On the Generalized Delay-Capacity Tradeoff of Mobile Networks with L\'evy Flight Mobility

In the literature, scaling laws for wireless mobile networks have been characterized under various models of node mobility and several assumptions on how communication occurs between nodes. To improve the realism in the analysis of scaling laws, we propose a new analytical framework. The framework is the first to consider a L\'{e}vy flight mobility pattern, which is known to closely mimic human mobility patterns. Also, this is the first work that allows nodes to communicate while being mobile. Under this framework, delays ($\bar{D}$) to obtain various levels of per-node throughput $(\lambda)$ for L\'evy flight are suggested as $\bar{D}(\lambda) = O(\sqrt{\min (n^{1+\alpha} \lambda, n^2)})$, where L\'evy flight is a random walk of a power-law flight distribution with an exponent $\alpha \in (0,2]$. The same framework presents a new tighter tradeoff $\bar{D}(\lambda) = O(\sqrt{\max (1,n\lambda^3)})$ for \textit{i.i.d.} mobility, whose delays are lower than existing results for the same levels of per-node throughput.


I. INTRODUCTION
Since the work in [1] that showed that mobility can be exploited to improve network throughput, there has been a plethora of work on this subject. A major effort in this direction has been in the design of delay tolerant networks (DTNs). However, this benefit in throughput comes at a significant delay cost. The amount of delays required to achieve a level of throughput for various mobility models such as i.i.d. mobility, random waypoint (RWP), random direction (RD), and Brownian motion (BM) have been extensively studied in [2]- [6]. Specifically, the delay required for constant pernode throughput has been shown to grow as Θ(n), which scales as fast as the network size n, for most mobility models including i.i.d. mobility, RWP, RD, and BM [2], [5], [7], [8]. Despite significant advances in the development of delaycapacity scaling laws, there has been considerable skepticism regarding the applicability of the results to real mobile networks because of various simplifying assumptions used in the analysis.
In this paper, we address two issues towards making the delay-capacity tradeoff analysis more realistic: 1) contacts among nodes in the middle of their movements and 2) Lévy mobility patterns of nodes in the network. In the literature, for mathematical simplicity, existing results have assumed that This work was supported in part by NSF grants CNS-1065136, CNS-1012700, CNS-0910868 and CNS-1016216, and ARO MURI Award W911NF-08-1-0238. Our considerations for improving the realism of delay-capacity tradeoff: (a) Conventionally, nodes A and B are assumed to have no contact opportunity, but we consider they can meet each other even during their movements. (b) We also consider that nodes follow Lévy flight mobility. nodes show slotted movements, and they do not communicate with each other while being mobile. Thus, they make contacts with other nodes and transfer data only at the edge of time slots. In other words, as shown in Fig. 1 (a), the opportunity for meeting other nodes during mobility has been ignored, although such opportunities can substantially reduce packet delivery delays. Also, in this work we focus on the Lévy flight model, which is widely accepted to closely mimic the actual movement of humans [9], [10]. The trajectory of this model is illustrated in Fig. 1 (b). To enhance the realism in the analysis of delay-capacity tradeoff, we develop a new analytical framework which takes both of these factors into account by developing a technique that characterizes the distribution of "first meeting time" among nodes conforming to Lèvy flight mobility in a two-dimensional space. It is important to note that the exact distribution of the first meeting time of Lévy flight even in a one-dimensional space has been an open problem even though it has applicability in a diverse set of research problems (e.g., characterization of particle movements and animal movements) in physics and mathematics. It is also informative to note that the distribution of the first meeting time of BM, which can be considered as an extreme case of Lévy flight, is also an open problem as noted in [11], [12].
Lévy flight, the mobility model we focus on in this paper, is a subset of Lévy mobility in which a node moves from position to position in a constant time. Another special case, Lévy walk, in which a node moves from one position to another in time proportional to the distance between the positions.
Except for the notion of the time required for each movement, Lévy flight and Lévy walk are fundamentally the same random walk whose flight length distribution asymptotically follows a power-law f α (z) ≈ 1/z α+1 , where z and α denote the flight length (i.e., moving distance of each slotted movement) and the power-law slope ranging 0 < α < 2, respectively. The heavy-tailed movements of Lévy mobility render the delay characterization extremely challenging. Our framework addresses these challenges using theories from stochastic geometry and probability, and provides a delay-capacity tradeoff for Lévy flight. Also, for a simpler i.i.d. mobility model, we provide a tighter delay-capacity tradeoff compared to existing studies using the same framework. Fig. 2 and Table I summarize the new tradeoffs identified using our analytical framework. The results show that the tradeoff for Lévy flight followsD(λ) = O( min{n 1+α λ, n 2 }) to obtain a per-node throughput of λ = Θ(n −η ) (0 ≤ η ≤ 1/2) as shown in Figs. 3 (a) and (b). These results are well aligned with the critical delay (i.e., minimum delay required to achieve λ = ω(1/ √ n)) suggested in [13]. Our tradeoffs show an important finding that the delay required to obtain constant per-node throughput (i.e., λ = Θ(1)) can be smaller than Θ(n) in mobile networks with mobility models such as Lévy flight with α < 1 and i.i.d. mobility. This is an important observation given that most of the existing studies present the delay required to obtain constant per-node throughput to be Θ(n) for almost all mobility models including the i.i.d. mobility.
Our tradeoff for Lévy flight becomes especially more interesting when we input α values from measurements presented in Table II into the tradeoff. This gives us a hint on how the performance of the network will scale in reality when the network consists of devices mainly carried (or driven) by humans. For α values between 0.53 and 1.81, the delays to obtain λ = Θ(1) are expected to lie between O(n 0.77 ) and O(n). This implies that in reality, a DTN mainly operated by human mobility may indeed experience less than Θ(n) delay in some areas. This observation of smaller delay suggests that mobile networks relying on opportunistic transmissions may have higher practical values in reality given that the delays have been overestimated by mobility and contact models with less realism.  The rest of the paper is organized as follows. We overview a list of related work in Section II and introduce our system models and definitions of performance metrics in Section III.    We then provide the intuition on how our analytical framework evaluates delay-capacity tradeoff using the properties of first meeting time of random walks in Section IV. Based on the understanding in Section IV, we analyze the tradeoffs of Lévy flight and i.i.d. mobility models in Sections V and VI, respectively. After briefly concluding our work in Section VII, we provide full details of proofs used in Sections V and VI through Appendices.

