Navigating a Mobile Social Network Yalin E. Sagduyu and Yi Shi Intelligent Automation Inc., Rockville, MD 20855, USA Email: {ysagduyu, yshi}@i-a-i.com Abstract—With the emergence of mobile social networks, there has been an increasing interest in studying the methods of using social network connections, such as social networkaware routing, to improve the performance of communication networks. This raises the need for new models of combined social and communication networks with realistic link reliability and mobility properties. Going beyond graph analysis of social networks, such models are useful to have a better understanding of the algorithmic aspects in mobile social networking, e.g., how to navigate a mobile social network and search for users with local information. The first step towards this goal is to develop a combined social and communication network model, where wireless communication becomes the underlay to route information with the aid of social connections. Random patterns of node mobility are included in this combined model, which is then used to analyze the average delay of search paths between source-destination pairs. By applying greedy routing over the combined network, analytical expressions are obtained to evaluate the average delay as a function of separation between sourcedestination pairs. The results quantify the dependence of the average delay on the network mobility and on the combined use of social and communication links. Search paths are typically prone to link failures due to message drops in social and communication links. Therefore, the network model is extended with probabilistic link failures and analytical expressions are obtained for the success probability on search paths. The analysis of average delay and success probability shows how social connections can be used to reduce the end-to-end delay and increase the end-to-end success probability in a mobile social network. Keywords—Social networks, communication networks, mobile networks, search, navigation, routing, delay, success probability.

I.

I NTRODUCTION

Mobile wireless communication has been continuously shaping the way people are connected from the early days of cellular phones to modern multimedia applications. Online social networks have added a new dimension for people to interact with each other and the traffic volume is gradually shifting to portable devices connected via mobile networks. Because of limited spectrum resources and growing demand for bandwidth, there arises the ultimate need in analyzing the effect of social networks on communication networks, e.g., how to exploit social links to improve routing performance as in social network-aware routing [1], [2]. Such study requires new models of combined social and communication networks that have realistic representations of network mobility and link reliability properties. With these models, it would be possible to analyze different algorithmic design aspects and enhance the commnuication performance (in a mobile social network). This material is based upon work supported by the Air Force Office of Scientific Research under Contract FA9550-12-C-0037. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Air Force Office of Scientific Research.

One important design problem is to determine how one user can search for others with the aid of both social and communication links in a mobile social network. The expected delay and success probability of such a search (or end-toend message delivery or navigation) can give insights into the design of mobile social networks and help improve the end-to-end routing performance. In a distributed setup, each node can apply greedy routing and search the next hop in the process of finding the destination node by using only local (one-hop) information about the neighbors [17]. This problem has been extensively studied for social networks with realworld experiments [3]–[5]. The algorithmic aspects have been analyzed with greedy routing [6]–[8] to minimize the hop distance locally on a social network graph. On the other hand, finding paths to deliver messages from sources to destinations is a standard problem in communication networks and it is handled by routing algorithms at the network layer. In mobile social networks, information can be transferred through both social and communication links. Therefore, joint analysis of social and communication networks is needed to reveal how one network affects the other [9]–[11]. One example of the combined network is that mobile users carry smart phones that establish social communications (e.g., messaging over a friendship network or online blogs) over the internet that is supported by the cellular backbone. In the meantime, they use Wi-Fi or Bluetooth to establish local wireless communications in ad hoc manner. Such a network structure finds applications in emergency communications, trust establishment, influence spread, counter-messaging, and reliable message delivery services. The combination of social and communication networks requires new models and analytical tools [12] and the introduction of mobility to the combined network raises further challenges: 1)

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social and communication networks involve different links with mostly different definitions of distances, e.g., social distances may reflect friendship status and communication distances may be rather geographic; links in both networks are subject to probabilistic failure events, e.g., a social link may fail when a node drops a message from an untrusted neighbor and a communication link may fail when a transmission is not successfully received in a wireless medium because of noise, fading, or interference effects; users may be mobile and their movements may change the distances to each other on the communication network.

