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Information Epidemics in Complex Networks with Opportunistic Links and Dynamic Topology Pin-Yu Chen and Kwang-Cheng Chen, Fellow, IEEE Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan Email : [email protected] and [email protected] Abstract—Wireless networks, especially mobile ad hoc networks (MANET) and cognitive radio networks (CRN), are facing two new challenges beyond traditional random network model: opportunistic links with random nature due to fading and dynamic channel access, and dynamic network topology due to mobility and time varying. For a more general networking of Internet users as social complex networks, such challenges of further importance remain open. A generic and reliable model is therefore required to capture the information dissemination dynamics (IDD) of machine-to-machine communications or interactions in relation-based social networks. Inspired from epidemiology, we investigate the IDD in well-known complex networks by modified Susceptible-Infected (SI) model, which is surprisingly suitable for above multiple scenarios. Systematically categorizing such networks and examining conditions to adopt SI model for IDD in complex networks, the fundamental properties including mixing type, vertex connectivity and giant component size provide valid insights for quantitative analysis. We also investigate the IDD in networks with time-varying topology and show that all individuals receive the information in dynamic sense, even if the giant component size is not compatible with the number of individuals in static sense. Consequently, we successfully establish such analytical model for characterizing the IDD in complex networks consisting of opportunistic links and time-varying topology, feasible for various random wireless networks and social networks.

I. I NTRODUCTION With the advancement of wireless communication and the Internet, information dissemination plays an important role in large-scale networks consisting of numerous individuals such as machine-to-machine communications and relationbased social networks. As a powerful tool for aggregating multiple networks of interests into an unified manner, complex network [1], [2] has evolved in recent years to model various network topologies observed in current networks. For instance, community resembles Erd¨os-R´enyi (ER) network, AS-level routers and the World Wide Web resemble power-law distributed networks, wireless sensors resemble lattice (regular) networks and relations between acquaintances resemble smallworld networks. Although various efforts have been done regarding the analysis of information dissemination dynamics (IDD) in certain type of network for a particular purpose such as Influenza, we may unify the IDD in different networks with the aid of complex network notions since many networks of interests share common features. Moreover, it is still desirable to understand over the topology of a random network evolving with time due to mobility, and opportunistic links (e.g. MANET,

CRN and social networks) [3]. It still opens to precisely describe the IDD concerning large-scale time-varying network topology. The approach of tracking adjacency matrix can perfectly capture the entire network topology but at the cost of high computation complexity for time-varying topology, and is hence hardly useful to investigate networks with mobility concern. To systematically establish a feasible, flexible and generic model representing the IDD in complex networks, we start from examining the fundamental properties of complex networks. Since the mixing type, vertex connectivity and giant component size are essential factors contributing to the IDD, based on these properties, we provide several definitions to categorize complex networks constructed by well-known models discussed in Sec. III. By leveraging the taxonomy, we greatly reduce the tremendous efforts for specifying every network of interests since the IDD of the same category share similar features. In wireless networks without clearly defined infrastructure or social networks, a message source (a machine or a person) originates the information, and its transportation much resembles spread of epidemics. Inspired from epidemiology [4], we model the IDD by Susceptible-Infected (SI) model. To deal with message dissemination in any form (e.g. data packet, virus, word or the Internet worm), we interpret it as information dissemination with informed rate λ, an adjustable parameter representing distinct characteristics, suitable for modeling opportunistic links appearing in complex networks. We relate IDD to SI model so as to put the IDD in different complex networks on the same scale for fair comparison. A modified SI model is further proposed to alleviate the degreecorrelation problem and better fit realistic scenarios. In reality, especially taking mobility into consideration, the dynamic (time-varying) network topology results in changing connectivity. Based on the quantitative analysis of IDD in static complex networks, we show that all individuals eventually receive the information in dynamic sense whereas it may not be true in static sense. And our model successfully captures the complicated IDD in dynamic complex networks. In Sec. VI, we use our model to evaluate the performance of epidemic routing (gossip protocol) in MANET simulated by Truncated L´evy Walk (TLW) model. These results illustrate that our model precisely characterizes the IDD without the knowledge of adjacency matrix.

