α = / (N )π (c) Average noe egree.8.2 α = α = / E(T )/τ (p /p = ) (b) Topology evolution spee.8 α =.2 α = / () Maximum elay Fig. 2. Influence of moel parameters on the elivery ratio. When unspecifie, N = 2, p = /2, p = /2, = 5. Maximum elay an average link lifetime are expresse in number of time steps. B. Influence of other parameters Number of noes. (Fig. 2a) The elivery ratio tens to as N increases. Inee, for a given source/estination pair, each new noe is a new potential relay in the epiemic issemination an thus helps the elivery ratio. Topology evolution spee. (Fig. 2b) Faster oscillations between an states make for a more ynamic network topology. This makes for shorter contact an inter-contact times (Section II) but increases contact opportunities. Small bunles (α ) take avantage of this an their elivery ratio increases as E(T ) ecreases. On the other han, excessive link instability rives the elivery ratio for larger bunles (α > ) to, because fewer links last longer than one time step. Average noe egree. (Fig. 2c) Greater connectivity increases the elivery ratio. The sharp slope of the curve when α is reminiscent of percolation in ranom graphs when the average noe egree hits. Maximum Delay. (Fig. 2) All else being equal, there is a threshol value beyon which almost all bunles are elivere. This can be linke to the space-time iameter of the unerlying topology [4]. V. EVALUATION The theoretical results from the previous section give us valuable insights into real-life scenarios. Although the ege- Markovian moel s iameter is significantly smaller than that of real-work networks ue to unwante small-worl properties, it accurately preicts, as we shall see in this section, the relations between elivery ratio, maximum elay an bunle size. A. Methoology Wireless connectivity traces involving mobile evices have typically been conucte using perioic Bluetooth scans [6], [], [2]. In this paper, we chose to stuy the Rollernet Delivery Ratio.8.2 Rollernet Theory Bunle Size (α) Fig. 3. Preicting elivery ratio for ifferent bunle sizes in Rollernet with a 5-minute maximum elay. trace [2], which captures the connectivity patterns in a rollerblaing tour, because of its very short sampling perio. Inee, the longer the sampling perio, the more likely link failures or short contacts will be misse. Furthermore, it becomes ifficult to claim that a contact translates into a link that lasts roughly as long as the sampling perio (one of our core theoretical assumptions). Therefore, in orer to compare theoretical an experimental results, we require traces with very short sampling perios. Other Bluetooth contact traces were consiere, such as the Reality Mining experiment conucte at MIT [] or the Infocom 25 traces from the Haggle Project [6]. Unfortunately, none of these ha a short enough sampling perio (6 an 2 secons respectively, compare to 5 secons for Rollernet). In a sense, the MIT an Infocom traces capture a subset of contact opportunities while Rollernet approaches the evolution of the connectivity graph. Since the ataset logs contacts between noes an not link urations, we assume that two noes in contact remain so for the entire sampling perio. Furthermore, we i not try to extrapolate aitional events (e.g., new contact opportunities an link failures) between multiples of the sampling perio. As in the Markovian network moel escribe previously, we again assume that all links have equal capacity. The first 3, secons of Rollernet trace were replaye. Every 5 secons for the first 2, secons, 6 source/estination pairs were ranomly selecte for a simulation of epiemic routing. The average link lifetime is 26.8 secons an the average noe egree Using the expressions from Section II, we erive p =.5 an p =.57. B. Results Preicting elivery ratio. Fig. 3 compares the measure elivery ratio in Rollernet to the moel s upper an lower bouns. Here we use the p an p values erive from the trace s average link lifetime an noe egree. Due to smallworl effects, our moel is overly optimistic, particularly for smaller bunle sizes (α 2). However it oes successfully boun the experimental values for larger bunle sizes. Smaller bunles increase elivery ratio. In Fig. 4, the elivery ratio is steay an close to before ropping sharply beyon a certain bunle size that epens on the target elay. Due to mobility, more than half of the links last less than 5 secons. Therefore, bunles of size greater than forgo many