These challenges motivate the need for a suitable model to combine social and communication networks incorporating mobility and link success/failure events. On one hand, wireless

communications with limited transmission range (because of limited transmission power) can be modeled as formation of communication links on a random geometric graph. On the other hand, social graphs are known to show small-world [13] and scale-free [14] network properties. To achieve this target model, social network model can randomly deploy two types of links, i.e., short-range connections (SRCs) and long-range connections (LRCs), as used in [15], [16] to model a static social network (without communication links or mobility patterns). SRCs potentially exist between a node and some of its neighbors that are socially separated within some range. LRCs are other social connections outside this range following some distributions, e.g., power-law distributions to model scale-free networks. Together with SRCs, LRCs model the small-world phenomenon by reducing the average path lengths between source-destination pairs. To construct the mobile social network model, we follow the following steps: 1)

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we integrate social links into communication network such that SRCs in the combined network follow from communication links and LRCs in the combined network are derived by mapping social links onto the communication network; we introduce probabilistic failures for social and communication links since some searches remain incomplete wit cut communications as observed in social experiments [3], [4] and in routine operation of mobile wireless networks; we introduce mobility by randomly varying node distances from one time instant to another.

This combined social and communication network model enables the analysis of the end-to-end delay and success probability under link failure and node mobility properties. The following results are obtained: 1) 2) 3)

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we obtain analytical formulations for both average delay and success probability as a function of separation between source and destination pairs; we quantify how mobility and link failure effects decrease routing performance (i.e., increase average delay and reduce success probability); we observe that the end-to-end delay remains finite, as the separation between users increases, thanks to LRCs from social links (this quantifies the smallworld phenomenon for mobile social networks); we illustrate how social links can be utilized to improve the information transfer through the combined network structures by reducing the average delay and possibly increasing the end-to-end success probability. II.

M OBILE S OCIAL N ETWORK M ODEL

A. Communication Network We consider a wireless network as the underlay communication network. Such a network can be modeled as a random geometric graph, where nodes are uniformly and independently distributed on a region (e.g., a disk with radius R) at a given time, and two nodes are connected if and only if the distance between them is less than a threshold rC (see Fig. 1-(a)). The

Fig. 1. (a) a communication network, (b) a social network, (c) a combined (social and communication) network, (d) time slot structure.

threshold rC is the transmission/reception range and typically depends on the transmission power, the signal-to-noise-ratio (SNR) requirement of wireless radios and the wireless channel characteristics, such as path loss, RMS delay spread, and interference, that are captured by probabilistic packet failures for communication links. The search problem we consider on the underlay communication network proceeds in a time slot fashion (see Fig. 1-(d)) as follows. A time slot includes four phases: (i) decision, (ii) transmission, (iii) move, and (iv) update. At the beginning of a time slot, each node knows exact distances of its neighbors to the destination, e.g., location information may be collected via GPS and exchanged among neighbors. This information exchange process can be done similarily as the broadcast process. Then the node currently having the message selects its next hop node by some routing algorithm (the “decision” phase) and transmits data to the selected node (the “transmission” phase). After data transmission, all nodes move by following some mobility model (the “move” phase). Finally, each node announces its new location to all its neighbors (the “update” phase) and a time slot is completed. B. Mobility Model We apply the following mobility model, which can be regarded as the random waypoint model with fixed moving

time for each move phase. In the move phase of a time slot, each node can move along any direction for up to distance rC . As a result, a node with distance d to the destination may change up to 2rC , due to the movement of itself and destination. As detailed in Section III-A, routing is designed to locally minimize the hop distance from the destination. The hop distance is defined as the minimum number of hops to deliver a message to the destination in the communication network by using communication links only. With mobility, the hop distance may change up to two hops in one time slot. We can determine the probability pi,ˆi that two nodes with distance i have new distance ˆi after they move. This probability depends on the initial distribution of nodes and the random mobility model, and can be determined by analyzing the statistical distribution of nodes on rings (with radius rC ) centered at the destination.

are mapped to the communication network (see Fig. 1-(c)). The following points are taken into account in the mapping process: •

Distances on social and communication networks are typically different from each other. That is, two nodes that are physically close in communication networks may not be close in social network and vice versa. Then, a SRC in social network may be mapped as a LRC in the combined network while a LRC in social network may be mapped as a SRC in the combined network. This approach distinguishes the distances according to their distributions (in social or communication network) on a common domain.