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II. S YSTEM M ODEL In terms of complex network, we denote individuals (e.g. wireless end terminals or the Internet users) the vertices and denote communication channels (e.g. physical channels or social relations) between any two individuals the edges on a undirected graph G(V, E), where V is the set containing all individuals, E is the set containing all communication channels and | V |= N is the total number of vertices. By epidemic terminology we denote a vertex is informed as it is infected, otherwise it is susceptible (uninfected). We assume that a vertex is capable of receiving multiple information from its neighbors and avoid collision. And the informed vertices adopt “consistent broadcasting” behavior so that an informed vertex tends to transmit the information to susceptible vertices in contact consistently, just as spread of epidemics. Consequently, we map the IDD to SI model [4]. Since the status of a channel is not always stable due to the channel access and its condition, capability of the vertex (network node) and Quality-of-Service (QoS) constraints, information may not be successfully transmitted, which defines opportunistic links in such networks. We introduce a twostate Markov chain for opportunistic links atop communication channels (i.e. on and off). Hence the informed rate (infection rate) λ is equivalent to the probability β transiting to on state independently among channels. And the IDD are equivalent to i(t), the fraction of informed vertices with respect to time. The following definitions and lemmas clarify the notions. Definition 1. For a communication channel between two vertices at arbitrary time t, a link is present with probability β that the information is transmitted from the informed end to the susceptible end. Lemma 1. The graph G(V, L) is a spanning subgraph of G(V, E), where L is the set containing all links at arbitrary time t.

Definition 2. The degree of vertex v of G(V, E) is the number of channels connecting to v, and the degree distribution Pk (t) of G(V, E) is the probability of having k channels for a vertex v ∈ V at arbitrary time t. We assume that there is at most one edge between any vertices pair and no self loops in G(V, E), by Definition 2 it is straightforward that the degree of a vertex is the number of distinct contacts (neighbors) of a vertex. Note that for dynamic topology the elements in E vary with time while the dynamic network still maintains the basic structure and properties. III. N ETWORK C ATEGORIES The mixing type (degree distribution), vertex connectivity (connection attribute) and size of the giant component (the connected subgraph containing a majority of V) are essential properties contributing to the IDD. To verify the significance, we provide several definitions regarding the above properties and categorize complex network models of interests, i.e., the power-law distributed networks, small-world networks, Erd¨ os − R´enyi (ER) networks, and lattice (regular) networks. Since almost all current networks are variations of the above network models, our taxonomy is a powerful tool for explaining the distinct IDD in complex networks. Definition 3. A network is homogeneous mixing if its degree distribution is centered at k¯ with variance σk 2 ≤ . Otherwise it is heterogeneous mixing. Definition 4. A network is connectible if every vertex has a positive probability to build a bidirectional (undirected) connection (channel) to any other vertices in the network. Based on Definition 4, we further define relevant concepts regarding the connecting probability and connection attribute. If the connecting probability is equal for every vertex, the network is equally connectible, otherwise it is unequally connectible. And If every vertex has a positive probability connecting to some fraction (not all) of vertices in the network, the network is partially connectible.

Proof: ∀ e ∈ E, a link l exists with probability β at arbitrary time t. We denote the relation by indicator function 1L (e). Since G(V, E) and G(V, L) have the same vertex set V, and L is a subset of E, G(V, L) is a spanning subgraph of G(V, E). Because there are only two types of vertices in G(V, E) i.e., the fraction of informed vertices i(t), and the fraction of susceptible vertices s(t), we denote i0 the fraction of initially informed vertices, by SI model we have

Definition 5. A network is saturated if the giant component size approaches total number of vertices in the network, i.e., G(V, E) is saturated if the giant component G(VGC , EGC ) has | VGC |∼ =| V |. Otherwise it is nonsaturated.

i(t) + s(t) = 1, i(0) = i0 di(t) ¯ = λki(t)s(t)

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It is obvious that G(V, E) is the giant component itself as long as it is a connected graph, and hence from Definition 5 the network is saturated.