II. RELATED WORK
In [14], it was shown that the per-node throughput of random wireless networks with n static nodes scales as O(1/ √ n). The result was later enhanced to Θ(1/ √ n) by exercising individual power control [15]. Grossglauser and Tse [1] proved that constant per-node throughput is achievable by using mobility when nodes follow ergodic and stationary mobility models. This contradicted the conventional belief that node mobility negatively impacts network capacity due to interruptions in connectivity.
Many follow-up studies [2], [4], [5], [7], [16]- [18] have been devoted to characterize and exploit the delay-capacity tradeoffs. In particular, the delay required to obtain constant per-node throughput has been studied under various mobility models [2]- [6]. The key message is that the delay of 2-hop relaying proposed in [1] is Θ(n) for most mobility models such as i.i.d. mobility, RD, RWP, and BM.
More realistic mobility models, Lévy mobility models known to closely capture human movement patterns, were first analyzed in [13] for a special case of the tradeoff. Using spatio-temporal probability density functions, the critical delay defined by the minimum delay required to achieve larger throughput than Θ(1/ √ n) is identified for Lévy flight as well as Lévy walk.
Existing results on delay-capacity tradeoffs for mobile networks have been built under the assumption that nodes are able to communicate with each other only at the edge of time slots for slotted movements. Also, there has been no framework which can fully understand the delay-capacity tradeoff for Lévy mobility. In this paper, we develop an analytical framework which handles both of these issues and are able to use this framework to characterize the delaycapacity tradeoff.

A. Network Model
We consider a wireless mobile network indexed by n ∈ N {1, 2, . . .}, where, in the nth network, n nodes move on a completely wrapped-around disc D (⊂ R 2 ) whose radius scales as Θ( √ n). 1 Without loss of generality, we set the radius and the center of the disc D as √ n and 0 (0, 0), respectively, i.e., D = {x ∈ R 2 | |x| ≤ √ n}. We assume that the density of the network is fixed to 1 as n increases. 2 We also assume that all nodes are homogeneous in that each node generates data with the same intensity to its own destination. The packet generation process at each node is independent of node mobility. The generated packets are assumed to have no expiration until their delivery and the size of each node's buffer is assumed to be unlimited. Each packet can be delivered by either direct one-hop transmission or over multiple hops using relay nodes.
To model interference in wireless networks, we adopt the following protocol model as in [6], [14]. Let X i (t) denote the location of node i (= 1, . . . , n) at time t (≥ 0). Let L (i,j) (t) |X i (t)−X j (t)| denote the Euclidean distance between nodes i and j at time t. Under the protocol model, nodes transmit packets successfully at a constant rate W bits/sec, if and only if the following is satisfied: for a transmitter i, a receiver j and every other node u = i, j transmitting simultaneously, where ∆ is some positive constant. In addition, the distance between nodes i and j at time t should satisfy L (i,j) (t) ≤ r, where r (> 0) denotes the maximum communication range. We assume the fluid packet model [6], which allows concurrent transmissions of node pairs (with the rate divided by the number of pairs) interfering each other. We denote by Π the class of all feasible scheduling schemes conforming our descriptions.