5 IEEE TRANSACTIONS ON COMMUNICATIONS, TCOM Fig. 4. values. Delivery Ratio.8.2 min 3 min 6 min Bunle Size (α) Rollernet: vs. bunle size for various maximum elay VI. CONCLUSION In this paper, we propose a new moel for epiemic propagation on ege-markovian ynamic graphs which capture the correlation between successive connectivity graphs. We fin a close-form expression of elivery ratio as a function of bunle size, maximum tolerate elay, an the ynamics of the unerlying evolving graph. In particular, we have shown that, given a certain maximum elay an noe mobility, bunle size has a major impact on the elivery ratio. Our theoretical insights on the interaction between these parameters are corroborate by experimental results on the Rollernet ataset. contact opportunities. However, longer maximum elays can compensate for this. This mirrors the theoretical results on size, elay, an mobility escribe in Section. IV-A. Boune gain from smaller bunles. In Fig. 4, when the maximum elay is minute, the maximum achievable elivery ratio is.95 no matter how small the bunles are. This boun on the gain achieve by smaller bunles appears because they hit the performance limit of epiemic routing. Inee, the best possible epiemic propagation of a message will, at each time step, infect a whole connecte component if at least one of its noes is infecte. A small enough bunle can sprea sufficiently quickly to achieve this, an thus even smaller bunles bring no performance gain. The same boune gain from smaller bunles is visible on Fig. on the = 4 curve. Tight elays require smaller bunles. The sharp elivery ratio rop in Fig. 4 occurs later for more relaxe elay constraints. A tight time constraint (less than a couple of minutes for example) forces the use of smaller bunles in orer to obtain an acceptable elivery ratio. On the other han, looser time constraints allow for more flexibility regaring bunle size. It is therefore possible to etermine the maximum bunle size for any given target elivery ratio. REFERENCES [] K. Fall, A elay-tolerant network architecture for challenge internets, in Proc. ACM SIGCOMM, 23. [2] A. E. Clementi, C. Macci, A. Monti, F. Pasquale, an R. Silvestri, Flooing time in ege-markovian ynamic graphs, in Proc. ACM PODC, 28. [3] J. Leskovec, J. Kleinberg, an C. Faloutsos, Graphs over time: Densification laws, shrinking iameters an possible explanations, in Proc. ACM SIGKDD, 25. [4] A. Chaintreau, A. Mtibaa, L. Massoulie, an C. Diot, The iameter of opportunistic mobile networks, in Proc. ACM CoNEXT, 27. [5] F. De Pellegrini, D. Miorani, I. Carreras, an I. Chlamtac, A graph-base moel for isconnecte a hoc networks, in Proc. IEEE INFOCOM, 27. [6] A. Chaintreau, P. Hui, J. Crowcroft, C. Diot, R. Gass, an J. Scott, Impact of human mobility on the esign of opportunistic forwaring algorithms, in Proc. IEEE INFOCOM, 26. [7] V. Leners, J. Wagner, S. Heimlicher, M. May, an B. Plattner, An empirical stuy of the impact of mobility on link failures in an 82. a hoc network, Wireless Communications, IEEE, vol. 5, no. 6, 28. [8] H. Cai an D. Y. Eun, Crossing over the boune omain: From exponential to power-law inter-meeting time in MANET, in Proc. ACM MobiCom, 27. [9] A. Vahat an D. Becker, Epiemic routing for partially-connecte a hoc networks, Tech. Rep., 2. [] X. Zhang, G. Neglia, J. Kurose, an D. Towsley, Performance moeling of epiemic routing, Computer Networks, vol. 5, no., pp , 27. [] N. Eagle an A. Pentlan, Reality mining: Sensing complex social systems, Personal an Ubiquitous Computing, vol., no. 4, 26. [2] P.-U. Tournoux, J. Leguay, F. Benbais, V. Conan, M. D. e Amorim, an J. Whitbeck, The accorion phenomenon: Analysis, characterization, an impact on DTN routing, in Proc. IEEE INFOCOM, 29.