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A. Homogeneous Mixing and Partially Connectible (HoMPC) Network

From (2) it is clear that i(t) approaches 1 as time evolves. However, conventional SI model is unaware of network topology and therefore can be applied only under certain constraints discussed in Sec. III. Throughout our work, we assume initially there is only one informed vertex, i.e., i0 = 1/N .

Because a lattice (regular) network is well structured that every vertex connects to its K nearest neighbors, it is a HoMPC network and the degree distribution is simply a Dirac delta function with magnitude equals 1 located at degree K. We treat an n-by-n lattice network on a two-dimensional plane

dt

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Lemma 2. If a network is denoted as G(V, E), and G(V, E) is a connected graph, then the network is saturated.

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(square lattice network) as a HoMPC network because the degree of every vertex except the marginal vertex is 4, and the number of marginal vertices are negligible as n increases. It is quite obvious that static wireless sensor network belongs to this category due to finite transmission range and regular topology. We will investigate its vulnerability of broadcasting protocol in Sec. IV. B. Homogeneous Mixing and Equally Connectible (HoMEC) Network Erd¨os-R´enyi (ER) model [5] assumes that a edge in G(V, E) is present with probability pER for total of N (N − 1)/2 possible edges. The degree distribution of an ER network is a binomial distribution B(N − 1, pER ). Since we are interested in large-scale networks, when N is relatively large, the degree distribution can be express by a Poisson distribution (N pER )k k¯k ¯ = e−kER ER (3) P (k) ∼ = e−N pER k! k! where k¯ER = N pER . The degree distribution is centered at k¯ER and a vertex has equal probability pER connecting to any other vertices in the network, thus ER network is a HoMEC network. C. Homogeneous Mixing and Unequally Connectible (HoMUC) network

to have great amount of neighbors, and a new vertex is hence prone to connect to those super vertices. Thus the power-law distributed network is a HeMUC network. We observe many HeMUC networks on the Internet such as links of AS-level routers and websites of the World Wide Web. Regarding the giant component size, we emphasize that if the giant component size is not compatible with the number of vertices, only the cluster including the initially informed vertex can possibly receive the information, and the condition is often ignored or implicitly excluded in many researches and simulations. From bond percolation theory, phase transition theory and complex network [1], [2], in 1995 and 1998 Molloy and Reed proved that a giant component emerges almost surely in random graphs with a given degree distribution P (k) when  k k(k − 2)P (k) > 0, provided that the maximum degree is less than N 1/4 [8]. For certain complex networks, Kesten proved that the critical probability in the square lattice network is 1/2 [9], from percolation theory the critical probability of an ER network is equivalent to having mean degree k¯ER = 1 [1], and Pastor-Satorras and Vespignani proved the absence of epidemic threshold in scale-free networks [10], indicating that a giant component is very likely to exist in scale-free network. For fair comparison, we emphasize the IDD in saturated network to ensure that the originator is within the giant component. We note that other important properties such as clustering coefficient and average path length are associated with the above definitions since cluster coefficient varies with vertex connectivity and the giant component has direct impact on the average path length.