•

We can either assume a random and uniform mapping from social network to communication network, or take into account the potential correlation of social and communication links (e.g., a SRC in social network may have high tendency to be geographically close and may become a SRC in the communication network). This way, we integrate both SRCs and LRCs in the social network into the communication network and construct the combined network.

•

When we map social links to the communication network, some of them coincide with communication links, whereas some of them become LRCs in the combined network. All LRCs are social links in the combined network and they are not supported by communication links. On the other hand, a SRC can be either a communication link only or a communication link plus a social link (if a social link is mapped to a SRC in the communication network). The numbers of SRCs and LRCs in the social network that become LRCs in the combined network follow the distributions QC,S and QC,L , respectively, and can be combined to determine the distribution QC of the total number of LRCs in the combined network (those LRCs are links pointing outside the transmission range as illustrated in Fig. 1-(c)). The combined network still has small-world and scale-free properties because of the distributions on LRCs.

•

A person may route a message over his/her social links, even if the source of this message is a stranger. This is because that a message is received either from this person’s friend (if a social link is used to reach this person) or from this person’s communication neighbor (if a communication link is used to reach this person) and thus it is easy to verify this neighbor’s credential. No matter which case, this person may forward the message.

C. Combination with Social Links In addition to the communication network, there is a social network (see Fig. 1-(b)), where nodes can communicate via social links. These social links are built on an additional network, e.g., an online social network. We consider the information transfer through the combined (communication and social) network, where the message can be carried by either communication or social links. Therefore, we need to combine the communication network with a social network that typically shows the following graph properties and we need to incorporate them in the mobile social network model: (i) Small-word phenomenon: Social actors are linked by short chains of acquaintances, e.g., in Milgram’s experiment the average number of hops for a source in Nebraska to reach a destination in Massachusetts via mail was found to be close to six (a.k.a. six degrees of separation). Several models have been proposed to capture the small-world phenomenon in social networks [13]. For instance, nodes are randomly deployed on a disk with unit radius in the Octopus model [15], [16]. Distance in a social network is a measurement on social closeness between two nodes. In general, two nodes with a smaller social distance are more likely to be friends (connected by a social link). The Octopus model [15], [16] was proposed as a simple model for social links. Under this model, if two nodes are within a social range rS , they are considered to be friends, i.e., connected by a SRC. Otherwise, they can be friends (connected by a LRC) with a small probability (see Fig. 1-(b)). The number of SRCs, nSRC , is a random variable with a small expected value due to limited rS while the number of LRCs, nLRC , follows a general distribution. With these LRCs, this approach generates small-world networks. (ii) Scale-free network: The degree distribution found in social networks typically follows a power law [14] such that there is a non-negligible number of nodes with significantly high degrees. In the above model, nLRC can follow any arbitrary distribution, including power law distribution (i.e., QS (nLRC ) ∼ nLRC −α with power law exponent α) to model scale-free networks. For the analysis of search problem in combined social and communication networks, we map each node i (a person) in a social network to a node i (this person’s communication device) in a communication network. As a result, social links

III.

AVERAGE S EARCH D ELAY N ETWORKS

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M OBILE S OCIAL

An important performance measure is the average delay when a source searches for a destination using intermediate hops. This average delay should be determined depending on the separation between the source-destination pair. Each node has only local information of the distance of its own and of its neighbors to the destination and the routing decisions are made individually using this local information only.

A. Greedy Routing 25

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The search problem has been considered in [6], [15], [16] with a greedy routing algorithm in static social networks. Each node on the path to a given destination node chooses its next-hop node with thmobile combined network. The same idea was applied in [17] to design routing algorithm for communication networks. In this paper, we consider a mobile combined network with different delay and success probability on communication and social links. Thus, when there is a tie between SRC and LRC, i.e., the best LRC decreases hop distance by one, a SRC is selected to break the tie (provided that SRCs are more reliable than LRCs). Also, when there is a tie between social and communication SRCs, a communication SRC is selected to break the tie (provided that communication links are more reliable than social links). Note that the focus here is not on developing another greedy routing algorithm but it is used as a suitable routing scheme operating with local information only when we derive the search delay and success properties in mobile social networks.

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Delay in different static/mobile communication/combined networks.

