1. Impact of Social Relation and Group Size in Multicast Ad Hoc Networks Yi Qin, Riheng Jia, Jinbei Zhang, Weijie Wu, Xinbing Wang Shanghai Jiao Tong University

2. 2 Outline Introduction   Previous works & Motivation System model and main idea The capacity upper-bound of multicast social networks The capacity achieving scheme Discussion Conclusion and future directions

3. Previous Works & Motivation Social networks   Social networks 3 Each user has multiple friends. The friends of each user are selected according to some factors such as friendship, common interest or alliance. The user transmits information to some or all of its friends.

4. Previous Works & Motivation Social networks   The history of social networks[1] 4 In the late 1890s, Ferdinand Tönnies and Émile Durkheim propose the idea of social networks in their theories and research of social groups [2]. Afterwards, major works about social networks can be found by several groups in psychology, anthropology, and mathematics working independently [3] [4]. Nowadays, social networks are widely studied based on the online social networks with millions of persons [5] [6]. [1] Wikipedia: social network. http://en.wikipedia.org/wiki/Social_network.http://en.wikipedia.org/wiki/Social_network [2] Tönnies, Ferdinand (1887). Gemeinschaft und Gesellschaft, Leipzig: Fues's Verlag. (Translated, 1957 by Charles Price Loomis as Community and Society, East Lansing: Michigan State University Press.) [3] Scott, John P., ”Social Network Analysis: A Handbook (2nd edition)”, Thousand Oaks, CA: Sage Publications, 2000. [4] Peter J. & Scott, John, “The Sage Handbook of Social Network Analysis”, 2011. [5] Y. Ahn, S. Han, H. Kwak, S. Moon, H. Jeong, “Analysis of topological characteristics of huge online social networking services,” in 16th international conference on World Wide Web, New York, USA, pp. 835-844, 2007. [6] D. Liben-Nowell, J. Novak, R. Kumar, P. Raghavan, A. Tomkins, R. Graham, “Geographic routing in social networks,” in Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 33, pp. 11623-11628, 2005.

5. Previous Works & Motivation The theory of social networks: social features   Social feature I : social group 5 The social group size describes the number of friends for each user. In [5] [7] [8], the authors analyzed the distribution of social group size based on the data of Cyworld, MySpace and orkutwith, each with more than 10 million users. The results showed that the probability satisfied the power-law distribution, which generally matched the fact. [7] P. Fraigniaud, G. Giakkoupis, “The effect of power-law degrees on the navigability of small worlds,” In Proc. 28th ACM Symposium on Principles of Distributed Computing, pp 240-249, 2009. [8] J. Kleinberg, “The small-world phenomenon: an algorithm perspective,” in 32nd Annual ACM Symposium on Theory of Computing, Portland, Oregon, USA, May 2000.

6. Previous Works & Motivation The theory of social networks: social features   Social feature II : social relation 6 The social relation reflects how users select friends in the network. From the experiments in [9] [10] about how people selected friends, some feasible social relation models are proposed such as distance-based model [9] and rank-based model [10]. [9] B. Azimdoost, H. Sadjadpour, J. Garcia-Luna-Aceves, “The Impact of Social Groups on The Capacity of Wireless Networks,” in Network Science Workshop, pp. 30-37, 2011. [10] R. Kumar, D. Liben-Nowell, et.al, “Theoretical Analysis of Geographic Routing in Social Networks,” in MIT-CSAIL- TR-2005-040, Jun. 2005.

7. Previous Works & Motivation The importance of multicast study   What is multicast? One source to m destinations 7 Xiangyang Li [11] [11] X. Li, “Multicast Capacity of Large Scale Wireless Ad Hoc Networks”, IEEE/ACM Trans. Networking, Vol.17, No. 3, pp. 950-961, Jan. 2008.(citation:132)

8. Previous Works & Motivation The multicast uses 8

9. Previous Works & Motivation Our view on the multicast social networks   According to the mentioned theory of social networks and multicast, we can conclude the relation between social features and multicast as follow: The social group size determines the multicast scale. The social relation determines the distribution of multicast destinations. 9 We investigate the impact of social features on network capacity in this paper.   The social features (social group size and relation) determine the network features (multicast scale and destination distribution) and further impact the network performance such as capacity.