B. Mobility Model
In this subsection, we mathematically describe the Lévy flight model and the i.i.d. mobility model. At time t = 0, node i chooses its location uniformly on the disc D (i.e., X i (0) ∼ Uniform(D)), which is independent of the others X j (0) for j = i. We assume that time is divided into slots of unit length and is indexed by k ∈ N. At the beginning of the kth slot (i.e., at time t = k −1), node i chooses its next slotted location X i (k) according to the associated mobility model. During the kth slot (i.e., during time t ∈ (k − 1, k]), node i moves from X i (k−1) to X i (k) with a constant velocity. Thus, and X i (k) as follows: (1)

Lévy Flight Model.
At the beginning of the kth slot (i.e., at time t = k − 1), node i chooses flight angle and flight length, denoted by θ i (k) (∈ (0, 2π]) and Z i (k) (> 0), respectively. During the kth slot (i.e., during time t ∈ (k − 1, k]), node i moves from X i (k − 1) to the selected direction θ i (k) of the distance Z i (k). Thus, the location X i (k) is determined as where The flight angle θ i (k) and the flight length Z i (k) are independent of each other and also independent of the previous locations X i (t) for the times t ∈ [0, k − 1] before they are generated. Hence, Each flight angle θ i (k) and flight length Z i (k) are independent and identically distributed across node index i and slot index k. Let θ and Z be a generic random variable for θ i (k) and Z i (k), respectively. Then, the flight angle θ is uniformly distributed over (0, 2π], and the flight length Z is generated from a random variable Z ⋆ having the Lévy α-stable distribution [20] by the relation Z = |Z ⋆ |. The probability density function of Z ⋆ is given by where ϕ Z ⋆ (t) E[e itZ ⋆ ] is the characteristic function of Z ⋆ and is given by ϕ Z ⋆ (t) = e −|st| α . Here, |s| > 0 is a scale factor determining the width of the distribution, and α ∈ (0, 2] is a distribution parameter that specifies the shape (i.e., heavytail-ness) of the distribution. The flight length Z for α ∈ (0, 1) has infinite mean and variance, while Z for α ∈ [1, 2) has finite mean but infinite variance. For α = 2, the Lévy α-stable distribution reduces to a Gaussian distribution with a mean of zero and variance of 2s 2 , for which the flight length Z has finite mean and variance. Due to the complex form of the distribution, the Lévy αstable distribution for α ∈ (0, 2) is often treated as a power-law type of asymptotic form: where we use the notation a(z) ∼ b(z) for any two real functions a(z) and b(z) to denote lim z→∞ [a(z)/b(z)] = 1 [20]. The form (5) is known to closely approximate the tail part of the distribution in (4), and a number of papers in mathematics and physics, e.g., [21], [22], analyze Lévy mobility using the form (5). For mathematical tractability, in our analysis we will also use the asymptotic form (5). Specifically, we assume that there exist constants c (> 0) and z th (> 0) such that i.i.d. Mobility Model. At the beginning of the kth slot (i.e., at time t = k − 1), node i chooses X i (k) uniformly on the disc D, which is independent of its previous locations X i (t) for the times t ∈ [0, k − 1] as well as the others X j (k) for j = i. Thus, X i (k) is independent and identically distributed across node index i and slot index k.

C. Contact Model
In our contact model, nodes are allowed to meet while being mobile. Hence, for a time t ⋆ in a domain {t | t ≥ 0}, we say that nodes i and j meet at time t ⋆ (or are in contact at time t ⋆ ) if they satisfy In the widely adopted contact model where nodes are allowed to meet only at the end of their movements (i.e., at slot boundaries), a meeting event can occur for a time k ⋆ in a domain {k | k ∈ {0} ∪ N} satisfying L (i,j) (k ⋆ ) ≤ r. We call this class of contact model slotted contact model throughout the paper.
Mobile nodes are exposed to more contact opportunities in our contact model compared to the slotted contact model.