C. Average Delay Analysis Results B. Delay Formulation The goal is to determine the average delay Tk to travel from any node with hop distance k to a given destination. The average delay is analyzed by considering the first hop event. With some probability Xk−1 , the first hop is a SRC and the hop distance is reduced by one from k to k − 1. Otherwise, the first hop is a LRC and the hop distance is reduced from k to i with some probability Yi , where i = 1, ..., k − 2. By using the distribution of LRCs, QC , in the combined network, we can apply basic geometry and statistics to compute the probability that a SRC or LRC selected by greedy routing is located in a ring with given distance from the destination. This way, the probabilities Xk−1 and Yi are determined. The delay for hop distance k ≥ 2 is expressed as X Tk = Xk−1 (dC + pk−1,kˆ Tkˆ ) ˆ k

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pi,ˆi Tˆi ) ,

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where T1 = 1, dC and dS are the delay on a communication link and a social link, respectively. The first term on the righthand side of Eq. (1) is the unit delay for the next hop. The second term is the product of probability and expected delay from the next hop node to the destination for the case that the next hop is a communication link (SRC in the combined network). The third term is the product of probability and expected delay from the next hop node to the destination for the case that the next hop is a social link that becomes LRC in the combined network. The probabilities in the second and third terms reflect how the distance between two nodes changes after they move, as discussed in Section II-B. By considering all possible values of k, we obtain H − 1 linear equations with H − 1 variables for delays Tk , where H is the maximum hop distance. We can solve this linear system by basic algebra (e.g., Gaussian elimination) and consequently determine Tk for any given hop distance k of source-destination pairs.

We first compare four different networks: (i) a static communication network, (ii) a static combined network, (iii) a mobile communication network, and (iv) a mobile combined network. In addition, we provide results comparing them with static social network analyzed in [15], [16]. We set rC = 0.05 for communication networks, set nSRC = 20 (SRCs are chosen from the links between nodes that are within some range rS ), set dC = dS = 1, and let nLRC (LRCs are chosen from the links between nodes that are outside some range rS ) follow a power law distribution with exponent α = 2 for social networks. The probabilities of distance changes are computed according to the random mobility model described in Section II-B (e.g., p10,8 ≈ 0.0183, p10,9 ≈ 0.2394, p10,10 ≈ 0.4612, p10,11 ≈ 0.2592, and p10,12 ≈ 0.0219). The average delay results are shown in Fig. 2. We can see that (i) in the combined network, social links can significantly decrease average delay because a social link may significantly decrease hop distance (generating the small-world effect such that average delay saturates as hop distance increases), (ii) in the combined network, mobility increases average delay but this increase is limited by the close to symmetric nature of mobility, and (iii) compared to the case using social network only (studied in [15], [16]), adding communication links decreases average delay in the static network case but average delay may increase with mobility especially for a small hop distance (however, this effect is reversed as hop distance increases). We also study the impact of power law exponent α on the average delay performance in mobile combined networks. Results for α = 1.2, 1.5, 1.8, 2.0 are shown in Fig. 3. Since a larger α means fewer nodes with a large number of LRCs, we can see that average delay is increasing with a larger α (when we observe small-world phenomenon). These effects are also observed in Fig. 3 for the case with static social network only. IV.

S EARCH S UCCESS

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M OBILE S OCIAL N ETWORKS

In the last section, we assumed that all chains (search paths) are eventually completed such that the average delay was the main performance measure of interest. This assumption should

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Success probability in different networks.

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be relaxed in mobile social networks, where dynamic and unreliable nature of links (corresponding to social connections and wireless channels) introduces potential message drops. A. Failures on Social and Communication Links A chain of search may not be completed because of the failure of links on the path from the source to its destination. Both social and communication links may fail. A node may drop a message from an untrusted neighbor in a social network. For instance, only 18 out of 96 message chains were completed in a real-world mail experiment (i.e., the success ratio is less than 20%) [3] and only 384 out of 24,163 message chains were completed in an online e-mail experiment (i.e., the success ratio is less than 2%) [4]. On the other hand, communication links may fail due to noise, fading and interference effects in the wireless medium. We assume that social links are less reliable than communication links since a person is typically less reliable than a machine (a radio or a computer). This is because that a person may not send/receive message (e.g., an email) unintentionally or intentionally while a machine will not refuse or forget to send/receive a packet (except node failures). In addition, machine may retransmit a packet multiple times while a person usually does not retransmit an email. We aim to model failures in chain completion by assigning a forwarding probability to each link. We distinguish the forwarding probabilities PfS and PfL for SRCs and LRCs, respectively, in a social network and PfC for communication links. This way, we can model the cases when communication links start failing and social links are used more frequently to sustain the routing performance. B. Success Probability Formulation Conditioned on this first hop event, the end-to-end success probability for k ≥ 2 can be expressed as Pk−2 P P Sk = Xk−1 kˆ pk−1,kˆ Skˆ PfC + i=1 Yi ˆi pi,ˆi Sˆi (2) (pS PfS + pL PfL ) ,

