Consiglio Nazionale delle Ricerche · Second, performance bounds under heterogeneous contact...

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Consiglio Nazionale delle Ricerche

Performance modelling of opportunistic forwarding with exact knowledge

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IIT TR-20/2011

Technical report

settembre 2011

Iit

Istituto di Informatica e Telematica

IEEE TPDS, VOL. XXX, NO. XXX, JANUARY XXX 1

Performance modelling of opportunisticforwarding with exact knowledgeChiara Boldrini, Marco Conti, Member, IEEE , and Andrea Passarella

Abstract—The Delay Tolerant Networking paradigm aims to enable communications in disconnected environments where traditionalprotocols would fail. Opportunistic networks are delay tolerant networks whose nodes are typically the users’ personal mobile devices.Communications in an opportunistic network rely on the mobility of users: each message is forwarded from node to node, accordingto a hop-by-hop decision process that selects the node that is better suited for bringing the message closer to its destination. Despitethe variety of forwarding protocols that have been proposed in the recent years, there is no reference framework for the performancemodelling of opportunistic forwarding. In this paper we start to fill this gap by proposing an analytical model for the expected delay andthe expected number of hops experienced by messages when delivered in an opportunistic fashion. This model seamlessly integratesboth social-aware and social-oblivious single-copy forwarding protocols, as well as different hypotheses for user contact dynamics.The proposed framework is used to derive bounds on the expected delay under homogeneous and heterogeneous contact patterns.We found that, in heterogeneous settings, finite expected delay can be guaranteed not only when nodes’ inter-meeting times followan exponential or power law with exponential cut-off distribution, but also when they are power law distributed, as long as weakerconditions than those derived by Chaintreau et al. [1] for the homogeneous scenario are satisfied.

Index Terms—Wireless communication, Routing protocols, Mobile Computing, Algorithm/protocol design and analysis, Mobilecommunication systems, Ubiquitous computing, Modelling techniques

F

1 INTRODUCTION

W ITH the advent of powerful and lightweight mo-bile devices, such as smartphones and tablets,

the ubiquitous networking vision is quickly becoming areality. A further step in the direction of communicatinganytime anywhere is represented by the Delay TolerantNetworking paradigm, which enables communicationsalso in disconnected environments. In such conditions,the main requirement of protocols for legacy Mobile AdHoc NETworks (MANET), i.e., the presence of an end-to-end path connecting the source and the destination of amessage, can hardly be satisfied. Typical delay tolerantnetworks are, e.g., networks made up of subnetworksconnected only by satellite links [2], or networks whosenodes are people moving around with their hand-helddevices [3]. The latter case is the scenario considered inthis paper. In the literature, such networks have beennamed Pocket Switched Networks (PSN [4]) or simplyopportunistic networks, because they opportunistically ex-ploit contacts between users.

In opportunistic networks, messages are dynamicallyhanded over from node to node upon contact, accordingto the store-carry-and-forward paradigm. Nodes carrymessages with them while they move across the net-work and with their movements they create transmis-

• C.Boldrini, M.Conti, and A.Passarella are with the Institute for Infor-matics and Telematics of the Italian National Research Council, Via G.Moruzzi 1, 56124 Pisa, Italy.E-mail: chiara.boldrini, marco.conti, [email protected]

• This work was partially funded by the European Commission under theSCAMPI (FP7-FIRE 258414) and RECOGNITION (FET-AWARENESS257756) projects.

sion opportunities that enable communications. Thus, inopportunistic networks the delay accumulated by themessages along the forwarding path critically dependson the way users move. The simplest exploitation ofcontact opportunities in order to forward messages isrepresented by Epidemic forwarding [5], which gener-ates and hands over a new copy of the message for eachnew encounter. The rationale behind this approach is toleverage as many routes to the destination as possible.Unfortunately, this greedy approach suffers from severeresource consumption and tends to overload the network[6]. Smarter strategies as to who to forward and howmany copies should be generated have been devisedsince then. According to the type of information usedwhen making forwarding decisions, these strategies canbe classified as partially social-aware [7] [8] and fullysocial-aware [9] [10] [11]. They leverage informationabout the users, their contact dynamics, the environmentthey operate in, the social relationships they share, inorder to select one (or a bunch of) best next hop. De-pending on the number of copies generated for the samemessage, forwarding protocols can also be classified intosingle-copy or multi-copy schemes. In the first case, atany time, in the network there is just one copy of themessage to be delivered, while in the second case morecopies are generated, hoping that at least one of themwill eventually reach the destination. While multi-copystrategies have been shown to improve the reliability ofdelivery, they are typically resource consuming.

Despite the variety of practical forwarding solutionsbased on different heuristics (such as encounter fre-quency and sociality metrics) no general framework has


been introduced so far for the analysis of opportunisticforwarding protocols in a structured way. Some modelsexist in the literature (e.g., [12], [13], [8], [14], [15]),but they are specific to the protocols being studied andcan hardly be re-used when the protocols are changed.The situation is even worse for social-aware schemes,which, despite their popularity, are typically difficultto model analytically. Moreover, the absence of a gen-eral consensus on some fundamental properties of usermovement patterns (e.g., the distribution of the inter-meeting times) makes it even more complex to founda model on a solid basis. In fact, the performance ofmessage forwarding closely depends on the users’ con-tact dynamics [1]. From the analysis of real movementtraces many hypotheses (e.g., [1], [16], [17], [18], [19],[20], [21]) have been made as to which distribution betterdescribes significant quantities such as the time betweenconsecutive contacts, or the duration of a contact, butwithout ultimate consensus.

The contribution of this paper is twofold. First, a gen-eral framework for the analysis of single-copy forward-ing schemes is introduced. This model, based on Markovchains, allows us to compute significant quantities, suchas the expected number of hops and the expected delay,that characterize the forwarding performance. This gen-eral framework also takes into account social-awareness,which can be incorporated seamlessly into the model.In addition, our framework is independent of specificmobility assumptions, thus it would remain usable evenif new insights on the way users move were provided.

Second, performance bounds under heterogeneouscontact dynamics are derived using the proposed frame-work, extending the results of Chaintreau et al. [1]. Intheir foundational work, Chaintreau et al. consider ahomogeneous network where inter-meeting times be-tween pairs of nodes are independent and identicallydistributed, according to a power law with shape α.Considering the Two Hop scheme (see Section 2 formore details) and Epidemic forwarding, they derive theconditions on the parameter α under which the expecteddelay experienced by messages diverges, i.e., is infinite.Comparing their findings with the estimated power lawexponents of real mobility traces, they conclude that theexpected delay for this class of forwarding protocols istypically infinite in opportunistic networks. In this paper,we extend the result of Chaintreau et al. [1] to any single-copy forwarding scheme for opportunistic networks. Theresult in [1] had a huge impact on the opportunisticnetworking literature because it implies that a largefamily of forwarding protocols cannot provide delayswith finite mean. However, Chaintreau et al. [1] left asan open point the analysis of heterogeneous networks,where inter-meeting times between node pairs are notidentically distributed. Focusing on such heterogeneoussetting, when node inter-meeting times are power lawdistributed, in this paper we find that forwarding withfinite expected delay is possible under weaker conditionsthan those derived for a homogeneous scenario. Given

that real networks have been shown to be typicallyheterogeneous [17], we conclude that expected delaybounds in opportunistic networks are actually much lesspessimistic than those derived by Chaintreau et al. [1].

The characteristics of single-copy schemes have beenanalytically studied in the literature for what concernssocial-oblivious strategies [8] [1], but, to the best of ourknowledge, the one proposed in this paper is the firstgeneral framework that takes into account the social-awareness of the forwarding process. Moreover, resultsobtained for single-copy schemes are important to multi-copy schemes as well. As an example, consider theperformance bounds of the expected delay. First, thedelay of single copy schemes marks the upper boundof the expected delay of the corresponding multi-copyversion of the same protocol. Second, bounds derivedfor single copy schemes can be extended to multi-copyschemes. In order to exemplify this extension, in Section7.1.1 we discuss the case of the multi-copy Two Hopscheme. However, an extensive coverage of multi-copyschemes is out of the scope of the paper and we leave itto future work.

The paper is structured as follows. In Section 2 wereview the state of the art on forwarding protocols andperformance modelling for opportunistic networks. InSection 3 we describe the scenario we consider andthe assumptions we make, based on which, in Section4, we define our general modelling framework. Afterdefining in Section 5 our reference forwarding schemes,in Section 6 the general framework is specialized underthe assumptions of power law and power law withexponential cut-off inter-meeting times. In order to ex-emplify how the proposed model can be used, we alsodiscuss its application to two case studies with differentmobility settings. Section 7 exploits our analytical modelto investigate the convergence properties of the expecteddelay when messages are forwarded in a network whichis heterogeneous from the contact dynamics standpoint.Section 8 concludes the paper.

2 RELATED WORK

2.1 Opportunistic forwardingAccording to the type of information that they exploitwhen making forwarding decisions, forwarding pro-tocols can be classified into social-oblivious, partiallysocial-aware and fully social-aware protocols [9]. In thefollowing we overview some of the most significantprotocols for each of these categories. For a more detailedsurvey, we refer the reader to Al Hanbali et al. [22].Social-oblivious protocols do not use at all informationon the way nodes meet or relate with each other. This isthe case of the Epidemic protocol [5], whose strategy isto generate and hand over a new copy of the messageto each node encountered, and of the Direct Trans-mission protocol [23], in which messages can only bedelivered to the destination when encountered directly.Their performance is typically poor because either theyconsume a lot of resources and overload the network


(Epidemic [9]) or they are not able to find a path to thedestination even when many are available (as shownin Section 6.4, the Direct Transmission strategy suffersfrom this problem). For this reason, they are typicallyused as a baseline for performance evaluation only. Morespecifically, Epidemic routing provides the minimumpossible delay in ideal settings with infinite resources,while Direct Transmission minimizes the number ofhops travelled by messages. In order to mitigate theside effects of Epidemic-style forwarding schemes inresource constrained environments, controlled floodingsolutions have been proposed. The Spray&Wait protocol[6] (where only L relays are used) and gossiping [24](where messages are forwarded with probability p uponencounter) are examples of limited flooding, and still canbe classified as social-oblivious protocols. Another popu-lar social-oblivious forwarding protocol is the Two Hopscheme [23], in which a message is forwarded by thesource node to the first node encountered, which is thenallowed only to pass the message directly to the destina-tion. The Two Hop strategy has been shown to guaranteethe maximum throughput capacity in a homogeneousnetwork [23]. Despite their appealing simplicity, thesesocial-oblivious protocols just make a random guess onwhich path towards the destination the message shouldfollow, and thus they are typically very far from beingoptimal in networks where the presence of humans, withtheir highly predictable movements, would provide thebasis for more accurate forwarding decisions.

Partially social-aware protocols leverage network-level information such as time since the last encounter(FRESH [25], Spray&Focus [6]), frequency of encounters(PROPHET [7]), and total number of encounters [26].This information is used to predict future meetingsbetween pairs of nodes and thus to select relays thatcan guarantee a quick delivery according to the heuristicin use. Partially social-aware protocols, however, do notallow for the intentional exploitation of the intrinsicsocial component in user mobility but only rely on verysimple metrics as the ones mentioned above.

Fully social-aware protocols explicitly exploit the so-cial structure of the network of users in order to makeforwarding decisions. This is because social-awarenessenables the prediction of user encounters [27], whichconstitute forwarding opportunities. One approach isbased on the exploitation of the roles of the nodes in thesocial graph associated with the network of users. Themain idea is that nodes that are more central in the socialgraph are likely to be better forwarders than the othernodes. BUBBLE [10], SimBet [11], and PeopleRank [28]rely on this approach. On the other hand, social context-aware protocols keep track of a variety of informationon the environment – context – the users live in (e.g., thepeople they meet, the friends they have, the places theyvisit) and use this information to quantify the ability ofnodes to deliver messages. HiBOp [9] and SocialCast [29]belong to this group.

2.2 Performance modelling

Performance modelling of opportunistic forwarding al-gorithms has been the subject of several papers. Zhanget al. [12], Haas and Small [24], and Groenevelt et al.[30] focus on the modelling of Epidemic-style routing,either by means of Markov chains or fluid (OrdinaryDifferential Equations) models. A class of two-hop for-warding schemes is studied by Al Hanbali et al. [13][31], again relying on Markov chain theory. A variety ofsingle-copy forwarding schemes have been analysed bySpyropoulos et al. [8] by means of random walks on agraph. Their approach shares many similarities with thispaper but, analogously to the contributions cited above,it relies on the exponential assumption for node inter-meeting times and assumes a homogeneous network,i.e., all node pairs being i.i.d. from the contact processstandpoint. In this paper, instead, we relax these assump-tions and we consider both heterogeneous mobility andvarious distributions for the inter-meeting times. As amatter of fact, homogeneous contact dynamics have beenshown to be unrealistic [17]: some users may cluster andmove together, others may never get in touch with eachother. For this reason, models taking into account nodediversity are needed.

To the best of our knowledge, heterogeneous contactpatterns have only been considered by Spyropoulos etal. [14], Lee and Eun [15], Boldrini et al. [32], and Ip etal. [33]. The latter, however, only considers two classesof nodes from the mobility standpoint, and focusesonly on Epidemic dissemination. Spyropoulos et al. [14]propose a more complete analysis, including multipleclasses and a variety of forwarding protocols. However,they still rely on the exponential assumption for inter-meeting times. Lee and Eun [15] study the performanceof a class of two-hop forwarding policies under het-erogeneous contact dynamics, but the distribution ofthe inter-meeting times is considered exponential. Andexponential inter-meeting times are again assumed byBoldrini et al. [32], where a simplified version of theframework discussed in this paper was presented.