D. Performance Metrics
The key performance metrics of our interest are per-node throughput and average delay as defined next: Definition 1 (Per-node throughput). Let Λ π:i (t) be the total number of bits received at the destination node i up to time t under a scheduling scheme π ∈ Π. Let λ π be the per-node throughput under π. Then, Definition 2 (Average delay). Let D π:i,v be the time taken for the vth packet generated from the source node i to arrive at its destination node under a scheduling scheme π ∈ Π. Let D π be the average delay under π. Then, In this paper, we focus on analyzing the scaling property of the smallest average delay achieving per-node throughput λ. We call this minimum average delay optimal delay throughout this paper. We focus on the throughput only in the range from Θ(1/ √ n) to Θ(1), since this range corresponds to the case where mobility can be used to improve the per-node throughput.
Definition 3 (Optimal delay). LetD(λ) be the optimal delay to achieve per-node throughput λ. It is then given bȳ

E. Throughput Achieving Scheme
We now consider a schemeπ that can realize per-node throughput λπ scaling from Θ(1/ √ n) to Θ(1). The schemeπ operates as follows: • When a packet is generated from a source node and the destination of the packet is within the communication range of the source node, the packet is transmitted to the destination node immediately. • Otherwise, the source node broadcasts the packet to all neighboring nodes within its communication range. Note that this broadcast is only performed by the source node when the packet is generated. • Any nodes carrying the packet can deliver the packet to the destination node when they are within the communication range of the destination node. • When one of the packets (including the original packet and the duplicated ones) reaches the destination node, all others are not considered for delivery.
By appropriately scaling r as a function of n, the schemeπ can achieve per-node throughput λπ ranging from Θ(1/ √ n) to Θ(1), as shown in the following lemma. Proof: If the network has been running for a long enough time, all nodes become to work as relay nodes and begin to have packets for all other nodes. Therefore, for a network with n/a n disjoint area where a n = Θ(r 2 ), all areas with more than two nodes can always be activated. Let b n denote the probability of having more than two nodes in an area. We then have b n = 1 − 1 − a n n n − n a n n 1 − a n n n−1 .
In addition, the total network throughput becomes b n n/a n and accordingly the per-node throughput is λπ = b n /a n . Without loss of generality, we assume r = n β (0 ≤ β ≤ 1/4). Then, the per-node throughput λπ is given by In the following, we will show that Hence, the per-node throughput λπ under the schemeπ scales as Θ(n −2β ). Note that Similarly, which gives By applying (8) and (9) to (7a), we have (7b). This complete the proof.
LetDπ(β) be the average delay underπ when r = Θ(n β ). Lemma 1 implies that by setting β = − log n ( √ λ), the schemeπ achieves the per-node throughput λπ = λ. Since the schemeπ is of the class Π, the order ofD(λ) can be obtained fromDπ(β) with the use of β = − log n ( √ λ), i.e., IV. PRELIMINARIES In this section, we provide the key intuition to understand how our analytical framework utilizes the properties of first meeting time in the derivation of delay-capacity tradeoffs under the Lévy flight and the i.i.d. mobility models. We then sketch the challenges residing in our framework and briefly describe our approach to address these challenges.

A. Delay Analysis with First Meeting Time
The first meeting time of two nodes moving in a twodimensional space, which is directly connected toDπ, is defined below: Definition 4 (First meeting time). For i = j, the first meeting time of nodes i and j, denoted by T (i,j) , is defined as Since T (i,j) is independent and identically distributed across pair index (i, j), we use T to denote a generic random variable for T (i,j) .
Let D (s,d) be a random variable representing the time taken by a packet generated from a source node s to arrive at a destination node d. Since the packet generation process is independent of node mobility, we consider that each packet is generated at time t = 0 without loss of generality. Then, the packet delay under the schemeπ, denoted by Dπ :(s,d) , can be expressed in terms of the first meeting time as where I(s) {i | L (s,i) (0) ≤ r} denotes a set of node indices that are within the communication range of the node s at time t = 0. Note that s ∈ I(s) by definition. Hence, the following equation represents the schemeπ described in Section III: From Definition 2, the average delayDπ can be obtained bȳ

B. Distribution of the First Meeting Time
In order to evaluate (11), the distribution of the first meeting time P{T (i,d) < τ } is essential. To obtain the distribution, we start from defining the following: let I (i,j) (k) (i = j, k ∈ N) be a random variable indicating the occurrence of a meeting event between nodes i and j during the kth slot (i.e., time t ∈ (k − 1, k]), i.e., For notational simplicity, throughout this paper, we omit (i, j) in I (i,j) (·) and L (i,j) (·), unless there is confusion. We then define a function H(k, l 0 ) for k ∈ N and l 0 ∈ (r, 2 √ n], which denotes the probability that nodes i and j are not in contact during the kth slot, conditioned on the fact that the initial distance between the nodes was l 0 and after that the nodes have not been in contact by time t = k − 1, i.e., Note that l 0 is upper bounded by 2 √ n since the radius of the disc D is set to √ n. We find that the distribution of the first meeting time T can be obtained from the function H(k, l 0 ) as shown in the following lemma.
The identity P{T > 0} = P{L(0) > r} shown in Lemma 2 has the following implications for P{T > 0}: (i) It is determined by the spatial distribution of nodes at time t = 0 (which is assumed to be uniform on the disc D). Hence, it is invariant for both the Lévy flight and the i.i.d. mobility models. (ii) It represents the probability that two arbitrary nodes are out of the communication range at time t = 0. Since P{T > 0} is frequently used throughput this paper, we define it as P o and summarize its implications using the following lemma.
Then, P o is bounded by Proof: See Appendix A.