where S1 = PfC , pS and pL are the probabilities of SRC and LRC, respectively. The first term on the right-hand side of Eq. (2) is the product of probability and expected success probability from the next hop node to the destination for the case that the next hop is a communication link (SRC in the combined network). The second term is the product of probability and expected success probability from the next hop node to the destination for the case that the next hop is a social link that becomes LRC in the combined network. We solve this linear system with H −1 variables and obtain Sk for all values of k. C. Success Analysis Results We again compare four different networks: (i) a static communication network, (ii) a static combined network, (iii) a mobile communication network, and (iv) a mobile combined network. The analysis does require a particular set of parameter values. To obtain numerical results, we set rC = 0.05 for communication networks, set nSRC = 20 and let nLRC follow a power law distribution with exponent α = 2 for social networks. For link success probabilities, we set PfC = 0.95, PfS = 0.7, and PfL = 0.3. The end-to-end success probability achieved on search paths is shown in Fig. 4. We can see that social links may not always improve success probability. The reason is as follows: on one hand, social links can decrease the number of hops to deliver a message, which can improve endto-end success probability, but on the other hand, social links are less reliable than communication links. Therefore, when hop distance from the source to the destination is small, social links may not help; but when hop distance is large, social links can help. We also see that mobility decreases success probability but the decrease is rather limited because the effect of mobility is close to symmetric in terms of increasing and decreasing node distances. In all cases of combined network, success probability does not drop to zero with the use of social links and saturates to a non-zero value as hop distance increases (pointing again at small-world effect of social links). Next, we evaluate the impact of social link reliability on mobile combined networks. Considering four sets of parameters with fixed PfC = 0.95: (i) PfS = 0.9 and PfL = 0.5, (ii) PfS = 0.8 and PfL = 0.4, (iii) PfS = 0.7 and PfL = 0.4, and (iv) PfS = 0.6 and PfL = 0.3, the end-to-end success probability results are shown in Fig. 5. We can see that if social

of average delay and success properties provide insights into the design, implementation and maintenance of mobile social networks. By combining social and communication links in a common graph model, we derive analytical expressions for the average delay and end-to-end success probability under realistic mobility and link reliability properties. The results highlight the use of social connections to improve routing performance in mobile social networks and can be further used to reverse-engineer network properties (e.g., degree distributions) from network measurements (e.g., average delay or success probability) in combined structures of social and communication networks.

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ACKNOWLEDGMENTS Fig. 5.

Success probability under different link success probabilities.

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R EFERENCES

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We would like to acknowledge the Air Force Office of Scientific Research for their support. We would like to thank Prof. Mung Chiang for his valuable information on the Octopus model, comments and suggestions.

Success probability under different communication parameters.

links are more reliable, then they can help improve success probability in end-to-end search. To further understand the potential benefits of social links, we consider the case that communication links start failing. Considering fixed social parameters PfS = 0.7 and PfL = 0.3 for social network and assuming that communication link reliability decreases from PfC = 0.8 to PfC = 0.5 for communication network, the end-to-end success probability results (with and without social links) are shown in Fig. 6. We can see that without social links, success probability quickly drops to zero as hop distance increases. By eliminating this effect, social links can help sustain reliable operation even when the use of only communication links cannot provide a reliable transmission. V.

C ONCLUSION

Online social networks revolutionized the way people are interacting and mobile wireless networks provide the necessary medium to connect devices that run the social network applications and enable these social interactions. Therefore, it is necessary to understand the joint operation of social and communication networks and capture their interplay in a realistic network model that can be used to analyze the performance of mobile social networks. One critical network application is navigating a network and searching for users with local information. The underlying performance measures

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