Small-world network generated by Watts-Strogatz (WS) model [6] is constructed from a regular ring lattice and randomly rewire each edge of the lattice with probability pW S such that self-connections and duplicate edges are excluded. The degree distribution of small-world network is IV. S TATIC N ETWORKS min(τ,j)   τ −n For static networks, we focus on the IDD in time-varying  j (pws j) (1 − pws )n pws j−n e−pws j (4) graph G(V, L) due to opportunistic links, where the elements P (k) = n (τ − n)! n=0 in L may change while E is time-invariant. With the aid of for τ = k − j ≥ 0 and P (k) = 0 otherwise. 2j is the number network categories in Sec. III, we compare the IDD in different of edges for every vertex before rewiring. Since the degree network categories with SI model and provide valid explanadistribution has a pronounced peak at k¯ = 2j and decays tions and proofs for the results. The simulations are the average exponentially for large k [1], and the connecting probability is ensembles over 100 times in saturated complex networks. In unequal except pW S = 1, the case of extreme randomness, we Fig. 1, the IDD in square lattice network accelerate with the regard small-world network as HoMUC network. Many social increase of λ but are quite slow compared with SI model. network models are extensions of WS model since HoMUC The results are reasonable since a HoMPC network such as square lattice network implicitly indicates that the IDD are property well explains the clustering (centrality) feature. limited by its structure and the total susceptible neighboring D. Heterogeneous Mixing and Unequally Connectible vertices at an arbitrary time t are actually much less than (HeMUC) Network ¯ giving rise to the “degreethe expected number N · i(t)k, correlated” problem for overestimating susceptible neighbors. A power-law distributed network has degree distribution P (k) ∼ k −r , if r is within the region 2 ≤ r ≤ 3, then it is also In Fig. 2 we run simulations in ER networks and observe called a scale-free network since the first and second moments that the IDD are in accordance with SI model. The SI model converge. Barab´asi and Albert observe two essential factors is slightly faster than the simulation curves due to degreeof scale-free network: growth and preferential attachment [7], correlation problem. The reason why SI model fits the IDD in and we can construct a scale-free network by Barab´asi-Albert ER model is that in (3) the parameters N and pER characterize (BA) model. In BA model, every new node connects its m the ER network since we have mean degree k¯ER = N pER . edges to existing nodes according to preferential attachment And the HoMEC property regards every vertex in the network rule, and the average degree of such network is 2m. The homogeneous or identical so that SI model, a simple first-order power-law distributed degree distribution suggests that most differential equation, captures the IDD. We prove it from (2) vertices have few neighbors while some “super vertice” tend by substituting k¯ER = N pER that

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where C = λpER . Given total number N and setting C constant, both SI model and simulation results show that the IDD are the same for ER networks with different mean degrees shown in Fig. 2. In Fig. 3 we run simulations in small-world networks by WS model with different rewiring probability pW S . Ranging from extreme regularity (pW S = 0) to extreme randomness (pW S = 1), the IDD accelerate due to the enhancement of connectivity and the small-world network transfers from HoMPC network (pW S = 0) to HoMUC network (pW S ∈ (0, 1)) and finally HoMEC network (pW S = 1). We also observe an interesting phenomenon for the IDD in scale-free network generated by BA model. The IDD curve exceeds the SI model at first, but at some instant t0 SI model transcends the curve. The result is actually not so surprising since for a HeMUC network with super vertices the information is more prone to be transmitted to super vertices than vertices with low degrees at early stages, which is quite plausible because an Internet worm or attack usually aims at routers or websites with high links. And the average degree of newly informed vertices in Fig. 4 verifies the statement. Since most of vertices have low degrees and the initially informed vertex is chosen at random, the average degree increases at early stages because the information is indeed transmitted to vertices with higher

degrees. Then the average degree decreases due to information dissemination to low-degree vertices and the average degree of newly informed vertices is below the mean degree at t0 . Despite that SI model may not be suitable for describing the IDD in scale-free networks, it still provides us useful insights for better understanding the IDD in networks with HeMUC property. For example, homogeneity still holds for the vertices of the same degree, and an enhanced epidemic model is further described in [10]. In order to bridge the gap between the SI model and the IDD in complex networks, we need to modify the SI model so as to fix the degree-correlated problem because SI model implicitly assumes that the neighbors of informed vertices are uncorrelated. For HoMEC networks, we provide an intuitive upper bound by modifying the SI model and take the degreecorrelation into consideration. Proposition 1. For a saturated and HoMEC network, there exists an upper bound given the informed rate λ and mean ¯ degree k. Since a HoMEC network can be characterized by its mean degree, and intuitively we know that a vertex except the originally informed vertex is informed unless at least one of its neighbors is informed, the susceptible neighbors of an ¯ We hence propose a informed vertex are actually less than k.