C. Technical Challenge and Approach
In our framework, characterizing the function H(k, l 0 ) in (12) which appears in the expression for P{T > τ } in (13) is the key to analyze the optimal delay. The major technical challenge arises from tracking meeting events in the middle of a time slot. The meetings over time are heavily correlated irrespective of mobility models. The correlation can be understood as follows: let us consider two consecutive slots, say the kth and the (k + 1)th slots, for ease of explanation. The occurrence of a meeting event during the kth slot (resp. the (k + 1)th slot) is determined by the locations of nodes i and j at the slot boundaries, i.e., at times t = k − 1, k (resp. at times t = k, k + 1). Hence, both I(k) and I(k + 1) depend on the values of X i (k) and X j (k), and accordingly the sequence {I(k)} k∈N is correlated in our contact model. Due to the complexity involved in the correlation, deriving the exact form of H(k, l 0 ) appears to be mathematically intractable. To address this challenge, we take a detour to derive a bound on H(k, l 0 ) using theories from stochastic geometry and probability. The detailed analysis of H(k, l 0 ) for the Lévy flight model and the i.i.d. mobility model is presented in Lemmas 5 and 10, respectively, which allow us to reach the conclusions of this paper.

V. DELAY ANALYSIS FOR THE LÉVY FLIGHT MODEL
In this section, we analyze the optimal delay under the Lévy flight model. We use the following four steps in our analysis: Step 1 (Formulation of the average delay using the first meeting time distribution): As shown in (11) Under the Lévy flight model, T (i,d) for i ∈ I(s) are heavily correlated since the next slotted location X i (k + 1) depends on the current location X i (k). Note that all the nodes i ∈ I(s) are in proximity of the node s, and thus min(T (i,d) ; i ∈ I(s) \ {d}) is not easily tractable. Therefore, we use the following bound to describē Dπ using T .Dπ For the simpler i.i.d. mobility model, we are able to derive a tighter bound on min(T (i,d) ; i ∈ I(s) \ {d}). We present the result in Step 1 of Section VI. Let ⌈T ⌉ denote the smallest integer greater than or equal to T . Then, since , and E[⌈T ⌉] is an upper bound onDπ, as shown in the following lemma.

Lemma 4. The average delayDπ of the schemeπ under the Lévy flight model is bounded bȳ
where T is the generic random variable for the first meeting time , which gives (16). Since the random variable ⌈T ⌉ takes on only nonnegative integer values, the expectation E[⌈T ⌉] can be obtained by where the second equality comes from the property that P{⌈T ⌉ ≥τ } = P{T >τ − 1} for allτ = 1, 2, . . .. Replacinĝ τ − 1 with τ gives the lemma. Step

Lemma 5.
Under the Lévy flight model, the function H(k, l 0 ) in (12) has the following characteristics: representing a flight differential between nodes i and j. 3 Then, geometrically the function H(1, l 0 ) can be viewed as the probability of the flight differential falling into a set S(l 0 ) (⊂ R 2 ) defined as follows. Let D r (u) (⊂ R 2 ) denote a disc of radius r centered at u ∈ R 2 , i.e., D r (u) {x ∈ R 2 | |x − u| ≤ r}. 3 The existence of the generic random variable for Let (v, w) denote a line connecting two points v, w ∈ R 2 . For a fixed l 0 ∈ (r, 2 √ n], define a set S(l 0 ) as An example of S(l 0 ) is shown in Fig. 4. The set S(l 0 ) has a connection with the function H(1, l 0 ) as follows: (iv) For k = 2, 3, . . ., each function H(k, l 0 ) is also bounded above byP . Thus, for all k ∈ N and l 0 ∈ (r, 2 √ n], we have (v) There exist constants c l , c u (> 0), and n th ∈ N such that for all n ≥ n th ,P is bounded above and below bŷ Based on the formula for P{T > τ } in Lemma 2 and (iv) in Lemma 5, we derive a bound on P{T > τ } in terms ofP as follows: for τ = 1, 2, . . ., we have from (iv) in Lemma 5 that Since (20) holds for all l 0 ∈ (r, 2 √ n], by integrating (20) over l 0 ∈ (r, 2 √ n], we have By combining (21) and Lemma 2, we have Since P{T > τ } = P o for τ = 0, the bound in (22) also holds for τ = 0. The above result is summarized in Lemma 6.