There are not many contributions that tackle the mod-elling of opportunistic forwarding relaxing the expo-nential assumption for inter-meeting times. The onlyexisting works that consider different distributions arethose by Chaintreau et al. [1] and Lee and Eun [34]. Thelatter is focused on capacity scaling issues, which arenot studied in this paper. The contribution of Chaintreauet al. [1] is foundational in the field of opportunisticnetworking. As anticipated, its main finding lies in thederivation of conditions on the power law exponent α ofinter-meeting times under which forwarding protocolscan provide finite expected delays. Summarizing theirresults for the single-copy scheme they consider, in ahomogeneous scenario with independent and identicallydistributed inter-meeting times, the single-copy TwoHop forwarding algorithm achieves finite expected delayif α > 2. In Section 7 we extend this result deriving


N number of nodes in the networkFX complementary cumulative distribution function

(CCDF) of random variable XX(x) probability density function of random variable XMij inter-meeting time for the i, j node pairRij residual inter-meeting time for the i, j node pairµij contact rate for the i, j node pairµij contact rate for the i, j node pair resulting from an

online estimation process, e.g., by means of pair-wiseexchange of history of encounters

fφi,d fitness of node i as a relay to destination d under

forwarding policy φpi,j transition probabilities of the forwarding Markov pro-

cesspforw(φ)i,j probability that node i hands over the message to node

j upon encounter when forwarding policy φ is in useT

forw(φ)ij time before node i hands over the message to node j

when forwarding strategy φ is in useT exiti time before node i hands over the message to any

other node or, equivalently, time before the forwardingMarkov process exits from state i

Ddi delay of a message generated by node i and addressed

to node d

Hdi number of hops travelled by a message generated by

node i and addressed to node dPi set comprising all nodes that can be encountered by

node iRφ

i set comprising all nodes that are potential relays fromnode i, i.e., pforw(φ)

ij > 0

TABLE 1Notation

a sufficient condition for the expected delay of anysingle-copy forwarding protocol to diverge. Moreover,we perform a study of the converge bounds of theexpected delay considering a heterogeneous scenarioand we find that convergence conditions are in this caseless restrictive than those derived for a homogeneousscenario.

3 NETWORK MODEL

We first introduce the network model and the notation(Table 1) that we use throughout the paper.

Our model considers a network with N mobile nodes.For the sake of simplicity, we hereafter assume thatmessages can be exchanged only at the beginning of acontact between a pair of nodes and that the transmis-sion of the relayed messages can be always completedwithin the duration of a contact. In addition, we assumethat each message is a bundle [2], an atomic unit thatcannot be fragmented. We also assume infinite bufferspace on nodes. Given that we are considering single-copy schemes, buffer size is not expected to be critical,at least from low to medium network load. All theabove assumptions allow us to isolate, and thus focus on,the effects of node mobility from other effects, and arecommon assumptions in the literature on opportunisticnetworks modelling (they are used in most of the liter-ature reviewed in Section 2).

Given that messages are handed over from node tonode before reaching their destination, the way nodesmove heavily affects the delay experienced by messages.As we assume that the transmission of a message can

always be completed during a pair-wise contact, theactual duration of the contact is not critical. Thus, themain role in the experienced delay is played by inter-meeting times, which are defined as follows.

Definition 1 (Inter-meeting Time): The inter-meetingtime Mij between node i and node j is defined asthe time between two consecutive meetings betweenthe same pair of nodes. If tf is the time at which acontact between node i and node j has just finished,the inter-meeting time Mij is given by:

Mij = mint>tf

{t− tf : ||Xi(t)−Xj(t)|| < r} (1)

where Xi(t) and Xj(t) denote the position of i and j attime t, and r is the transmission range1.In the following we denote as µij the rate of inter-meeting times of the process of encounters between twonodes i and j. We also assume that the network isstationary, thus inter-meeting rates do not vary with time(i.e., µij(t) = µij). By definition, µij = 1

E[Mij ], where

E[Mij ] denotes the expectation of the inter-meeting timeMij between node i and node j. As we assume thatinter-meeting times between every specific node pair i, jare independent and identically distributed, the meetingprocess between node i and node j can be modelled asa renewal process [35].

The message generation and process and the mobilityprocess are independent. We also assume that nodes donot keep track of the time since the last encounter withany other node. This means that when a node generatesa new message (or it receives a new message to relay),the time since the last encounter with any other node isunknown. For this reason, in our analysis we will oftenuse the concept of residual inter-meeting time.

Definition 2 (Residual Inter-meeting Time): Assumingthat node i and node j are not in contact at time tr,the residual inter-meeting time Rij(t) between them isgiven by the time interval between tr and the first timenode i and node j come into each other’s range again,i.e.:

Rij = mint>tr

{t− tr : ||Xi(t)−Xj(t)|| < r}, (2)

where Xi(t) and Xj(t) denote the position of i and j attime t, and r is the transmission range.

There has been an intense debate in the researchcommunity about the probability distribution that betterdescribes the inter-meeting times between users. Chain-treau et al. [1] found that inter-meeting times could bedescribed by a power law distribution. After analysingboth the same traces and an additional one, Karagianniset al. [16] suggested that a power law distribution witha final exponential cut-off could better match the actualshape of the inter-meeting times. According to Gao et

1Without loss of generality, here we assume a deterministic unitdisk graph model for radio propagation. In other words, nodescan communicate only if their current distance is smaller than thetransmission range. This is a common assumption in the literatureon opportunistic networks. The proposed framework still applies forevery other model of radio propagation.


GFED@ABC1 GFED@ABC2 ... GFED@ABCi

pdid !!

pdi2

xxpdi1

yy... GFED@ABCd

Fig. 1. Fragment of the Embedded Markov Chain (validfor all i = d)

al. [18], the same traces support instead the hypothesisof exponentially distributed inter-meeting times. Alongthese contributions, also other hypotheses have beenstudied (such as LogNormal [17] and Double ParetoLogNormal [36]). This brief overview suggests the needfor a more careful and deeper statistical analysis ofcontact traces, which is clearly out of the scope of thiswork. In the following, we restrict our analysis to ex-ponential, power law, and power law with cut-off inter-meeting times, which have stood out as the most popularassumptions for the distribution of inter-meeting timesin the literature.

4 A GENERAL FRAMEWORK FOR MODELLINGTHE FORWARDING PROCESS

As discussed above, there is no final agreement on theprobability distribution that better describes the inter-meeting process between pairs of mobile users. For thisreason, we choose to make our analytical framework asgeneral as possible. Due to its flexibility, we use a semi-Markov process with N states to model the opportunisticforwarding process. A semi-Markov process is one thatchanges state in accordance with a Markov chain (calledembedded or jump chain) but where transitions betweenstates can take a random amount of time with an ar-bitrary distribution [35]. As such, it is fully describedby the transition matrix associated with its embeddedchain and by T exit

i ,∀i = 0, · · · , N , where T exiti denotes

the distribution of the time that the semi-Markov processspends in state i before making a transition.

We express our semi-Markov process associated withthe single-copy message forwarding process in terms ofthe embedded Markov chain in Figure 1. Assuming thatnode i is currently holding a message whose destina-tion2 is d, the probability pdij that node i will delegatethe forwarding of the message to another node j is afunction of both the likelihood of meeting node j andthe probability that node i will hand over the messageto node j according to the forwarding policy in use.The transition matrix T associated with the processof forwarding a message from a source node i to thedestination node d is given below, where, as an example,

2The chain is different for different destinations, because theconvenient relays are generally not the same. However, for the sake ofreadability, in the following we drop superscript d

d = N .

T=

0 p12 . . . p1,N−1 p1,Np21 0 . . . p2,N−1 p2,N

......

. . ....

......

......

. . ....

0 0 . . . 0 1

The state associated with the destination node d isabsorbing, because in state d the forwarding process iscompleted. Please note, however, that there is no guaran-tee that such absorbing state is eventually reached, dueto the potential presence of other closed classes in theforwarding Markov chain.

Once the forwarding Markov process is completelydefined in terms of transition probabilities and exittimes, we can exploit well known algorithms for Markovchain transient analysis in order to compute significantproperties of the forwarding process. In the following,we describe how to compute the expected delay and theexpected number of hops travelled by messages. Proofsfor this section can be found in Appendix A.

Lemma 1 (Expected delay): The expected delay E[Ddi ]

for a message generated by node i and addressed tonode d can be obtained from the minimal non-negativesolution, if it exists, to the following system:{

E[Ddi ] = 0 i = d

E[Ddi ] = E[T exit

i ] +∑

j =d pijE[Ddj ] ∀i = d,

(3)

where T exiti is the time interval before the Markov chain

exits from state i and pij gives the probability of atransition from state i to state j.

Lemma 2 (Expected number of hops): The expectednumber of hops E[Hd

i ] travelled by a message generatedby node i and addressed to node d can be obtained, ifit exists, from the minimal non-negative solution to thefollowing system:{

E[Hdi ] = 0 i = d

E[Hdi ] = 1 +

∑j =d pijE[Hd

j ] ∀i = d,(4)

where pij denotes the probability of a transition fromstate i to state j in the Markov chain.

In order to solve Equations 3 and 4, we need to firstcompute T exit

i and pij for all i, j pairs and for each of theforwarding policies in use. In the following we provide ageneral formulation for both T exit

i and pij , which will bespecialized later in the paper based on the distributionof inter-meeting times considered.

Theorem 1 (Exit Time): The time required for the chainto exit from state i, which corresponds to the time beforenode i hands over the message to any of the nodes ofthe network when the forwarding protocol in use is φ,is given by:

T exiti = min

j =i{T forw(φ)

ij }, (5)

where the random variable Tforw(φ)ij denotes the time

interval since node i receives (or generates) the message


to the time it hands it over to node j. Tforw(φ)ij is

characterized by the following probability density:

P(T

forw(φ)ij = t

)= p

forw(φ)ij P (Rij = t) +

+

+∞∑n=2

[(1− p

forw(φ)ij

)n−1

pforw(φ)ij ·

· P

(Rij +

n−1∑m=1

M(m)ij = t

)], (6)

where Rij denotes the residual inter-meeting time be-tween node i and node j, and M

(1)ij , ...,M

(m)ij are m

i.i.d. random variables describing the inter-meeting timebetween node i and node j. Probability p

forw(φ)ij denotes

the probability that node i will hand over the messageto node j upon encounter and it is dependent on theforwarding strategy φ in use.

Theorem 2 (Transition probability): The transition prob-ability pij , or equivalently the probability that node ihands over the message to node j when the forwardingstrategy φ is in use, is given by:

pij = P (Tforw(φ)ij < T

forw(φ)i−others) (7)

where Tforw(φ)ij is defined as in Equation 6 and T

forw(φ)i−others

as Tforw(φ)i−others = minz =j{T forw(φ)

iz }. In other words, pij isequal to the probability that a forwarding event fromnode i to node j happens before a forwarding event fromnode i to any other node.

As highlighted by Theorems 1 and 2, T exiti and pij

depend on i) the forwarding policy φ in use and ii) thedistributions of inter-meeting times Mij , which in turncharacterize the distribution of residuals Rij . Bullet i)is discussed in the next section, where the reference for-warding policies considered in this paper are introduced.Bullet ii) is discussed in Section 6, where the generalmodel presented above is specialized for the power lawand power law with exponential cut-off distributions.

5 REFERENCE FORWARDING STRATEGIES

Providing a model that is simple but at the same timecomplete enough to correctly describe the variety of ex-isting single-copy forwarding approaches is not an easytask. In order to accomplish this goal, we abstract thevariety of protocols described in Section 2 into the twomain categories of social-oblivious (or blind) and social-aware forwarding protocols. For these categories, weconsider the following policies, which identify importanttraits of existing forwarding strategies. More specifically,among the social-oblivious schemes we consider thefollowing three policies.

Definition 3 (Direct Transmission): The source node canonly deliver the message to the destination itself.

Definition 4 (Always Forward): The source node handsover the message to the first node encountered, and sodoes each intermediate node. The process stops whenthe message is delivered to the destination.

Definition 5 (Two Hop): The source node hands overthe message to the first node encountered. If this firstencounter is with the destination, the forwarding processis completed. Otherwise, the relay node is allowed tohand over the message only to the destination, if evermet.Such social-oblivious policies have been commonly usedin the literature as baseline references [8] [23] [1]. TheDirect Transmission and the Always Forward policiesrepresent the two end points in the single-copy forward-ing spectrum. The Two Hop scheme can be consideredas an intermediate solution between these two extremesand it has been extensively used by Chaintreau et al. [1],to which we compare our results in a later section.

With regards to social-aware schemes, a common fea-ture of all these algorithms is that a message (be it onthe source node or on an intermediate relay) is handedover to another node only if the latter has a higherprobability (we call it fitness) of bringing the messagecloser to its destination than the node currently hold-ing the message. In the following, we consider fitnessfunctions computed using only information on contactsbetween nodes, which have a direct dependence onthe inter-meeting time distribution. This lets us clearlyshow what is the impact of the contact dynamics on theperformance of opportunistic forwarding protocols. Forthe sake of completeness, in Appendix B we then discusshow the proposed analytical framework can be appliedto more complex and popular social-aware policies, suchas BUBBLE, SimBet, and HiBOp. Our two simplifiedreference social-aware policies are the following.