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modified SI model with effective mean degree k¯ef f that di(t) = λk¯ef f i(t)[1 − i(t)] = λ[k¯ − f (t)]i(t)[1 − i(t)] (6) dt where f (t) is a time varying function accounting for the number of informed neighboring vertices in average. If we simply set f (t) = 1, then we have an tight upper bound for a static saturated and HoMEC network. Proposition 1 can also be applied to HoMUC and HoMPC networks by estimating f (t). To analyze the vulnerability of broadcasting protocols in static wireless sensor networks, we assume the sensors are uniformly and randomly distributed on a plane with node density σ, the communication radius of a sensor R and the radius of the circle containing the informed 2 sensors r(t). Since σπr(t) = N · i(t), we have f (t) = k¯ · θin = k¯ ·

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where θin is the ratio of inoperative sensors (sensors which have no susceptible √ neighbors) to number of informed sensors at time t, c = 2 σπR and σπR2 is usually negligible compared with N i(t). Applying f (t) to the modified SI model, we thus obtain the same result derived in [11]. V. DYNAMIC N ETWORKS For dynamic networks we refer to networks whose network topology changes with time while maintaining the basic structures and properties. i.e., the elements in E vary with time according to the connection attribute. The time-varying topology provides great chances for information dissemination to the entire network, and is therefore suitable for describing complicated interactions within large-scale networks with mobility support (e.g. routing protocols in MANET). Even given the condition that a network is nonsaturated (e.g. MANET in sparsely populated area), we expect that a virtual giant component (the connected subgraph of possible connections containing a majority of V) exists and the network is virtually saturated since all vertices eventually receive the information due to change of connectivity. Proposition 2. A dynamic network is virtually saturated in the sense that the virtual giant component size approaches the number of vertices.

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To focus on the IDD in dynamic networks, we set the informed rate λ = 1 so that L = E in order to justify Proposition 2. We run simulations in networks that are nonsaturated, especially in ER networks and square lattice networks since the two networks represent extreme randomness and extreme regularity structures, providing us great insights about the impacts of dynamic topology on the IDD. Whereas the giant component of HeMUC network such as scale-free network usually exists so that the dynamic HeMUC networks are beyond our scope. In square lattice networks we assume there are initially K degrees for each vertex and we randomly remove pLa fraction of edges so that the mean degree k¯La = KpLa . At each time step, the remaining edges rewire while keeping the regular structure. In Fig. 5 we show that the dynamic square lattice network is virtually saturated even if pLa is below the critical probability 1/2. We also observe that when pLa = 1/2, the giant component size is roughly 20 percent of total vertices for static case, suggesting that the existence of giant component is not a sufficient condition for a saturated network. On the other hand, for dynamic case the network is virtually saturated as long as pLa is larger than zero. For modified SI model f (t) =

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in slotted time period. The discrepancy increases due to the finite number of vertices (border effect) and uncertainty about the number of susceptible vertices at each time step. The critical mean degree of an ER network is 1 since intuitively for HoMEC network a vertex has to connect to at least one vertex to form a giant component. In Fig. 6 we observe the same outcomes as Fig. 5 that a dynamic ER network is virtually saturated but not necessarily saturated. And the SI model failing to describe the IDD of nonsaturated static networks fits the IDD of dynamic ER network with proper estimation of f (t). The results reveal that a equally connectible network does not guarantee G(V, E) a connected graph. We also point out that lessening the informed rate λ contributes only to longer latency for information dissemination. The IDD in these two networks justify Proposition 2. And we conclude Proposition 3. The information dissemination dynamics of a

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dynamic HoMEC or HoMPC network can be characterized by modified SI model.