Lemma 6. Under the Lévy flight model, the CCDF of the first meeting time T is bounded by
whereP and P o are defined in (18) and (14), respectively.

Lemma 7. The expectation E[⌈T ⌉] under the Lévy flight model is bounded by
Proof: Using Lemma 6, we can give a bound on E[⌈T ⌉] in Lemma 4 as By (v) in Lemma 5, we haveP < 1 for any r > 0. Thus, the expectation E[⌈T ⌉] is bounded by the geometric series which converges to Po 1−P .
Step 3 (Analysis of the delay scaling): In Lemma 1, we have analyzed the order of the per-node throughput λπ of the schemeπ. The results in Lemma 4,(v) in Lemma 5, and Lemma 7 allow us to analyze the order of the average delayDπ, which is shown in Lemma 8.
In addition, under the Lévy flight model, we have a trivial upper bound for all α ∈ (0, 2] as Combining (23) and (24) yields our lemma.
Step 4 (Analysis of the delay-capacity tradeoff): In the last step, we derive the delay-capacity tradeoff under the Lévy flight model. By combining the capacity scaling in Lemma 1 and the delay scaling in Lemma 8, we get the following theorem.

VI. DELAY ANALYSIS FOR THE i.i.d. MOBILITY MODEL
In this section, we provide detailed analytical steps for obtaining the optimal delay under the i.i.d. mobility model. We again follow the four steps analogous to those used for the Lévy flight model.
where T ⋆ v (v = 1, . . . , m) denotes the first meeting time of the node d and the vth node in the set I(s), provided that |I(s)| = m and d / ∈ I(s). Let T 1 , . . . , T m be m independent copies of the generic random variable T . Then, we can derive a bound on T ⋆ v in terms of T v as follows: Here, i v denotes the vth index in the set I(s) and d = denotes "equal in distribution". The last equation comes from the aforementioned nature of the i.i.d. mobility model in which the locations of nodes are reshuffled at every time slot.
We define a functionŪ : Note that discretization of a random variable T v to ⌈T v ⌉ is for mathematical simplicity and it does not affect the result (i.e., order of the optimal delay) of this paper. The function U (m) works as a tight upper bound onDπ as shown in the following lemma.
Step   (12) has the following characteristics: (i) Let ∆X be a generic random variable for X i (t) − X j (t) representing a location differential between nodes i and j. 4 Then, geometrically the function H(1, l 0 ) can be viewed as the probability of the location differential falling into a set where the definitions of (·, ·) and D · (·) can be found in Lemma 5. An example of S ⋆ (l 0 ) is shown in Fig. 5. The set S ⋆ (l 0 ) has a connection with the function H(1, l 0 ) as follows: (v)P is bounded above and below for all n ∈ N bŷ Similarly to Step 2 in Section V, we derive a bound on P{T > τ } in terms ofP as follows: from (iv) in Lemma 10 and Lemma 2, we have Since P{T > τ } = P o for τ = 0, the bound in (37) also holds for τ = 0. The above result is summarized in Lemma 11. whereP and P o are defined in (18) and (14), respectively.
Proof: Combining Lemma 2 and (iv) in Lemma 10 gives Lemma 11. The detailed derivation was described earlier in (37).
Using Lemma 11, we can give a bound on the function U (m) in Lemma 9 as By (v) in Lemma 10, we have (P ) m < 1 for any r > 0. Thus, U (m) is bounded by a convergent geometric series and we summarize the result in Lemma 12.
where γ is a constant in (0, 1) and γr 2 implies the γ fraction of the average number of nodes within the communication range of a source node. In (40), we used the property that U (m) is a nonincreasing function of m. Hence, by Lemma 9 and (40), the average delayDπ of the schemeπ under the i.i.d. mobility model is bounded by: The results in (41), Lemmas 3 and 12, and (v) in Lemma 10 allow us to analyze the order of the average delayDπ, which is shown in Lemma 13. Proof: Here, we provide a sketch of the proof with details given in Appendix C.

Step 4 (Analysis of the delay-capacity tradeoff):
In the last step, we derive the delay-capacity tradeoff under the i.i.d. mobility model. By combining the capacity scaling in Lemma 1 and the delay scaling in Lemma 13, we get the following theorem.