Definition 6 (Direct Acquaintance): The source and eachintermediate relay hand over the message to the firstencountered node having a higher fitness, where thefitness fDA

i,d of a generic node i for a message withdestination d is defined as the estimated frequency µid

of a direct meeting with the destination d (Equation 8).fDAi,d = µi,d,∀i = d (8)

Definition 7 (Social Forwarding): Messages are deliv-ered through a path with positive gradient of fitness,where the fitness fSF

i,d of node i for a message addressedto node d is computed (Equation 9) as the weighted sumof the fitness for a direct acquaintance (fDA

i,d ) and thefitness for an indirect meeting (f I

i,d):

fSFi,d = βfDA

i,d + (1− β)f Ii,d, where 0 < β < 1. (9)

Component f Ii,d is a measure of the probability of being

indirectly connected to the destination or, in other words,of the likelihood of being connected to nodes that havehigh delivery probability for destination d. In the gen-eral case, it can be recursively defined as the weightedaverage of the social fitness of the encountered nodes,which implies:

f Ii,d =

∑j∈Pi

wij ·(γfDA

j,d + (1− γ)f Ij,d

), where 0 < γ < 1.

(10)


In Equation 10, Pi denotes the set of nodes that canbe encountered by node i, and fDA

j,d and f Ij,d are the

direct and indirect fitness values of node i’s neighbourj. Component f I

j,d ensures that high fitness values arealso indirectly detected over multi-hop paths. We definewij as µij∑

j∈Piµij

, thus wij weights the information about jbased on the relative frequency of meeting j with respectto all other nodes. The rationale is that the informationabout j is as useful as node i is able to exploit it, i.e.,as likely it is that node i can exploit node j as relay.Parameter γ is a weight that can be tuned in order toprioritize what neighbour j directly sees (γ → 1, in thiscase) or what the neighbours of j see (γ → 0, in this case).Parameter γ can be in general different from β in order toweight differently the fitness values associated directlywith node i itself and those related to its neighbours. Forthe sake of simplicity, in the following we assume γ = 1.

Differently from the Direct Acquaintance policy, theSocial Forwarding strategy is able to detect not onlydirect meetings with the destination, but also meetingswith people that have a high probability of deliveringthe message to the destination. This strategy enablesthe exploitation of the delivery skills that are presentin the environment surrounding the users, and not onlyof those of the user itself. In Section 6.4 we show howimportant it can be to exploit this feature.

If we assume a stationary mobility process and thatnodes have an exact knowledge of the portion of thenetwork they get in touch with (i.e., accurate informationon their neighbourhood but no global knowledge), nodeswill be able to estimate with no error their expected inter-meeting rate with the other neighbours. Thus, whencomparing its fitness value to that of another node, ageneric node i will always make the same decision,either to forward, or not, to another node j. Instead,when nodes do not have an exact knowledge, the for-warding decisions may vary depending on the actualrate estimated by the node. We refer to the first caseas deterministic forwarding (or forwarding with exactknowledge) and to the second as forwarding with errors.In this work we focus on deterministic forwarding, whilethe case of forwarding with estimation errors is left asfuture work.

6 FORWARDING WITH EXACT KNOWLEDGE

When the nodes of the network have exact knowledge,they all know exactly the expected inter-meeting ratewith their neighbours. Being all rate estimates µij ex-act (i.e., µij = µij) during the forwarding process, allforwarding decisions are deterministic: a generic node ican identify with certainty who is a better next hop andthus to whom a message should be handed over. Thisimplies that the forwarding probability p

forw(φ)ij can be

either 1 or 0. As a consequence, Theorems 1 and 2 canbe simplified as follows.

Corollary 1 (T exiti with Deterministic Forwarding): The

time required for the chain to exit from state i when

a deterministic forwarding process is in use (i.e.,pforw(φ)ij ∈ {0, 1}) is given by:

T exiti = min

j =i

{Rij | i, j : pforw(φ)

ij = 1}, (11)

where Rij denotes the residual inter-meeting time be-tween node i and node j, and p

forw(φ)ij is the probability

that node i will hand over the message to node jaccording to the forwarding policy φ in use.

Corollary 2 (pij with Deterministic Forwarding): Thetransition probability pij under deterministic forwardingis given by:

pij = P

(Rij < min

z =i,j

{Riz | i, z : p

forw(φ)iz = 1

})∀j : pforw(φ)

ij = 1,∀z : pforw(φ)iz = 1 (12)

where again Rij denotes the residual inter-meeting timebetween node i and node j, minz =i,j{Riz} denotes theresidual inter-meeting time between node i and anyother node z different from j, and p

forw(φ)ij (pforw(φ)

iz ) isthe probability that node i will hand over the message tonode j (z) according to the forwarding policy φ in use.

Corollaries 1 and 2 can now be used in order to derivethe expected delay and the expected number of hopsfor a given forwarding process. However, first we haveto define p

forw(φ)ij for each of the reference forwarding

policies in Section 5. In the following we denote with dthe destination of the message, and with s the source ofthe message.

Proposition 1 (pforw(φ)ij for Direct Transmission): The

probability pforw(DT )ij that node i hands over the

message to node j under the Direct Transmission policyis:

pforw(DT )ij =

{1 j = d0 otherwise

.

Proposition 2 (pforw(φ)ij for Always Forward): The prob-

ability pforw(AF )ij that node i hands over the message to

node j under the Always Forward policy is:

pforw(AF )ij =

{0 i = j1 i = j

.

Proposition 3 (pforw(φ)ij for Two Hop): The probability

pforw(2H)ij that node i hands over the message to node j

under the Two Hop policy is:

pforw(2H)ij =

{1 i = s ∨ (i = s ∧ j = d)0 otherwise

.

Proposition 4 (pforw(φ)ij for Social-aware Strategies):

Under the Direct Acquaintance strategy, the probabilitypforw(DA)ij that node i hands over the message to node

j is:

pforw(DA)ij =

{1 fDA

i,d < fDAj,d

0 otherwise.


Analogously, for the Social Forwarding scheme pforw(SF )ij

is:

pforw(SF )ij =

{1 fSF

i,d < fSFj,d

0 otherwise.

Fitness components fDAi,j and fSF

i,j are defined in Equa-tions 8 and 9.

In the remaining of the section we specialize Corollar-ies 1 and 2 for the case of power law and power law withexponential cut-off distribution of inter-meeting times.More specifically, in the following we derive the exit timeand the transition probabilities, which are a function ofthe mobility considered (which in turn affects Rij) and ofthe forwarding strategy φ in use (which affects p

forw(φ)ij ).

For convenience of notation, in the following we denotewith Rφ

i the set of potential relays under forwardingstrategy φ when the message is on node i, i.e., Rφ

i =

{j : pforw(φ)ij = 1}. We anticipate that we are not able

to obtain closed form solutions in the power law withexponential cut-off case. Nevertheless, the result that weobtain is sufficient to derive the convergence conditionson the expected delay when inter-meeting times featurea power law with exponential cut-off (Section 7.2). Thedefinition of an approximated model for the power lawwith exponential cut-off case is left as future work.

6.1 The Exponential Case

In this section we revisit the model proposed in Section4 assuming that the inter-meeting time Mij between ageneric pair of nodes i, j is exponentially distributedwith rate λij . In this case, the rate µij of the inter-meetingtime exactly coincides with the rate λij of the exponentialdistribution describing Mij . Let us start our analysis withthe computation of the expected time E[T exit

i ] requiredto exit state i for the chain in Figure 1.

Theorem 3 (Exit time): When inter-meeting time Mij

follows an exponential distribution with rate λij , T exiti ,

the time before the semi-Markov process exits state i,follows an exponential distribution with rate

∑j∈Rφ

iλij .

T exiti ’s expected value is thus given by the following:

E[T exiti ] =

1∑j∈Rφ

iλij

(13)

Proof: In order to apply Corollary 1, which definesT exiti under deterministic forwarding, we need to com-

pute the distribution of Rij , the residual inter-meetingtimes between node i and node j. Based on the memo-ryless property of the exponential distribution, we knowthat such Rij follows an exponential distribution withthe same rate λij , i.e., Rij ∼ Exp(λij). If we substitutethe CCDF of Rij to Equation 11, we obtain FT exit

i(t) =

e∑

j∈Rφiλijt. This implies that the exit time from state i

is again exponentially distributed with rate∑

j∈Rφiλij .

From standard probability theory, the expectation of anexponential random variable follows directly (Equation13).

Theorem 3 proves that, under the exponential assump-tion for inter-meeting times, the semi-Markov processthat describes the forwarding evolution becomes a Con-tinuous Time Markov process, in which T exit

i follows anexponential distribution.

Below we derive the transition probabilities associatedwith the chain in Figure 1.

Theorem 4 (Transition probabilities pij): Transition prob-abilities pij for all j ∈ Rφ

i are given by:

pij =λij∑

z∈Rφiλiz

, (14)

where λij denotes the rate of encounters between nodei and node j. Probabilities pij are equal to zero for allj ∈ Rφ

i .Proof: Equation 14 follows from the application of

Corollary 2. From standard probability theory we knowthat the minimum of a set of n exponential randomvariables is again a random variable with rate equalto the sum of the rates of the n random variables.Thus, minz{Riz} ∼ Exp(

∑z λiz). Then, we have to com-

pute the probability that Rij is smaller than minz{Riz}(P (Rij < minz{Riz}) = P (Rij − minz{Riz} < 0)). Thisis a well known result from standard probability theoryand the solution is given in Equation 14.

Theorems 3 and 4 completely define the forwardingMarkov process in the case of inter-meeting times expo-nentially distributed. Thus, it is now straightforward tocompute the expected delay and the expected numberof hops travelled by messages using Lemmas 1 and 2.

6.2 The Power Law Case

In this section we revisit the analytical framework pro-posed in Section 4 when the inter-meeting times betweena generic pair of nodes i and j follow a power law(Pareto3) distribution with shape αij and scale tminij .In the following we use the definition of the Paretodistribution which allows for values arbitrarily close tozero and whose CCDF is shown in Equation 15. Theexpected value of such distribution is

tminij

αij−1 .

FMij (t) =

(tminij

t+ tminij

)αij

(15)

This version of the Pareto distribution is usually denotedas American Pareto [37]. We refer the interested readerto Appendix C for a throughout study of our analyt-ical framework when the alternative definition of thePareto distribution, usually denoted as European Pareto,is used. Please note that being the American Pareto aEuropean Pareto shifted by tminij to the left, both Paretodefinitions share the same requirements for having finiteexpectation, as discussed in more detail in Appendix C.Thus, the following remark holds.

3In the following we will use the terms ”power law” and ”Pareto”interchangeably.


Remark 1: The Pareto distributions introduced aboveare defined for αij > 0 (due to the required PDF nor-malization [38]), and their mean is finite when αij > 1.

Recall from Corollaries 1 and 2 that, when consideringthe case p

forw(φ)ij ∈ {0, 1}, T exit

i and pij are expressed interms of the residual inter-meeting times Rij , i.e., thetime until the next contact between node i and node jstarting from a random time t. When inter-meeting timesfeature an American Pareto distribution, we can directlyapply the formula in Karagiannis et al. [16] that re-lates inter-meeting times and residuals. More specifically,from an American Pareto random variable with shapeαij and scale tminij we obtain residuals that feature anAmerican Pareto distribution with shape αij−1 and scaletminij . Similarly to the reference literature [1] [16], forease of computation in the following we restrict to thecase of power law random variables having the samescale, i.e., tminij = tmin,∀i, j.

Remark 2: The Pareto distribution of Rij is defined forαij > 1 (due to the required PDF normalization), and itsmean is finite when αij > 2.

From a mathematical standpoint, Corollaries 1 and 2are mainly based on the computation of the minimummini Xi of a set of random variables {Xi}i and thecomputation of P (X1 < X2), i.e., the probability thata random variable X1 is smaller than another randomvariable X2. When Xi features a Pareto distribution withshape αi and scale tmin for all i values, it is possible toprove (see Appendix D) that mini Xi follows a Paretodistribution with shape

∑i αi and scale tmin, while

P (X1 < X2) is equal to α1

α1+α2. Using these results, in

Theorem 5 we derive the exit time T exiti .

Theorem 5 (Exit time): When inter-meeting times Mij

follow a power law distribution with shape αij and scaletmin for all i, j pairs, and forwarding scheme φ is in use,the time T exit

i before the semi-Markov process exits statei follows a Pareto distribution with rate

∑j∈Rφ

iαij − n

(where n denotes the cardinality |Rφi | of the set Rφ

i )and scale tmin. From standard probability theory, theresulting expectation of T exit

i , when finite (see Remark3), is thus:

E[T exiti ] =

tmin∑j∈Rφ

iαij − n− 1

. (16)

Proof: The time before exiting state i is given bythe time before handing over the message to any ofthe potential relays. This is equivalent to T exit

i =minj∈Rφ

i{Rij}, which is power law distributed with

shape∑

j∈Riαij − n and scale tmin. Then Equation 16

follows from the definition of the expected value of thePareto distribution.

Remark 3: The Pareto distribution of T exiti is defined

for∑

j∈Riαij > n (due to the normalization), and its

expectation is finite when∑

j∈Riαij > n + 1, with n =

|Rφi |.Finally, we derive the transition probabilities of the

Markov chain in Figure 1 under the power law assump-tion for inter-meeting times.

Theorem 6 (Transition probabilities pij): When inter-meeting times Mij are power law distributed withshape αij and scale tmin, and forwarding strategy φ isin use, transition probabilities pij are given by:

pij =αij − 1∑

z∈Rφiαiz − n

, (17)

where n = |Rφi |.