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Proposition 4. For a (virtually) saturated and homogeneous mixing network, the time T needed to inform iT fraction of vertices can be obtained from (6) and T = i−1 (iT ) if i−1 (·) exists. Because the cumulative informed fraction approaches 1 in a (virtually) saturated network, Proposition 4 serves a more accurate benchmark to any broadcasting mechanisms in complex networks since it greatly mitigates the biased estimation of susceptible neighbors due to degree correlation problems. VI. A PPLICATIONS Regarding the IDD in both static and dynamic complex networks, there are wide applications associated with the propositions. In social network, the rumor spread or propagation of the Internet worm within a small-world network is upper-bounded by our model, where the trust of the rumor or the activity of the Internet worm relates to the informed rate. In machineto-machine communications either with or without mobility support, including cognitive radio networks (CRN), MANET, wireless sensor networks (WSN), and robotic networks, the informed rate is determined by channel condition and access, capability of devices and QoS constraints. And traditional methodology does not conduct such universal study. Leveraging our model, we evaluate the performance of epidemic routing via broadcasting in large-scale MANET where each node has the same transmission range R and is uniformly and randomly distributed on a D × D square field in wrap-around condition. The network is thus a dynamic HoMEC network since every mobile node may have chances to meet others within a distance R, and the average number of neighbors k¯ equals DN2 · πR2 for each node. By TLW model [12], we set the step length exponent s = 1.5 and pause time exponent ϕ = 1.38, which fit the trace-based data of human mobility pattern collected in UCSD and Dartmouth [13]. And we incorporate the packet successful transmission rate to informed rate λ. In Fig. 7 we show that the MANET is nonsaturated at every snapshot but virtually saturated, and our model captures the complicated interactions among numerous mobile nodes precisely. VII. CONCLUSION Our main contributions are to establish a generic and reliable model for evaluating the IDD in complex networks with opportunistic links and dynamic topology, and to examine the sufficient conditions to adopt the model for IDD in both static and dynamic networks. With the proposed taxonomy regarding the mixing type, vertex connectivity and giant component size, we first quantitatively analyze the IDD in static network

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categories, then we prove and show the flexibility of the proposed model, and a tight upper bound for the IDD of static and saturated network. The results and observations serve as the framework for investigating the IDD in networks of time-varying topology. We illustrate that a dynamic network is virtually saturated and our model characterizes the IDD even if the network is nonsaturated. For the performance evaluation of epidemic routing in large-scale MANET simulated by TLW model, our model captures the IDD precisely without the knowledge of tine-varying adjacency matrix. We hence successfully devise an analytic model suitable for various scenarios, especially for WSN and MANET. R EFERENCES [1] R. Albert and A. L. Barab´asi, “Statistical mechanics of complex networks,” Reviews of Modern Physics, vol. 74, no. 1, pp. 47–97, Jan. 2002. [2] M. E. J. Newman, “The structure and function of complex networks,” SIAM Review, vol. 45, no. 2, pp. 167–256, Mar. 2003. [3] K. C. Chen, B. K. Cetin, Y. C. Peng, N. Prasad, J. Wang, and S. Y. Lee, “Routing for cognitive radio networks consisting of opportunistic links,” Wiley Wireless Communications and Mobile Computing, 2009. [4] W. Kermack and A. McKendrick, “A contribution to the mathematical theory of epidemics,” in Proc. Roy. Soc., vol. A, no. 115, 1927, pp. 700–721. [5] P. Erd¨os and A. R´enyi, “On random graphs, I,” Publicationes Mathematicae (Debrecen), vol. 6, pp. 290–297, 1959. [6] D. J. Watts and S. H. Strogatz, “Collective dynamics of ’small-world’ networks,” Nature, vol. 393, no. 6684, pp. 440–442, Jun. 1998. [7] A. L. Barabasi and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509–512, Oct. 1999. [8] M. Molloy and B. Reed, “A critical point for random graphs with a given degree sequence,” Random Structures and Algorithms, vol. 6, no. 2-3, pp. 161–180, 1995. [9] H. Kesten, “The critical probability of bond percolation on the square lattice equals 1/2,” in Commun. Math. Phys., no. 74, 1980, pp. 41–59. [10] R. Pastor-Satorras and A. Vespignani, “Epidemic spreading in scale-free networks,” Physical Review Letters, vol. 86, no. 14, pp. 3200–3203, Apr. 2001. [11] P. De, Y. Liu, and S. Das, “An epidemic theoretic framework for vulnerability analysis of broadcast protocols in wireless sensor networks,” IEEE Transactions on Mobile Computing, vol. 8, no. 3, pp. 413–425, Mar. 2009. [12] Networking Research Lab, Department of Computer Science, North Carolina State University. [Online]. Available: http://netsrv.csc.ncsu.edu/twiki/bin/view/Main/WebHome.html [13] S. Kim, C.-H. Lee, and D. Y. Eun, “Superdiffusive behavior of mobile nodes and its impact on routing protocol performance,” IEEE Transactions on Mobile Computing, vol. 9, no. 2, pp. 288–304, Feb. 2010.

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