VII. CONCLUDING REMARKS
In this paper, we developed a new analytical framework that substantially improves the realism in delay-capacity analysis by considering (i) Lévy flight mobility, which is known to closely resemble human mobility patterns and (ii) contact opportunities in the middle of movements of nodes. Using our framework, we obtained the first delay-capacity tradeoff for Lévy flight and derived a new tighter tradeoff for i.i.d. Our result is well aligned with the critical delay suggested in [13]. For i.i.d. mobility, our analysis providesD(λ) = O( max(1, nλ 3 )) as shown in Fig. 2 (b). These tradeoffs are especially remarkable in both Lévy flight and i.i.d. mobility for the constant pernode throughput (i.e., λ = Θ(1)) as they demonstrate that the delay can be less than Θ(n), which has been widely accepted for most mobility models. Our future work includes (i) an extension of our framework to analyze the delay-capacity tradeoff under Lévy walk and (ii) another extension to capture correlated movement patterns among nodes.

APPENDIX A PROOF OF LEMMA 3
By the definition of P o in (14), we have Let F Xi(0) (·) denote the CDF of X i (0). Then, by conditioning on the values of X i (0), the probability P{L (i,j) (0) ≤ r} in (46) can be rewritten as where the last equality comes from the independence between X i (0) and X j (0). Note that, since X j (0) ∈ D with probability 1 and X j (0) ∼ Uniform(D), the probability P{X j (0) ∈ D r (x)} in the integral in (47) is given by where Area(S) denotes the area of a set S ⊂ R 2 . An example of D∩D r (x) is shown in Fig. 6. From the figure, it is obvious that Area(D ∩D r (x)) is nonincreasing as x approaches to the boundary of the disc D. Hence, (48) is bounded above by In addition, it is bounded below by where θ 2 cos −1 ( r 2 √ n ) (See Fig. 6). Since √ n ≥ r, we have θ ≥ 2 cos −1 ( r 2r ) = 2π 3 . Hence, the inequality in (50) is further bounded by By substituting (49) and (51) into (47), we have which, combined with (46), gives

APPENDIX B PROOFS OF LEMMAS FOR THE LÉVY FLIGHT MODEL
Here, we give detailed proofs of Lemmas 5 and 8, which are used for analyzing the optimal delay under the Lévy flight model. To prove Lemma 5, we need the following Lemmas 14, 15, and 16.

Lemma 14.
For i = j and k ∈ N, let where V · (k) (representing the kth flight of a node ·) is defined in (3). Then, under the Lévy flight model, ∆V (i,j) (k) has the following properties: and slot index k. Hence, we use ∆V to denote a generic random variable for ∆V (i,j) (k).  (1) and (2)). Since V i (k) is independent of F u (k − 1), it is independent of X u (t). By the same reason, V j (k) is independent of X u (t). Therefore, the difference V i (k)−V j (k) is independent of X u (t).
(ii) Since each of the flight angle θ u (k) and the flight length An example of vectors satisfying the condition in (54): for a given (v i , v j , v), the angle ∠∆V (i,j) is determined by the angle ∠V j . Since Z u (k) is independent and identically distributed across node index u and slot index k, the random variable V u (k) ( (Z u (k) cos θ u (k), Z u (k) sin θ u (k))) is also independent and identically distributed across u and k. Therefore, the difference V i (k) − V j (k) is identically distributed across pair index (i, j) and slot index k. However, it is not necessarily independent across (i, j) while it is independent across k for a fixed (i, j). (iii) To prove (iii), it suffices to show that for any v ≥ 0, where 0 < θ ≤ 2π. In the following, we will prove (52). For simplicity, we omit the slot index k in V · (k) and ∆V (i,j) (k) in the rest of this proof. By conditioning on the values of (|V i |, |V j |), we can rewrite the probability on the left-hand side of (52) as follows: For a fixed v ≥ 0, consider an event An example satisfying (54) is shown in Fig. 7. Under the con- is determined by the angle ∠V j as the figure shows. Since Since the above equality holds for any (v i , v j ) satisfying (54) for a given v, the probability in (53) boils down to the following: This completes the proof.

Lemma 15.
Suppose k ∈ N and l ∈ (r, 2 √ n]. Then, for any sets L(·) ⊂ [0, 2 √ n] satisfying we have under the Lévy flight model the following: The definitions of ∆V and S(l) can be found in Lemma 5.