Proof: From Corollary 2 we know that pij is equal tothe probability that T

forw(φ)ij is smaller than T

forw(φ)i−others.

Under deterministic forwarding, Tforw(φ)ij is equal to

Rij , which follows a power law distribution with shapeαij − 1. T forw(φ)

i−others is defined as minz =j{T forw(φ)iz }, where

again Tforw(φ)iz is equal to Riz . Using the rule for deriving

the minimum of a set of power law distributed ran-dom variable, we obtain that T forw(φ)

i−others features a Paretodistribution with shape

∑z∈Rforw(φ)

i ,z =jαiz − (n − 1).

Then, Equation 17 follows directly after applying the rulefor deriving P (X1 < X2) when both random variablesfeature a Pareto distribution.The expected delay and the expected number of hopscan be computed after substituting Equations 16 and 17into Lemmas 1 and 2.

6.3 The Power Law with Exponential Cut-Off CaseIn this section we consider the case of inter-meetingtimes Mij following a power law with an exponentialcut-off. Let us assume the shape of Mij to be αij , thepower law scale tmin, and the rate of the exponential cut-off to be λij . Using a standard notation (see, e.g., [38]),we define the PDF of Mij as Mij(t) = k′ ·t−αij−1 exp−λijt,

where k′ =λ−αijij

Γ(−αij ,λijtmin)is the normalization constant

and Γ(s, x) =∫∞x

ts−1e−tdt is the upper incompleteGamma function. Using standard probability theory, weobtain the following CCDF that characterizes the inter-meeting times Mij :

FMij (t) =Γ(−αij , λijt)

Γ(−αij , λijtmin). (18)

Remark 4: The PDF and the expectationΓ(1−αij ,λijtmin)

λijΓ(−αij ,λijtmin)of a random variable featuring a

power law distribution with exponential cut-off arealways defined for all values of αij , λij , and tmin greaterthan zero.

As anticipated, in this case we are not able to comeup with a closed form solution for the expected delayand the expected number of hops using our exact model.Approximate solutions could be derived, but we leavethis as future work. However, we are able to derive aclosed-form solution for the residual inter-meeting timeRij , which we omit as it not used further in the paper(it can, however, be found in Appendix E). For such Rij ,the following remark holds:

Remark 5: The probability density function associatedwith Rij is continuous and integrable over the domainfor all values of αij and λij greater than zero. Similarly,the expectation is defined for all values of αij and λij

greater than zero.


Lemma E1 in Appendix E shows that in the power lawwith exponential cut-off case the residual inter-meetingtime neither follows any well known distribution norhas a convenient mathematical form. This implies thatmanipulating the residuals Rij as we did for powerlaw inter-meeting times is not feasible and thus it isnot possible to use an exact analytical model in thepower law with exponential cut-off case. However, ourexact model can still be used to discuss the convergenceconditions for the expected delay in the power law withexponential cut-off case (Section 7.2).

6.4 Using the framework: two case studies

In this section we exemplify how the proposed frame-work can be used by discussing two case studies, andthe performance of the Direct Transmission, AlwaysForward, Two Hop, Direct Acquaintance, and SocialForwarding schemes in such cases. Due to space limi-tations, in the following we focus only on power lawinter-meeting times. Under the assumptions in Section3 the proposed analytical model is exact, thus it is notcompared with simulation results, which would simplygenerate totally overlapping curves.

In the following we consider 15 nodes, which movearound in the network and exchange messages accordingto our reference forwarding policies. We consider thecase of a heterogeneous network, in which we equallydistribute our 15 nodes into 3 communities (hereafter de-noted as C1, C2, and C3). We consider each communityas being a complete subgraph, meaning that all nodeswithin each community share a social link with eachother. We also add social links between nodes in differentcommunities. As we assume that nodes’ movements aretriggered by their social relationships, these nodes willcommute between different communities, and for thisreason we denote them as travellers. This is an example ofsocial-oriented mobility models, which are currently oneof the most important approaches in the literature [39][40]. In the following, we consider two different scenar-ios, each of which is characterized by a different socialstructure connecting the nodes in different communities.More details on these social structures will be providedin the corresponding sections.

We define node mobility according to the followingalgorithm. For nodes that have only social relationshipswith members of their own community, we assume thateach pair of nodes connected by a social link meetsaccording to inter-meeting time Mij , with default shapeα. If two nodes do not share a social link, they neverget in touch with each other. Without loss of generality,in the following we set tmin to 1 second and α to 3.5(which guarantees finite expectation for both the inter-meeting times and their residuals). For nodes that areconnected with more than one community, we mimic thefact that the user divides its time between these groupsby increasing its expected inter-meeting time with themembers of these communities. So, basically, we keepconstant the average number of peers encountered by

Fig. 2. Scenario 1

each node in any time interval, be it a traveller or alocally roaming user. Thus, for a generic node j that isin touch with n communities (or, equivalently, which isconnected to nodes associated with n distinct communi-ties), we force its expected inter-meeting times with anyother node in those communities to be n times greaterthan that of another node i that is only connected withjust one community. Thus, by imposing tmin

α′−1 = n tmin

α−1 ,the shape α′ for traveller node j will be equal to α−1+n

n .For each of the reference forwarding schemes we plot

the histogram of the expected delay and of the expectednumber of hops computed for any pair of nodes. Inthe case of 15 nodes, there are n(n − 1) = 210 nodepairs, for which we extract 210 values of expected delayand 210 values of expected number of hops solvingthe system of equations in Lemmas 1 and 2. The y-axis in all histograms shows the frequency of expecteddelay values normalized by the total number of expecteddelay samples (210, in this case). Bin width is chosenfor each scenario in order to ensure the significance andreadability of plots.

6.4.1 Scenario 1: travellers in each communityWe start by considering the case of all three commu-nities being directly connected by moving nodes. Morespecifically, focusing on community C1, we add one linkconnecting one node in C1 with one node in C2 and onelink connecting one node in C1 with one node in C3.Using the same approach we connect one node in C2 toone node in C1 and one node in C2 to one node in C3,and the same is done for C3. As we assume that nodemovements are triggered by their social relationshipswith the other nodes of the network, community C1will have two travellers visiting the other communities:specifically, one traveller goes to C2 and back, the othergoes to C3 and back. The travellers in C2 and C3 havean analogous behaviour (Figure 2). This configurationensures that the network is connected because it existsat least one multi-hop path between any pair of nodes.This allows us to show that, despite the network beingconnected, not all forwarding strategies are able to de-liver messages between any node pair.

Figure 3 shows the forwarding performance as far asdelay is concerned. Specifically, we compute from themodel the expected delay E[Dij ] for all pairs i, j, andwe plot in Figure 3 the distribution of the expected delay


(across all pairs). The Direct Transmission scheme sufferswhen the source and the destination of the message donot get in touch with each other directly, thus producinginfinite delays. This is because, with Direct Transmission,nodes can only deliver their messages directly to thedestination, thus missing all the opportunities offeredby relaying: when the destination is never met, themessage cannot be delivered. However, relaying doesnot always guarantee a better performance in terms ofexpected delay, as the Two Hop case in Figure 3 shows.Recall that the expected delay is a weighted average ofthe expected delay of each possible path. Thus, if thereexists even a single path with infinite expected delay,the overall expected delay will diverge. This is exactlywhat happens with the Two Hop strategy: due to theblind selection of the next hop, messages can take awrong path at the first hop, and then they get stuckthere because the intermediate relay node never meetsthe destination. In this scenario, such sequence of eventsis possible for all i, j source-destination pairs such thateither (i) source node i and destination node j neitherare traveller nor are in the same community or (ii) sourcenode i is a traveller. In both cases there are some pathsthat achieve a finite expected delay, but there are alsopaths with infinite expected delay, and the latter dragthe overall expected delay to infinite. Comparing theTwo Hop scheme with the Direct Transmission strategy,in case (i) the fraction of node pairs that experience aninfinite expected delay is the same under both protocols.In the second case, instead, i.e., when source node i is atraveller, among the possible paths that are added by theTwo Hop scheme with respect to the Direct Transmissionstrategy, there are some characterized by an infinitedelay, and those paths drag to infinite the expected delayfor the Two Hop scheme, even if the direct encounterbetween the traveller and the destination would havea finite expectation. As an example of the first case,consider a message with source node in community C1and destination node in community C2. In addition,assume that the source and destination nodes are nottravellers. If the first encounter of the source node is withthe traveller connecting C1 and C3, the message will behanded over to this node. However, this traveller nevergets in touch directly with the destination in communityC2, and the message will never be delivered. As forthe second case, when the traveller is the source ofthe message (with destination in community C1, forexample), there is always a non-negligible probabilitythat, at the time the message is generated, the traveller isroaming in a community (C3, for example) different fromthe one in which the destination resides. In this case, themessage will be handed over to the first encounterednode, which, in our example, belongs to C3 and whichwill never meet the destination.

Direct Acquaintance, Social Forwarding, and AlwaysForward are able to exploit the social bridges betweencommunities and to hand over the message to the con-venient node. The Always Forward approach, however,

0

0.2

0.4

0.6

0.8

1

DT 2H AF DA SF

Nor

mal

ized

freq

uenc

y

Intervals [s]:[0,1)[1,2)[2,3)[3,4)[4,5)

Inf

Fig. 3. Distribution of the Expected Delay for Scenario 1

0

0.2

0.4

0.6

0.8

1

DT 2H AF DA SF

Nor

mal

ized

freq

uenc

y

Intervals [# hops]:[1,2)[2,3)

[11,12)[12,13)[13,14)[14,15)[18,19)[21,22)[22,23)

Inf

Fig. 4. Distribution of the Expected Number of Hops forScenario 1

DT 2H AF DA SF

Exp. Delay [s] ∞ ∞ 1.6112 2.28701 2.28701Exp. Num. Hops ∞ ∞ 16.8745 1.77143 1.77143

TABLE 2forwards totally at random, and many hops may berequired before the message eventually finds, by chance,its destination (Figure 4). Social strategies are insteadable to choose the relays providing the best trade-offbetween low delay and efficient use of resources. Notealso that in this scenario Direct Acquaintance and SocialForwarding show the same performance. In fact, theyonly differ when transitivity of contacts needs to beexploited for successful delivery, which is the case ofthe scenario discussed in the next section.

The expected delay and expected number of hops av-eraged across all node pairs are summarized in Table 2.

6.4.2 Scenario 2: travellers in a single communityIn this section we use the same scenario as in Section6.4.1, except that we assign travellers only to communityC1 (Figure 5). As in the previous case, the network isconnected. However, while in Section 6.4.1 all communi-ties were directly connected by means of traveller nodes,here C2 and C3 cannot communicate directly, and theyhave to exploit the forwarding capabilities of the visitingtravellers from C1.

Figure 6 shows the expected delay experienced bymessages in this scenario. The Direct Transmission, TwoHop, and Direct Acquaintance schemes are not able todeliver a subset of messages. In the case of the DirectTransmission scheme the reason lies in the absence ofdirect contacts between the source of a message and itsdestination. The Two Hop scheme again suffers from theproblem of messages that move away from their sourcenode and get stuck at intermediate relays. In the caseof the Direct Acquaintance policy, losses are due to thefact that a node hands over a message to another node


Fig. 5. Scenario 2

0

0.2

0.4

0.6

0.8

1

DT 2H AF DA SF

Nor

mal

ized

freq

uenc

y

Intervals [s]:[0,1)[1,2)[2,3)[3,4)[4,5)[5,6)[6,7)[8,9)

Inf

Fig. 6. Distribution of the Expected Delay for Scenario 2

0

0.2

0.4

0.6

0.8

1

DT 2H AF DA SF

Nor

mal

ized

freq

uenc

y

Intervals [# hops]:[0,3)[3,6)

[9,12)[15,18)[24,27)[27,30)[33,36)[36,39)[39,42)[48,51)[57,60)

Inf

Fig. 7. Distribution of the Expected Number of Hops forScenario 2

that has a higher probability of meeting the destination,measured in terms of direct encounters only. The travellerthat visits C2 does not meet any nodes of C3 directly,thus it is not considered a good relay for destinations inC3 by the Direct Acquaintance scheme. However, thattraveller will meet in C1 the other traveller that visitsC3 and thus it can be considered, indirectly, a goodforwarder for C3 by nodes that roam only in C2. For thisreason, a more efficient strategy should also consider thetransitivity of opportunities (e.g., node a meets b, whichin turn meets c, thus a can be considered a good relay fordestination c). This transitivity of encounters is detectedby the Social Forwarding strategy, which, for this reason,is able to deliver all messages to their destinations. TheAlways Forward strategy is, as before, able to deliverall messages, but using many relays (Figure 7), evenmore than in the previous scenario. The reason is that,being the forwarding opportunities so limited, with theAlways Forward strategy the destination is typicallyfound by chance after many (bad) relays have been used.

Summary results for the expected delay and the ex-pected number of hops averaged across all node pairsare shown in Table 3.

DT 2H AF DA SF

Exp. Delay [s] ∞ ∞ 3.7167 ∞ 4.59114Exp. Num. Hops ∞ ∞ 35.1955 ∞ 2.35238

TABLE 37 BOUNDS ON THE EXPECTED DELAY

In addition to performance evaluation, the model pre-sented in the paper can be used to derive bounds onthe performance of forwarding protocols, and thus toinvestigate whether opportunistic forwarding strategies,and social-aware strategies in particular, are able toprovide finite expected delays. To this aim, in this sectionwe provide a formal discussion on the convergenceconditions for the expected delay under power lawand power law with exponential cut-off distribution ofinter-meeting times. Please note that all the distributionsconsidered in the following share the same scale tmin.

Our reference point will be the work by Chaintreau etal. [1], where the convergence of the expected delay hasbeen studied considering homogeneous inter-meetingtimes in the power law case (using an approach differentfrom the one used in this paper). For the sake of compari-son with [1], we also assume that the probability that twonodes meet is greater than zero for all node pairs. Thisensures that, in principle, all nodes can meet with eachothers. With respect to the forwarding process describedby the Markov chain in Figure 1 and developed in theprevious sections, this means that the only absorbingstate is the destination, for any forwarding strategy inuse. Therefore, the cases of deadlock and infinite delaydiscussed in Section 4 are not possible anymore. The onlycause of infinite delay are therefore the distributions ofinter-contact times. Extending the study of this specificrelationship with respect to [1] is exactly the goal of thissection.

7.1 Bounds under Power Law Inter-Meeting TimesIn this section we study analytically the convergencebounds for the expected delay when inter-meeting timesfollow a Pareto distribution (a validation by means ofsimulations is provided in Appendix G). Please note thatthe analysis presented hereafter holds true regardlessof the definition (American or European) of the Paretodistribution considered, because these two definitionsshare the same convergence conditions on their expectedvalue (see Appendix C). In the following we assume thatαij > 1 for all i, j node pairs, so that the residual inter-meeting times are defined (see Remark 2).

Chaintreau et al. [1] took the Two Hop scheme as rep-resentative of social-oblivious approaches and studiedits convergence properties in a homogeneous networkwhere αij = α for all i, j node pairs. More specifically,as far as single-copy forwarding schemes are concerned,the main finding was that only for α > 2 the TwoHop scheme can provide delays with finite expectation.For the sake of comparison, we thus first consider ahomogeneous network (αij = α,∀i, j) and we confirm inTheorem 7 the condition α > 2 necessary and sufficientfor the Two Hop scheme to achieve finite expected delay.


Theorem 7 (Two Hop scheme in homogeneous network):In a homogeneous network where the inter-meetingtime Mij follows a power law distribution with shapeα for all i, j node pairs, the Two Hop relaying protocolis able to provide finite expected delays if and only ifα > 2.

Proof: We distinguish between the case in whichthe destination is the first node met (i.e., the messageis forwarded from source node s to destination node ddirectly), and the case in which the source node s firstmeets another generic node (i.e., the forwarding path iss → j → d, with j being any another node differentfrom s and d). In the former case, we need to considerthe expected residual time before the destination is en-countered. The expectation of the residual requires α > 2to be finite (see Remark 2). When the message follows atwo-hop path from node s to node j and then to node d,the delay at the first hop depends on minj∈Ps−{d}{Rsj},which is the time before the first node is encountered.Please recall that Ps denotes the set of nodes that can bemet by node s and that Rsj in this case follows a powerlaw distribution with shape α−1. In addition, please notethat |Ps−{d}| = N −2. Based on the property discussedin Section 6.2, the minimum of N−2 power law randomvariables with the same shape α−1 is a power law withshape (N−2)(α−1). The requirement for the expectationof minj∈Ps−{d}{Rsj} to be defined is (N − 2)(α− 1) > 1,which is equivalent to α > 1 + 1

N−2 . This is a weakercondition than α > 2, because 1 + 1

N−2 is smaller than 2for all N > 3. Then, from j to d, the delay is given bythe residual inter-meeting times between j and d, whoseexpectation is finite when α > 2. Thus, overall, havingfinite expected delays implies condition α > 2. We caneasily show that also the dual condition holds true. Infact, if α ≤ 2, the destination can never be encounteredwith a finite expected delay (Remark 2), and thus thenecessary condition α > 2 for the Two Hop strategyfollows. This is in accordance with previous results [1].

Theorem 7 confirms one the main findings of Chain-treau et al. [1]: the Two Hop scheme is not able to delivermessages with finite expected delay as long as α ≤ 2. InTheorem 8 we extend this result by providing a sufficientcondition for the expected delay of any single-copyforwarding scheme to diverge when a homogeneouscontact process is considered.

Theorem 8 (Single-copy schemes in homogeneous networks):In a homogeneous network where the inter-meetingtime Mij follows a power law distribution with shapeα for all i, j node pairs, the expected delay of anysingle-copy forwarding protocol diverges if α ≤ 2.

Proof: Regardless of the specific forwarding algo-rithm in use, the condition for the message to be even-tually delivered is that the destination is encountered bythe node currently holding the message. If we denote thisnode with j, the expected delay for the last hop is givenby the expectation of the residual inter-meeting timeRjd between j and d. The expectation of this residual

diverges when α ≤ 2 (Remark 2). If the expected delayof the last hop diverges, the whole expected delay willdiverge, thus proving that α ≤ 2 is a sufficient conditionfor the expected delay to diverge in a homogeneousscenario for any single-copy forwarding scheme.

Clearly, there can exist less restrictive conditions beforethe last hop . However, the finiteness of the expecteddelay over the whole path is only guaranteed if themost restrictive condition is satisfied. As an example,consider the case of the Always Forward policy. At eachhop but the last one, the message is handed over to thefirst out of N − 1 nodes that can be encountered, thusthe delay component for each hop but the last one isdescribed by minj∈Pi{Rij}, where i can be any nodeof the network but the destination of the message. Therequirement for the expectation of minj∈Pi{Rij} to bedefined (see Remark 2) is (N − 1)α− (N − 1) > 1, whichis equivalent to α > N

N−1 . Expression NN−1 is smaller

than 2 for all N ≥ 3. Thus, conditions less restrictivethan α > 2 can hold for intermediate hops. However, ifonly condition α > N

N−1 applied, the forwarding processcould end up in a forwarding loop in which the expecteddelay for each hop is finite, but, as the destinationcannot be reached within a finite expected time, theexpected number of hops happens to be infinite. Thus,the convergence over the whole path is guaranteed onlyif the less restrictive conditions are satisfied.

When the inter-meeting times between any pair ofnodes are i.i.d., social-aware policies are not very help-ful. In fact, the strength of social-aware policies lies insmartly exploiting node diversity. When inter-meetingtimes are i.i.d., all relays are equally good as next hopbecause they all meet with each other at the same rate.If the fitness values of nodes are all the same, theforwarding algorithm has to break the tie by imposingan additional forwarding rule (e.g., never forwarding,randomly forwarding, or always forwarding upon en-counter with a node having the same fitness for thedestination). Thus, social-aware policies are equivalentto social-oblivious strategies in this case. Let us nowconsider a heterogeneous network, for which we caneffectively compare the Two Hop relaying bounds withthe bounds guaranteed by social-aware forwarding. Thefollowing theorem holds.

Theorem 9 (Two Hop scheme in heterogeneous networks):In a heterogeneous network where the inter-meetingtime Mij between any generic i, j node pair follows apower law distribution with shape αij , the Two Hoprelaying protocol is able to provide finite expecteddelays for messages generated by the source node s forthe destination node d if and only if both the followingconditions hold true:

C1∑

j∈Psαsj > 1 + |Ps|, where Ps denotes the set

of all nodes that can be encountered by node s;C2 αjd > 2, ∀j ∈ Ps − {d}.

Proof: The source node s can either deliver themessage directly to the destination or hand it over to an


intermediate relay. The time before the source node re-leases the message is distributed as minj∈Ps{Rsj}, whichis the time before the first node (possibly includingthe destination) is encountered. From Section 6.2 weknow that minj∈Ps{Rsj} features a Pareto distributionwith shape

∑j∈Ps

(αsj − 1), which, according to Remark2, should be greater than 1 in order to have finiteexpectation. This implies

∑j∈Ps

αsj > 1 + |Ps|, thusobtaining condition C1. However, if the node to whichthe message has been handed over is not the destinationbut another generic node j, the expected delay from jto d can be finite only if the expectation of Rjd is finite,i.e., if αjd > 2. Given that node j can be any node apartfrom s and d, condition αjd > 2 must hold for all nodesj different from s and d, and thus sufficient condition C2is proved. Conditions C1 and C2 are not only sufficientbut also necessary conditions for the expected delay tobe finite. In fact, if condition C1 is not satisfied, when theTwo Hop scheme is used, a message can never leave itssource node within a finite expected time, and thus itsoverall expected delay will not converge. Analogously,if condition C2 is not satisfied, no intermediate relaycan deliver the message to the destination within afinite expected time, and thus there is no two-hop pathachieving a finite expected delay. Given that the TwoHop scheme cannot control whether to choose a one-hopor a two-hop path, the presence of potential two-hoppaths with infinite expected delay forces the expecteddelay to be infinite.

Corollary 3: The minimum requirement4 for the ex-pected delay to converge under the Two Hop schemeis that there exist i) at least one node z ∈ Ps such thatαsz > 2, and ii) αjd > 2 for all j ∈ Ps − {d}.

Proof: Condition C1 in Theorem 9 has multiple solu-tions. Here we are interested in the one that imposes theloosest constraints on αsj values. Recall that we haveassumed αij > 1, for all i, j node pairs, so that theresidual inter-meeting time is defined. As for C1, theworst case scenario for the convergence conditions ofthe expected delay is when all, but one, intermediaterelays from s to d are arbitrarily close to 1. In this case,we thus have αsj = 1 + ϵ for all j in Ps − {d, z}, withϵ → 0, and αsz > 1 + ϵ. We want to investigate theconditions on αsz for having finite expected delay inthis case. Applying condition C1 in this scenario, weobtain (1 + ϵ)(|Ps − {d, z}|) + αsz > 1 + |Ps − {d}|. If wedenote |Ps −{d}| with n′, the cardinality of |Ps −{d, z}|can be expressed as n′ − 1. After expansion, we getαsz > 1 + 1 + ϵ + n′ − n′ − ϵ, from which αsz > 2follows. Condition C2 in Theorem 9 must be taken asit is, because even if there exists a node z such thatαsz > 2, the Two Hop scheme has no means for selectingit, and will just blindly hand over the message to thefirst node encountered (thus requiring αjd > 2 for allpotential encounters j).

4By minimum requirement we indicate a sufficient condition thataffects the minimum amount of nodes, and thus it is convenient toverify it even when the total number of nodes is large.

Corollary 3 states that the Two Hop scheme achievesfinite expected delay only if the source node meetsat least one node with finite expected residual inter-meeting time and all intermediate nodes meet the desti-nation with finite expected residual inter-meeting time.Let us now analyze how this changes when social infor-mation is exploited in the forwarding process.

Theorem 10 (Social-aware schemes in heterogeneous net.):In a heterogeneous network where the inter-meetingtime Mij between any generic i, j node pair follows apower law distribution with shape αij , the social-awarestrategies in Definitions 6-7 are able to provide finiteexpected delays if and only if

∑j∈Rφ

iαij > 1 + |Rφ

i | forall i ∈ Rφ

s ∪ {s}.Proof: The proof exploits the ordering guaranteed by

the social-aware policies. Specifically, when social-awarepolicies are used, messages are forwarded along a pathwith increasing fitness. For the sake of simplicity, in thefollowing we assume that there cannot be two nodeswith the same fitness value. Recalling that Rφ

i denotesthe set of potential relays when the message is on node i,or in other words the set of nodes that are more likely tomeet the destination with respect to node i, we have that,for a generic path with increasing fitness {s, i, · · · , j, z, d}(Figure 8), the relation Rφ

s ⊃ Rφi ⊃ · · ·Rφ

j ⊃ {d} holds.The embedded Markov chain describing the forwardingprocess is regular and has a finite state space. Moreover,all states, apart from the absorbing one, are transientand are visited at most once (this is guaranteed by thefact that messages follow a path with increasing fitness).We know that the time before the forwarding processexits a generic state i is distributed as minj∈Rφ

i{Rij},

which, following the same line of reasoning used in theproof of Theorem 9, has a finite expectation as long as∑

j∈Rφiαij > 1+ |Rφ

i |. When using social-aware policiesthe Markov chain has no loop and all possible routescan have at most a finite number of hops equal to |Rφ

s |.Thus, if condition

∑j∈Rφ

iαij > 1 + |Rφ

i | holds true forall the possible states i of the forwarding Markov chain,the overall expected delay will converge. This conditionis both necessary and sufficient. In fact, assuming that itdoes not hold for a given node i, there will be a possiblepath with infinite expected delay, and thus the overallexpected delay will diverge.

Corollary 4: The minimum requirement for the ex-pected delay to converge under the social-aware schemesdefined in Definitions 6-7 is that there exists, for alli ∈ Rφ

s ∪{s}, at least one node j ∈ Rφi such that αij > 2.

Proof: See Appendix F.Corollary 4 derives for social-aware schemes conver-

gence conditions on the power law shape αij that drasti-cally improve those for social-oblivious schemes derivedby Chaintreau et al. [1] for a homogeneous network andin Corollary 3 for a heterogeneous network. In fact, therequirement for the expected delay to converge statedby Corollary 4 is that each node i can hand over amessage to at least another node j in its inner circleRφ

i (Figure 8) with finite expected residual inter-meeting


s

i

z

j

d

!

"

!

#

!

$!

…

Fig. 8. Social forwarding at a glance

time. However, as far as convergence bounds of theexpected delay are concerned, an optimal forwardingpolicy should be able to achieve a finite expected delayalso in the worst case scenario, i.e., when only one routeexists that can provide a finite expected delay. Corol-lary 4 tells us that the social-aware strategies defined inDefinitions 6-7 are not able to guarantee that such routeis chosen, and in fact the convergence condition musthold for all potential relays. In Theorem 11 we showthat it is possible to modify the social-aware strategiesDefinitions 6-7 so that they can handle correctly alsosuch worst case scenario. More specifically, we modifyour social-aware strategies assuming that nodes are ableto detect the peers for which they do not have finiteexpected residual inter-meeting times. These peers arenot considered at all as possible relays. Under theseassumptions, the following theorem holds.