A. Proof of (i)
To prove (i), we need the following: for any x ∈ [0, 1], The proof of (87) is given at the end of this section. From (19a) in Lemma 5 with r = n β , we have for all n ≥ n th the following: Since n β−1/2 2 ∈ [0, 1] for any β ∈ [0, 1/4] and n ∈ N, we further have from the lower inequality in (87) that Hence, we have Using a similar approach as above, from (19b) in Lemma 5 and the upper inequality in (87), we have for all n ≥ n th the following: Hence, we have lim sup Combining (88) and (89) proves (i).

B. Proof of (ii)
Without loss of generality, we assume P{Z α > z th } = 1. (In this proof, subscript α is added to all random variables to specify the underlying parameter α of the Lévy flight model.) Then, from (6), we have P{Z α > z} = ( zth z ) α for all z ≥ z th , which gives for any 0 < α 1 ≤ α 2 ≤ 2 and z ≥ z th the following: The inequality in (92) shows that for any t 2 > t 1 ≥ 0 having a sufficiently small difference ǫ t 2 − t 1 > 0, we get Note that since P{T α1 > t} = P{T α1 > t, T α1 > t − ǫ} for t ≥ ǫ, we can express P{T α1 > t} in a nested form as Using the nested form continuously, we have Hence, by applying (93) to (94), we have Note that P{T α1 > 0} = P{L α1 (0) > r}. In addition, since X i (0) ∼ Uniform(D) for all i = 1, . . . , n regardless of α, we have P{T α1 > 0} = P{T α2 > 0}. Thus, the right-hand side of (95) boils down to P{T α2 > t}, and consequently P{T α1 > t} ≤ P{T α2 > t} for all t ≥ 0.
where X · (t) denotes the location of a node · at time t. Then, under the i.i.d. mobility model, ∆X (i,j) (t) has the following properties: (i) ∆X (i,j) (k) (k ∈ N) is independent of X u (t) for all u = 1, . . . , n and t ∈ [0, k − 1]. (ii) ∆X (i,j) (t) is identically distributed across pair index (i, j) and time t (≥ 0). Hence, we use ∆X to denote a generic random variable for ∆X (i,j) (t).
(ii) For any i = j and t ≥ 0, X i (t) and X j (t) are independent and identically distributed. Therefore, the difference X i (t)−X j (t) is identically distributed across pair index (i, j) and time t. However, it is not necessarily independent neither across (i, j) nor across t. (iii) To prove (iii), it suffices to show that for any x ≥ 0, where 0 < θ ≤ 2π. By noting that ∠X i (t) ∼ Uniform[0, 2π] for any i = 1, . . . , n and t ≥ 0 and using a similar approach as in the proof of (iii) in Lemma 14, we can prove (iii) in Lemma 17. Due to similarities, we omit the details.
we have under the i.i.d. mobility model the following: The definitions of ∆X and S ⋆ (l) can be found in Lemma  Proof: Using a similar approach as in the proof of Lemma 15, we can prove Lemma 18. The difference is that the key idea of this proof is to use the following equality: for any k ∈ N, l ∈ (r, 2 √ n], θ ∈ (0, 2π], and F (k − 1), we have where the definition of F (k − 1) can be found in (57). Then, similarly to the proof of Lemma 15, using the key equality in (99) we can prove Lemma 18. Due to similarities, we omit the details.
In the following, we show (99). We first consider the event {I(k) = 0}. Since (61) also holds for the i.i.d. mobility model, by the same reason in the proof of Lemma 15, the event {I(k) = 0} occurs if and only if the following event occurs: We next consider the event {I(k) = 0} conditioned by ∆X(k − 1) = le jθ and F (k − 1). Under these conditions, (100) is reduced to the following: An example of S ⋆ (l, θ) is shown in Fig. 11. Hence, the probability on the left-hand side of (99) becomes By (i) in Lemma 17, ∆X(k) is independent of ∆X(k − 1) and F (k − 1), and thus we have In addition, by (ii) in Lemma 17, P{∆X(k) ∈ S ⋆ (l, θ)} = P{∆X ∈ S ⋆ (l, θ)}.
In addition, for any u ∈ D and sufficiently small ǫ (> 0), the area Area(R (x,ǫ) (u)) is calculated as By applying the combined result of (107) and (108) to (106), we have This proves the lemma.

A. Proof of (i)
Similarly to the proof of (i) in Lemma 5, we can prove (i) in Lemma 10 by applying Lemma 18 with k = 1, l = l 0 , and L(0) = (r, 2 √ n]. Due to similarities, we omit the details.
Hence, we have From Lemma 3, we also have P c o = 1 − P o ≤ r 2 n . Hence, the term 2(n − 2)P c o in (131) is bounded above by 2(n − 2)P c o ≤ 2(n − 2) r 2 n ≤ 2r 2 , from which we have Thus, by (132) and (133), the argument of the exponential function in (131)