Theorem 11 (Modified social-aware schemes in het. net.):In a heterogeneous network where inter-meeting timeMij between any generic pair i, j of nodes follows apower law distribution with shape αij , the modifiedsocial-aware strategies are able to provide finite expecteddelays if there exists at least one route such that αij > 2for all relaying pairs i, j from s to d defined by themodified social-aware strategy in use.

Proof: The proof follows that of Theorem 10. In brief,modified protocols allow any node i currently holdingthe message to hand it over to another node j only ifαij > 2. Thus, a message can never go from i to anothernode z for which αiz < 2. This ensures that all possiblehops have finite expected residual inter-meeting time.If there is just one path that guarantees finite expecteddelay, that will be the only path that can be traversed bya message delivered according to the modified social-aware policies.

Theorem 11 is an important result that tells us that, ina heterogeneous opportunistic environment, forwardingwithin a finite expected time interval is possible, as longas there exists at least one path with finite expectationfrom the source to the destination and smart forwardingstrategies are used. Being real networks heterogeneousin the node contact dynamics [17], we can conclude thatforwarding with finite expected delay in opportunisticnetworks is possible when conditions more optimisticthan those derived by Chaintreau et al. [1] for a homo-geneous scenario and social-oblivious schemes hold true.

7.1.1 From single-copy to multi-copy schemesConvergence bounds on the expected delay derived forsingle-copy forwarding schemes can be used to deriveconvergence bounds in the case of multi-copy schemes.Clearly, an extensive coverage of this problem is out ofthe scope of the paper. However, in the following wediscuss the case of the multi-copy Two Hop scheme, inorder to give a flavor of such an extension. Accordingto the multi-copy version of the Two Hop forwardingscheme [1], the source node hands over a copy of themessage to the first m encountered nodes, which willthen be only allowed to deliver the message directly tothe destination, if ever met. We also distinguish betweentwo cases. In the memoryless case, the source node doesnot keep a record of the relay nodes used so far, and thustwo consecutive encounters with the same node will endup in the message being copied again to the same relay.In the memoryful case, a relay node cannot be used morethan once. Before discussing the memoryless case, weintroduce the following general results.

Lemma 3: Let us define random variable X(n) as beingthe minimum of n independent random variables , i.e,X(n) = mini={1,...,n} Xi. Regardless of the distributionof Xi, it holds that X(n) ≥ X(n+1), i.e., P (X(n) > x) ≥P (X(n+1) > x),∀n ≥ 1. In addition, if Xi can only takenon-negative values, then E[X(n)] ≥ E[X(n+1)],∀n ≥ 1.

Proof: P (X(n) > x) ≥ P (X(n+1) > x) fol-lows from the fact that probabilities are constrainedwithin zero and one, whatever the Xi. In fact,P (X(n) > x) =

∏i={1,...,n} P (Xi > x) and

P (X(n+1) > x) =∏

i={1,...,n+1} P (Xi > x). Thus,P (X(n) > x) ≥ P (X(n+1) > x) is equivalent to∏

i={1,...,n} P (Xi > x) ≥∏

i={1,...,n+1} P (Xi > x),or, alternatively,

∏i={1,...,n} P (Xi > x) ≥ P (Xn+1 >

x)∏

i={1,...,n} P (Xi > x). The latter is always verifiedbecause P (Xn+1 > x) ∈ [0, 1] by definition. In summary,P (X(n) > x) decreases as we increase n, i.e., as thecardinality of the set of random variables consideredincreases. For Xi taking only non-negative values, theexpectation E[X(n)] can be computed as

∫∞0

P (X(n) >x)dx. Exploiting the fact that integration preserves theordering of functions (i.e., if f(x) ≥ g(x) in a genericinterval [a, b], then

∫ b

af(x) ≥

∫ b

ag(x)), we also obtain

that E[X(n)] ≥ E[X(n+1)],∀n ≥ 1.Lemma 4: Consider random variable M , featuring a

power law distribution with shape α and scale t0, andits residual R. Denote with M t1 (wrt Rt1 ) the randomvariable obtained when conditioning M (wrt R) to begreater than t1. Then, for all α > 1 and t0 > 0, thefollowing stochastic ordering applies:

M < M t1 < M t2 < Rt2 < Rt3 ,where t0 < t1 < t2 < t3.(19)

Proof: Recall that, given two random variables Xand Y , X < Y if P (X > x) < P (Y > x). Let usfirst focus on M < M t1 . Due to the scale invarianceof the power law distribution, M t1 is again powerlaw distributed, with shape α and scale t1 (t0 < t1).


When comparing the CCDF of M and M t1 we obtain( t0t0+t )

α < ( t1t1+t )

α for all t1 greater than t0. In fact,function f(t∗) = t∗

t∗+t is monotonically increasing witht∗ when t > t∗, and exponentiation with α > 1 preservessuch property. Applying the same reasoning, we alsoobtain M t1 < M t2 . Next, we compare M t2 and Rt2 . Wehave ( t2

t2+t )α < ( t2

t2+t )(α−1), because t2

t2+t belongs to theinterval [0, 1]. Finally, using again the same approach, wehave ( t2

t2+t )(α−1) < ( t3

t3+t )(α−1), for all t3 > t2.

Theorem 12 (m-copy memoryless Two Hop): In a hetero-geneous network where the inter-meeting time Mij be-tween any generic i, j node pair follows a power lawdistribution with shape αij , the memoryless multi-copyTwo Hop relaying protocol is able to provide finiteexpected delay for messages generated by the sourcenode s for destination node d if and only if conditionsC1 and C2 in Theorem 9 hold true.

Proof: In the memoryless case, the source node handsover m copies of the message, one for each of the firstm nodes encountered. However, the source node doesnot keep track of the relay nodes already exploited,thus it can happen that more than one copy is relayedto the same node. Let us focus of the first hop, i.e.,on the delivery from the source node to the m relays.The delivery of the first copy is subject to the samecondition C1 derived in Theorem 9, because the timeto relaying is described by minj∈Ps Rsj and thus weneed

∑j∈Ps

αsj > N for convergence. We now considerthe delivery of a generic copy k, with k > 1. Afterdelivering the (k − 1)-th copy, the delivery of copy kstarts. Let us denote with tk−1 the time at which the(k − 1)-th copy is handed over, and with t0 the timeat which the message is generated by the source node.In addition, we define Pk

s as the set of nodes that havenot been yet used as relays when delivering the k-thcopy. We have that the time to relaying is described bymin

{{Rtk−1−t0

sj }j∈Pks, {M tlast(j)

sj }j∈Ps−Pks

}, where last(j)

denotes the last time node j has been used as relayby the source and R

tk−1−t0sj (wrt M

tlast(j)

sj ) the randomvariable obtained when conditioning Rsj (wrt Msj) tobe greater than tk − t0 (wrt tlast(j)). We have to considerthe inter-meeting time distribution Msj rather than theresidual Rsj because, after a generic node j has beenused as relay, the time since the last encounter betweens and j is not anymore a random time with respect tothe meeting process. Using the results in Lemma 4, thefollowing inequality can be derived:

min{{Rtk−1−t0

sj }j∈Pks, {M tlast(j)

sj }j∈Ps−Pks

}<

min{{Rtk−1−t0

sj }j∈Ps

}. (20)

Note, in fact, that tlast(j) < tk−1 − t0, thusM

tlast(j)

sj < Rtk−1−t0sj . From Remark 2, we have that

min{{Rtk−1−t0

sj }j∈Ps

}has a finite expectation as long as∑

j∈Psαsj > N , which is the same condition that applies

for the first copy. Summarizing, we have that the first

copy is relayed within a finite expected time if only if∑j∈Ps

αsj > N , and that the other m − 1 copies arerelayed within a finite expected time if

∑j∈Ps

αsj > N .Thus, overall,

∑j∈Ps

αsj > N is a necessary and suffi-cient condition for the convergence of the expected delayat the first hop, and it is the same as condition C1 inTheorem 9.

Let us now focus on the second hop. Given that thesource node can select the same relay more than once,the number m′ of distinct nodes actually carrying a copyof the message can range from 1 to m. However, ofall the possible combinations, we are interested in theworst one from the point of view of the convergenceof the expected delay. In fact, if it exists even a singlepossible realization that provides an infinite expectation,the whole expectation will diverge. The worst case corre-sponds to the set of relays having the lowest cardinality(Lemma 3), i.e., to m′ = 1. Given the blind selection ofrelays performed by the Two Hop scheme, this uniquerelay j can be any of the initial N − 1 potential relays.Thus, the expected delay at the second hop is finite aslong as the expectation of Rjd is finite for all j ∈ Ps.From Remark 2, this implies αjd > 2 for all j ∈ Ps−{d},which is equivalent to condition C2 in Theorem 9.

In the following we derive the convergence conditionsfor the expected delay under the memoryful m-copy TwoHop scheme. To this aim, using Lemma 3, in Theorem 13we prove the existence of an operating point m∗ for theTwo Hop scheme such that, when m ≤ m∗, all m copiesare delivered to their m relays within a finite expectedamount of time, while for m > m∗ copies exceeding m∗

will experience an infinite expected delay. In the rest ofthe section we assume that the m-copy Two Hop schemeis operating at m ≤ m∗.

Theorem 13: In a heterogeneous network where theinter-meeting times Mij between any generic i, j nodepair follow a power law distribution with shape αij andthe memoryful m-copy Two Hop forwarding protocol isin use, there exists a characteristic value m∗ such that,when m ≤ m∗, all m copies are delivered to their mrelays within a finite expected amount of time, whilefor m > m∗ copies exceeding m∗ will experience aninfinite expected delay. The value of m∗ can be obtainedas follows:

m∗ =

{0 if

∑j∈Ps

αsj ≤ N

argmaxm{m+∑N−1

i=m α∗i > 1 +N} o.w.

,

(21)where α∗

i denotes the i-th largest αsj with j ∈ Ps.Proof: According to the memoryful multi-copy Two

Hop relaying protocol, at the first hop m copies arerelayed to the first m distinct encountered nodes. Thus,the delivery process at the first hop is a selection withoutrepetitions: every time a relay is selected, it is removedfrom the set of future relays for the same message. Letus define Pk

s the set of relays still available to s whenthe source node is delivering the k-th copy, t0 the timeat which the message is generated at the source, and


tk the time at which the k-th copy is delivered. Giventhat we assume (see beginning of Section 7) that theprobability that any two nodes meet is greater thanzero, we have that |Ps| = N − 1 and |Pk

s | = N −1 − (k − 1) = N − k. The time before the k-th copyis relayed is given by minj∈Pk

s{Rtk−1−t0

sj }, where againwe denote with R

tk−1−t0sj the random variable obtained

when conditioning Rsj to be greater than tk−1−t0. Thus,from Remark 2, we know that convergence is ensured aslong as

∑j∈Pk

sαsj > 1 + |Pk

s |, with |Pks | = N − k. This

condition must be satisfied for all k. However, Lemma3 tells us that the smaller the cardinality of the set ofrandom variables of which we take the minimum, theslower the convergence. This implies that the strictestcondition for the convergence of the expected delay ofthe first hop is imposed by the m-th copy, i.e., by theone that sees that narrower set of nodes left for relaying.Thus, if we are able to define a convergence conditionfor the m-th copy, then it follows that the finiteness ofthe expected time to relaying for all previous copies isautomatically guaranteed.

Let us thus focus on the relaying of the m-th copy.When the (m − 1)-th copy has been delivered, thereare N − 1 − (m − 1) = N − m potential relays leftfor the m-th copy. The identities of these N − m po-tential relays depend on the previous evolution of theforwarding process (i.e., which nodes have already beenused). More specifically, there can be

(N−1N−m

)possible

combinations. If we denote with νi the i-th of thesecombinations, the time to the next encounter is describedby minj∈νi R

tm−1−t0sj . Using Remark 2, we have that the

convergence condition for the expected delay at the firsthop is given by

∑j∈νi

αsj > 1 +N −m. This conditionmust hold for all possible combinations, i.e, for all νisuch that i ∈ {1, ...,

(N−1N−m

)}, thus obtaining the following

system of inequalities:{∑j∈νi

αsj > 1 +N −m}i∈{1,...,(N−1

N−m)}(22)

In order to find a solution to this system, let us de-fine a mapping f∗ that goes from set {αsj}j∈Ps to set{α∗

i }i∈{1,...,|Ps|}, where α∗i corresponds to the i-th largest

αsj in Ps (implying α∗i ≥ α∗

i+1, for all i ∈ {1, ..., |Ps|}).For a given value of m, set {α∗

i }i∈{m,...,N−1} denotes theset of the smallest N − m values of αsj with j ∈ Ps.We argue that the solution to the system of inequalitiesin Equation 22 is given by the solution to inequality∑N−1

i=m α∗i > 1 + N − m. In fact, for all νi,

∑j∈νi

αsj ≥∑N−1z=m α∗

z . Thus, if∑N−1

z=m α∗z > 1+N−m, then

∑j∈νi

αsj

will be also greater than 1+N−m. In practical terms, thisis equivalent to deriving conditions for the worst caseonly, where the worst case corresponds to only relayswith the lowest alpha values being left for the relayingof the m-th copy. Clearly, if the worst case satisfies theconvergence condition, then all other cases will alsosatisfy the convergence condition. Thus, Equation 23provides the convergence conditions for the first hop,

i.e., for the delivery of the message from source node sto all m relaying nodes.

N−1∑i=m

α∗i > 1 +N −m (23)

Equation 23 tells us how the inter-meeting times shouldbe distributed (i.e., which shape they should have) inorder for the Two Hop scheme to achieve a finite ex-pected delay at the first hop when using m copies.Equation 23 can also be used in the opposite way. Infact, Equation 23 also tells us, given the distribution (i.e.,the shape) of inter-meeting times, how many copies canbe generated at the first hop while still maintaining afinite expected delay. In this case, it is more convenientto rewrite Equation 23 as follows:

m+N−1∑i=m

α∗i > N + 1 (24)

Function g(m) = m+∑N−1

i=m α∗i (corresponding to the left-

hand side of Equation 24) is always greater than zero,and decreases as m increases. In fact, consider movingfrom m to m+ 1. Function g(m+ 1) can be rewritten asm + 1 +

∑N−1i=m α∗

i − α∗m, but 1 − α∗

m is always smallerthan zero, as we have assumed αij > 1 for all i, j nodepairs. This implies that the left-hand side of Equation 24decreases as m increases. Let us assume for the momentthat condition in Equation 24 is verified for m = 1. Then,as m increases, either g(m) always remains above N −1,or there will be an intersection point between the twocurves. In the first case, m∗ = N − 1, because conditionin Equation 24 is always verified for all m ≤ N−1. In thesecond case, m∗ corresponds to the maximum value ofm that still satisfies Equation 24, i.e., to the intersectionpoint between g(m) and N − 1. If condition in Equation24 is not verified for m = 1, not even a single copy canbe delivered with finite expected delay at the first hop.This happens when condition C1 in Theorem 9 is notsatisfied, i.e., when

∑j∈Ps

αsj ≤ N . In this case, m∗ = 0.

Finally, in Theorem 14 we provide the convergenceconditions for the overall expected delay under thememoryful m-copy Two Hop scheme operating at m ≤m∗.

Theorem 14 (m-copy memoryful Two Hop): In a hetero-geneous network where the inter-meeting times Mij

between any generic i, j node pair follow a powerlaw distribution with shape αij , the memoryful m-copyTwo Hop forwarding protocol operating at m ≤ m∗ isable to achieve a finite expected delay if and only if∑N−1

j=N−m α′j > 1 +m, where α′

j denotes the j-th largestαjd with j ∈ Ps (thus

∑N−1j=N−m α′

j is the sum of the msmallest αjd with j ∈ Ps).

Proof: Given that the memoryful m-copy Two Hopforwarding protocol is operating at m ≤ m∗, the conver-gence of the expected delay at the first hop is guaranteedby definition. Let us then focus on the second hop, i.e., on


the delivery of the m copies to destination d. Again, thepossible sets of nodes holding the m copies are given bythe combinations of N−1 nodes grouped into subsets ofm elements. If again we denote the i-th combination withνi, the convergence condition following from Remark 2is given by

∑j∈νi

αjd−m > 1, for all i ∈ {1, ...,(N−1m

)}. In

order to find a solution to this system, let us again definea mapping f ′ that this time goes from set {αjd}j∈Ps

to set {α′i}i∈{1,...,|Ps|}, where α′

i corresponds to the i-th largest αjd with j in Ps. For a given value of m,set {α′

i}i∈{N−m,...,N−1} denotes the set of the smallestm values of αjd with j ∈ Ps. Using the same argumentdiscussed above, we force the convergence condition inthe worst case only, as the convergence in all other caseswill automatically follow. Thus, we obtain

∑N−1i=N−m α′

i >1 +m.

Corollary 5 (m-copy memoryful Two Hop in homo. net.):In a homogeneous network where the inter-meetingtimes Mij follow a power law distribution with shapeα for all i, j node pairs, the m-copy Two Hop strategy(m ≤ m∗) achieves a finite expected delay for a messagefrom source node s to destination node d if and only if

α >1

N −m+ 1. (25)

In addition, m∗ is given by:

m∗ =

⌊N − 1

α− 1

⌋(26)

Proof: It follows from Theorem 13 and 14 after simplesubstitutions.Please note that the necessary and sufficient conditionin Equation 25 extends the sufficient condition providedby Chaintreau et al. [1]. In fact, Chaintreau et al., underthe assumption N > 2m (which we have relaxed), derivethat the m-copy Two Hop scheme (m ≤ m∗) achieves afinite expected delay in a homogeneous setting as longas α > 1 + 1

m . Exploiting assumption N > 2m, we havethat N −m > m, thus 1

N−m < 1m , and 1+ 1

N−m < 1+ 1m .

Thus, when condition α > 1+ 1m is verified, also Equation

25 holds true. This further confirms our results.

7.2 Bounds Under Power Law with Exponential Cut-Off Inter-Meeting TimesIn this section we study the expected delay bounds inthe case of inter-meeting times following a power-lawdistribution with an exponential cut-off. All proofs forthis section can be found in Appendix F. They all exploitthe fact that the residual inter-meeting times have in thiscase a finite expectation (Remark 5). For the Two Hopand social-aware schemes the following theorems holds.

Theorem 15 (Two Hop relaying in heterogeneous networks):In a heterogeneous network where the inter-meetingtimes Mij between any generic i, j node pair followsa power law with exponential cut-off distribution withshape αij and rate λij , the Two Hop relaying protocolis always able to provide finite expected delays.

Proof: This follows directly from Remark 5. Theresidual inter-meeting time Rij and its expectation are

always defined for all αij and λij greater than zero. Thus,focusing on all possible forwarding paths (i.e., s → d ands → j → d), we have that Rsd, Rsj , and Rjd have finiteexpectations for all s, j, d.

Theorem 16 (Social-aware strategies in heter. networks):In a heterogeneous network where the inter-meetingtimes Mij between any generic i, j node pair followsa power law with exponential cut-off distribution withshape αij and rate λij , the social-aware strategies inDefinitions 6-7 are always able to provide finite expecteddelays.

Proof: Again we consider the direct acyclic graph,rooted in the source node, in which nodes are orderedbased on their fitness value. All intermediate hops inthe direct acyclic graph have a finite expected delay bydefinition because, when inter-meeting times follow apower law with exponential cut-off, their residuals havefinite expectation, and thus the overall path is associatedwith a finite expected delay.

Summarizing, Theorems 15 and 16 tell us that, wheninter-meeting times follow a power law with exponentialcut-off distribution, a single-copy forwarding algorithm,either social-oblivious or social-aware, can always bedesigned that achieves finite expected delay.

8 CONCLUSION

In this paper we have proposed a general frameworkbased on semi-Markov processes for modelling the for-warding process in opportunistic networks. Besides be-ing independent of any specific forwarding policy, theframework is also independent of the specific hypothe-sis on the distribution of inter-meeting times betweenpairs of nodes, making it general enough to be usedalso when such hypothesis is changed. We have usedthe model to compare the forwarding performance ofsocial-oblivious and social-aware strategies in terms ofexpected delay and expected number of hops. Finally,using this model we have derived the bounds on theexpected delay under heterogeneous contact dynamics.Specifically, we have found that finite expected delaycan be provided under heterogeneous contact dynamicsby imposing significantly less restrictive conditions thanthose derived by Chaintreau et al. [1] for social-obliviousstrategies and a homogeneous setting.

APPENDIX APROOFS FOR SECTION 4

Lemma 1 (Expected delay): The expected delay E[Ddi ]

for a message generated by node i and addressed tonode d can be obtained from the minimal non-negativesolution, if it exists, to the following system:{

E[Ddi ] = 0 i = d

E[Ddi ] = E[T exit

i ] +∑

j =d pijE[Ddj ] ∀i = d,

(3)

where T exiti is the time interval before the Markov chain

exits from state i and pij gives the probability of atransition from state i to state j.


Proof: The expected delay from node i to node dis equivalent to the expected hitting time on d fromstate i. As we recall from Markov process analysis [41],the expected hitting times E[Dd

i ], i.e., the expected timeneeded to go from state i to state d, are the minimalnon-negative solutions to the system in Equation 3.

Lemma 2 (Expected number of hops): The expectednumber of hops E[Hd

i ] travelled by a message generatedby node i and addressed to node d can be obtained, ifit exists, from the minimal non-negative solution to thefollowing system:{

E[Hdi ] = 0 i = d

E[Hdi ] = 1 +

∑j =d pijE[Hd

j ] ∀i = d,(4)

where pij denotes the probability of a transition fromstate i to state j in the Markov chain.

Proof: The expected number of hops travelled bya message is equivalent to the expected number ofstates visited in the embedded chain in Figure 1 beforereaching d, i.e., the expected hitting time for the embed-ded discrete Markov chain. Thus, the expected numberof hops E[Hd

i ] is given by the minimal non-negativesolutions to the system in Equation 4, where 1 accountsfor exiting state i.

Theorem 1 (Exit Time): The time required for the chainto exit from state i, which corresponds to the time beforenode i hands over the message to any of the nodes ofthe network, is given by:

T exiti = min

j =i{T forw

ij }, (5)

where the random variable T forwij denotes the time in-

terval since node i receives (or generates) the message tothe time it hands it over. T forw

ij is characterized by thefollowing probability density:

T forwij (t) = p

forw(φ)ij Rij(t) +

+

+∞∑n=2

[(1− p

forw(φ)ij

)n−1

pforw(φ)ij ·

· P

(Rij +

n−1∑m=1

M(m)ij = t

)], (6)

where Rij denotes the residual inter-meeting time be-tween node i and node j and M

(1)ij , ...,M

(m)ij are m

i.i.d. random variables describing the inter-meeting timebetween node i and node j. Probability p

forw(φ)ij denotes

the probability that node i will hand over the messageto node j upon encounter and it is dependent on theforwarding strategy φ in use.

Proof: Let us focus on a i, j node pair and assumethat at time tr node i has a new message to be relayed(either newly created or received from another peer).Every time there is a new encounter between node iand node j, node i hands over the message to nodej with probability p

forw(φ)ij . Let us denote with T forw

ij

the time interval before node i hands over the messageto another tagged node j, assuming no interactions

between node i and other nodes different from j. Instead,when p

forw(φ)ij > 0, if node i hands over the message at

the n-th encounter, the time between message reception(generation) at i and message relaying from i to j is givenby:

T ijforw(n−th) = Rij +

n−1∑m=1

Mij

The PDF of the sum of n i.i.d. random variables is givenby their n-th convolution [42]. Given that at each meetingnode i forwards the message to node j with probabilitypforw(φ)ij , we can use the geometric distribution to model

the probability of handing over the message at exactlythe n-th encounter with j. As a consequence, accordingto the law of total probability [35], the overall timeinterval since node i receives (generates) a new messageto the time when node i hands the message over hasprobability density:

T forwij (t) = p

forw(φ)ij Rij(t) +

++∞∑n=2

[(1− p

forw(φ)ij

)n−1

pforw(φ)ij ·

· P

(Rij +

n−1∑m=1

M(m)ij = t

)],

However, T forwij considers i, j node pair in isolation,

but in the general case the chain in Figure 1 exits fromstate i when node i hands over the message to the firstencountered node. Thus, the time required to exit statei is given by the minimum of the random variablesdescribing the forwarding time T forw

ij for all possible i, jpairs. By standard probability theory [42], we know thatthe CCDF of the minimum of n independent randomvariables is equal to the product of their CCDF. As T exit

i

can only take positive values, its expectation can becomputed by integrating the CCDF FT exit

i.

Theorem 2 (Transition probability): The transition prob-ability pij is given by:

pij = P (T forwij < T forw

i−others) (7)

where T forwij is defined as in Equation 6 and T forw

i−others asT forwi−others = minz =j{T forw

iz }.Proof: From Theorem 1 we know that the time before

node i hands over the message to node j is defined byT forwij . The forwarding process from node i to node j

competes with all the other forwarding processes fromnode i to any other node. More specifically, a mes-sage can be handed over from node i to node j withprobability p

forw(φ)ij upon meeting only if the meeting

with node j is the ”first to arrive” with respect to allthe other nodes. If we denote with T forw

i−others the timebefore the new message is forwarded by node i to anyother node different from j, we have that T forw

i−others =

minz =i,j{T i−zforw}. Then, the forwarding process between

node i and node j is the first to arrive only if T forwij <

T forwi−others. The likelihood of this event gives the transition


probability pij from state i to state j in the Markov chaindescribed in Figure 1:

pij = P (T forwij < T forw

i−others) = P (T forwij − T forw

i−others < 0)

The PDF of the difference Y −X between two randomvariables X and Y is given by their cross-correlation X⋆Y [42].

Transition probabilities expressed by means of Equa-tion 7 are well-formed according to Markov chain theory,i.e.,

∑j pij = 1. We can easily prove that by rewriting∑

j pij using Equation A as follows:∑j

pij =∑j

P (T forwij < min

z =j{T forw

iz })

Trivially, this sum adds up to one by definition.

APPENDIX BMODELLING WELL KNOWN SOCIAL-AWAREPROTOCOLSIn Section 5 we touched on the ability of the proposedanalytical framework to represent a variety of forward-ing solutions. For the sake of completeness, here wediscuss how the model can be applied to some wellknown social-aware policies proposed in the literature,specifically, BUBBLE [10], SimBet [11], and HiBOp [9].Given the generality of the framework, it is sufficient toshow how these algorithms can be mapped into appro-priate definitions of the fitness of nodes as forwarders.

B.1 BUBBLEThe BUBBLE forwarding strategy is a combination ofthe LABEL and the RANK policies. In LABEL, nodesare assumed to be tagged with a label that identifiesthem as belonging to the same organization. A messageis handed over upon encounter only if the peer sharesthe same label as the destination. According to thisdefinition, the fitness of a node as a forwarder underthe LABEL scheme is given by:

fLABELi,d =

{1 L(i) = L(d)0 otherwise

where L(i) gives node i’s label. Under the RANK policy,messages are forwarded along a path of increasing nodecentrality. If we denote with ci the node centrality ofnode i as defined in [10], we obtain the following:

fRANKi,d = ci

In BUBBLE, the authors distinguish between global rank-ing and a local ranking, the latter being a node’s central-ity value with respect to the community it belongs to.Thus, we hereafter use f

RANK(global)i,d and f

RANK(local)i,d

to differentiate the two rankings. The LABEL fitness andthe RANK fitness (global and local) are then compared inorder to select the best relay. More specifically, a messageis forwarded to nodes with higher f

RANK(global)i,d as long

as a no node belonging to the destination’s communityis found. Then, messages are handed over following anincreasing path of fRANK(local)

i,d .

B.2 SimBetIn SimBet [11], the fitness of a generic node i as aforwader for destination d is measured based on itsego-betweeness Beti and its similarity Sim(i, d) withrespect to the destination. The ego-betweeness expressesthe centrality of the node in its ego network, whilethe similarity metric measures the number of commonneighbors. We can now define fBet

i and fSimi,d , as the

fitness of node i according to its betweeness and itssimilarity to node d. fBet

i and fSimi,d can be computed

directly from Equations 5 and 6 in [11]. We thus obtain:

fSimBeti,d = αfSim

i + βfBeti,d

B.3 HiBOpThe modelling of context-aware protocols like HiBOp[9] introduces additional complexity. So far, the fitnessvalues have been depending only on node encounters,from which statistics on the meeting patterns or socialnetwork characteristics were extracted. On the contrary,in context-based forwarding protocols, nodes are en-riched with a description of the environment the usersoperate in (e.g., the place they live, the company theywork for, what they do in their leisure time) and thisinformation is used to make more accurate predictionson the future encounters among nodes. Typically, thecontext is described by means of atomic pieces of in-formation that we hereafter call attributes. Each attributeAi takes a value from a set VAi of the possible valuesfor that attribute. As an example, attribute city cantake values New York, Paris, Rome, and so forth. Theattribute values describing each node are collected in atable, called Identity Table (IT), which is exchanged uponcontacts with other nodes. Using statistics on the neigh-bors’ Identity Tables collected during pairwise meetings,nodes dynamically build their context-awareness andstore this information into two other tables: the Cur-rent Context table contains information on the directencounters, while the History table stores in an aggregatemanner statistics on the context of the direct encounters.The overall forwarding fitness is then a composition ofthe fitness values computed for each of these tables,which we denote as f IT

i,d , fCCi,d , and fH

i,d. Without provid-ing further details on the way the protocols works (forwhich we refer the interested reader to [9]) we hereafterprovide a convenient formulation for computing thesefitness values. The Identity Table fitness is measuredbased on the correspondence between node i’s IT and thedestination’s IT. Assuming that each IT is composed ofK attributes, the IT fitness can be computed as follows:

f ITi,d =

∑Kk=1 wk1Ak(d)(Ak(i))∑K

k=1 wk

.

Ak(i) denotes the value of the k-th attribute in node i’sIT and wk the weight assigned to each attribute. Theindicator function 1Ak(d)(Ak(i)) returns one when thevalue of the k-th attribute is the same in both node i’s


and node j’s identity table, zero otherwise. The CurrentContext (CC) fitness can be computed as follows:

fCCi,d = max

j∈Pi

f ITj,d

Finally, assuming P(k)op gives the combination of the

different statistics in the History tables as far as thek-th attribute is concerned, the History fitness can becomputed as follows:

fHi,d =

∑Kk=1 wkP

(k)op 1Ak(d)(Ak(i))∑Kk=1 wk

,

where again Ak(i) denotes the value of the k-th attributein node i’s IT, wk the weight assigned to each attribute,and 1Ak(d)(Ak(i)) is an indicator function that returnsone when the value of the k-th attribute is the same inboth node i’s and node j’s identity table.

APPENDIX CTHE EUROPEAN PARETO DISTRIBUTION

The notation presented in Section 6.2 is commonly re-ferred to as American Pareto distribution. There existsalso the European version of the power law distribution,which writes as follows:

FE(t) =

(t

tmin

)−α

(C1)

Basically, being X a random variable following a Eu-ropean power law with scale tmin and scale α, thenY = X − tmin is an American power law randomvariable.

Remark C1: The expectation of a random variable fea-turing a European Pareto distribution with PDF definedas in Equation C1 is finite and equal to tmin · α

α−1 whenα > 1.

In order to apply the analytical model proposed inSection 4 to the case of inter-meeting times featuring aEuropean Pareto distribution, we first need to computethe residual inter-meeting time, for which the followingtheorem holds (see [43] for the proof).

Theorem C1: When inter-meeting time M features aEuropean Pareto distribution with scale α and scale b(FM (t) =

(bt

)α), the residual inter-meeting time R is

distributed as follows:

FR(t) =

t−αtαb + 1 t > 0 ∧ t ≤ b

1α

(bt

)−1+αt > b

0 otherwise(C2)

Remark C2: The expectation of the residual of a Euro-pean Pareto distribution with scale α is finite for all αvalues greater than 2.Given that the constraints on the α values are the sameas those we discussed in Section 6.2, the discussionin Section 7 holds true also when using the Europeanversion of the power law distribution. On the other hand,the analytical model proposed in Section 6.2 cannot be

directly applied in this case. In fact, manipulating theresiduals Rij as we did for American power law inter-meeting times is not feasible, given that, according toCorollaries 1 and 2, we would have to multiply theCCDFs in Equation C2 with each other. However, it isstill possible to use an approximate model. In fact, it

is straightforward to prove that FR(t) <(

ttmin

)−(α+1)

,which is the PDF of a European Pareto random variable.By approximating the residual with a European Paretorandom variable, we are able to use the analytical modeldiscussed in Section 6.2.

APPENDIX DPROPERTIES OF POWER LAW DISTRIBUTIONSUSED IN THE PAPER

In this appendix we provide a general form for theminimum and difference between two power law dis-tributed random variables. For the ease of computation,and without loss of generality, here we restrict to thecase of power law random variables having the samescale, i.e., tminij = tmin,∀i, j. The following lemmas holdtrue both for the American and the European Paretodistribution.

Lemma D1 (Minimum of n Pareto Random Variables):The random variable X defined as X = mini{Xi}, whererandom variables Xi follow a power law distributionwith scale αi and scale xmin, is distributed according toa power law distribution with scale

∑i αi.

Proof: From standard probability theory we knowthat the CCDF of mini{Xi} is equal to

∏i FXi . When

multiplying the CCDF of n power law random variableshaving the same scale, we again obtain a power law withscale equal to the sum of the scales of the n power lawrandom variables.

Remark D1: The Pareto distribution resulting from theminimum of n Pareto distributions, each with its ownscale αi, is defined for

∑i αi > 0 (due to the PDF

normalization), and its mean is defined when∑

i αi > 1.Lemma D2 (Comparison between two Pareto R.V.):

Let us consider two random variables, X1 and X2,following a power law distribution with scale α1 andα2, respectively. Then, the probability that X1 is lowerthan X2 is given by:

P (X1 < X2) =α1

α1 + α2(D1)

Proof: We can rewrite P (X1 < X2) using the law oftotal probability:

P (X1 < X2) =

∫ +∞

xmin

P (X1 < X2|X2 = y)P (X2 = y)dy

=

∫ +∞

xmin

P (X1 < y)P (X2 = y)dy (D2)

Equation D1 is the solution to the above integral,computed after substituting the PDF and the CDF of thepower law random variables into Equation D2.


APPENDIX ERESIDUAL INTER-MEETING TIME FOR POWERLAW WITH EXPONENTIAL CUT-OFF CASE

Lemma E1 (Residual Inter-meeting Time): Assumingthat the inter-meeting time Mij follows a power lawwith exponential cut-off with shape αij , scale tmin, andrate λij , the CCDF of the residual inter-meeting timeRij is:

FRij(t)=

1 +tλij

αij− e−tminλij t(tminλij)

1−αij

tminαijΓ(1−αij ,tminλij)t>0∧t<tmin

e−tλij(

ttmin

)1−αij(tminλij)

1−αij

αijΓ(1−αij ,tminλij)·

· [−1+etλij (tλij)−1+αij (αij+tλij)Γ(1−αij ,tλij)]

αijΓ(1−αij ,tminλij)t>tmin

(E1)Proof: In Appendix C we discussed how to com-

pute the residuals for distributions having a scale tmin

different from zero. Thus, Equation E1 simply followsfrom the application of this method to the power lawwith exponential cut-off distribution. In order to checkthat the result is correct, we study the behaviour ofthe tail of the distribution. As we are focusing on thetail, we consider the case t > tmin. For the convenienceof the reader, in the following we drop subscript i, j.After collecting all constant terms together, we obtainFRij (t) = −K1e

−tλt1−α+K2e−tλt1−αeλt(tλ)αΓ(1−α, tλ).

We exploit the asymptotic relation between the Upper In-complete Gamma function and the exponential function,according to which, for large x, Γ(s, x) ∼ xs−1e−x [44].Thus, we obtain FRij (t) ∼ −K1e

−tλt1−α+K ′2e

−tλt1−α =K3e

−tλt1−α, where for large t it goes to zero faster thana power law.

APPENDIX FPROOFS FOR SECTION 7

Corollary 4: The minimum requirement for the ex-pected delay to converge under the social-aware schemesdefined in Definitions 6-7 is that there exists, for alli ∈ Rφ

s ∪{s}, at least one node j ∈ Rφi such that αij > 2.

Proof: The proof goes along the same line of theproof of Corollary 3. We solve the condition of Theo-rem 10 by imposing the loosest constraints on αij values,i.e., αij = 1 + ϵ for all j in Rφ

i − {d, z} (with ϵ → 0)and αiz > 1 + ϵ. We obtain αiz > 2. Given that thiscomputation must be repeated for all potential relays,i.e., for all nodes i belonging to Rφ

s , Corollary 4 follows.

Theorem 15: In a heterogeneous network where theinter-meeting times Mij between any generic i, j nodepair follows a power law with exponential cut-off distri-bution with shape αij and rate λij , the Two Hop relayingprotocol is always able to provide finite expected delays.

Proof: This follows directly from Remark 5. Theresidual inter-meeting time Rij and its expectation arealways defined for all αij and λij greater than zero. Thus,focusing on all possible forwarding paths (i.e., s → d and

s → j → d), we have that Rsd, Rsj , and Rjd have finiteexpectations for all s, j, d.

Theorem 16: In a heterogeneous network where theinter-meeting times Mij between any generic i, j nodepair follows a power law with exponential cut-off dis-tribution with shape αij and rate λij , the social-awarestrategies in Definitions 6-7 are always able to providefinite expected delays.

Proof: Again we consider the direct acyclic graph,rooted in the source node, in which nodes are orderedbased on their fitness value. All intermediate hops inthe direct acyclic graph have a finite expected delay bydefinition because, when inter-meeting times follow apower law with exponential cut-off, their residuals havefinite expectation, and thus the overall path is associatedwith a finite expected delay.

APPENDIX GBOUNDS EVALUATION

In the following we confirm the results discussed inSection 7.1 showing via simulations the advantages ofsocial-aware strategies in heterogeneous networks. Weconsider a network made up from 10 nodes, which inprinciple can all meet with each other. However, all pairsare characterized by a shape equal to 1.1 (recall thatfor αij ≤ 1 the expected inter-meeting time is infinite),except pairs i, 1 (and conversely 1, i), which are associ-ated with an α value of 3 (implying finite expectationof the inter-meeting time). Thus, there always exists apath with finite expected delay from a generic node ito any other node j, and this path always goes throughnode 1, because node 1 is the only node having finiteinter-meeting times with all other nodes. According toTheorems 9 and 11, the delay under the Two Hopstrategy will diverge, while it will be finite under themodified social-aware strategies. To check this result, werun a set of simulations in which each pair of nodescommunicates (i.e., a message is generated according toa Poisson process with mean 1 second). In order forthe comparison to be fair, we run 20000s of simulatedtime and we considered only the messages generatedin the first 10000s in our statistics. The 10000s packetlifetime has been chosen in order to be significantlygreater that the expected delay from node i to nodej (please note that E[Mi1] + E[M1j ] = 3s, given thatwe set tmin = 1s). Among social-oblivious forwardingstrategies we consider the Two Hop relaying protocol,while modified Social Forwarding is chosen as represen-tative of social-aware schemes. Results are shown withconfidence bands at 95% confidence level. Such bandsare quite narrow and thus barely visible in the plot.

The inability of the Two Hop scheme to providefinite expected delay clearly emerges from Figure 9.The modified Social Forwarding strategy completes theforwarding process within 10 seconds for the greatmajority of messages, and guarantees zero packet loss.On the contrary, there is a non-negligible amount of


0

0.2

0.4

0.6

0.8

1

∞ 1 100 10000

P (

X >

x)

Delay [s]

SF2H

Fig. 9. Delay CCDF under heterogeneous mobility

packet loss (around 10%) for the Two Hop scheme, whichaccounts for the infinite expected delay. These losses areimputable to messages that get stuck on a node thatmeets the destination with an infinite expectation.

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