10. Previous Works & Motivation Main challenge and contribution 10

11. Previous Works & Motivation Main challenge and contribution   The distribution of destination is analyzed for multicast scenario in social networks. Different from the unicast scenario in previous works, the investigated distribution is related with the multicast scale.   The previous study of Euclidean Minimum Spanning Tree (EMST) is under the assumption that the node distribution is unrelated with the network scale. In this paper, a more general theorem is proposed for the case that the node distribution is related with the network scale, which further perfects the theory of EMST.   The capacity upper-bound is calculated based on the EMST. An EMST-based transmission scheme is proposed, which is proved to achieve the capacity upper-bound of the social network.   Finally, the impact of social features on network capacity is analyzed by comparing the social network capacity with traditional ad-hoc network. 11

12. 12 Outline Introduction System model and main idea The capacity upper-bound of multicast social networks The capacity achieving scheme Discussion Conclusion and future directions

13. System model The two-layer model 13   In this paper, we adopt the two-layer network model, which includes 1) 1)Social layer: This layer captures the social relation among individuals, which is not related with the network topology. 2) 2)Networking layer: This layer reflects the network topology based on the node positions.

14. System model The two-layer model 14   Social layer For user i, the probability that it has q i friends satisfies where β>0 is a constant and G n is This model is verified by the study of large scale social networks [5] [7] [8]. Moreover, we assume that each user transmits information to all of its friends in this paper.

15. System model The two-layer model 15   Networking layer For user i, the probability that user j is one of its q i friends satisfies Denoting P as the set consisting all the nodes and d ( i, j ) as the distance between user i and j, the rank of j with respect to i is defined as where α >0 is a constant and H n is defined as This model is verified by the study of large scale social networks [10].

16. System model The network model 16   Static networks   Total number of nodes : n   Multicast   Protocol model   Transmission range:

17. System model Capacity definition 17 Per-node Throughput: For a given scheme, we define the per-node throughput as the maximum achievable transmission rate. In t time slots, we assume that there are M ( i, t ) packets transmitted from node i to its destination(s). Firstly, the long term per-node throughput is defined as Afterwards, the per-node throughput of this model for a given scheme is defined by the maximum T ( n ) satisfying Per-node Capacity: For a given network, the per-node capacity of it is defined as where is a scheme for the network, is the set of all possible schemes, and is the per-node throughput of scheme.

18. Main idea of this paper 18 Two-layer social network model The distribution of destinations Build Euclidean Minimum Spanning Tree (EMST) among source and its destinations Calculate the capacity upper-bound based on the total length of EMST Propose a capacity upper-bound achieving scheme Conclude the impact of social features on network capacity

19. 19 Outline Introduction System model and main idea The capacity upper-bound of multicast social networks   The distribution of destinations   Build the EMST   Calculate the capacity upper bound The capacity achieving scheme Discussion Conclusion and future directions

20. 20 The distribution of destinations The rank distribution   Based on the networking layer model   Firstly, we analyze the rank distribution of the q i destinations Lemma 5: If i is a source node with q i destinations, and j is one of them. The probability that Rank i ( j ) = X satisfies when 0≤ α <1, and when α =1, and when α >1.

21. 21 The distribution of destinations The rank distribution   Proof of lemma 5   We sort the nodes based on the ranks respect to i as v 1,···, v n-1, where Rank i ( v X ) = X. The set of destinations of i is denoted as Q i, where | Q i |= q i. Therefore, the probability that Rank i ( j ) = X is where

22. 22 The distribution of destinations The rank distribution   Proof of lemma 5 Defining then the rank distribution can be expressed as We prove that when, where, and is only related with q i and n. Therefore, according to the fact that, the lemma 5 can be proved.

23. 23 The distribution of destinations The position distribution   Based on the rank distribution, we analyze the position distribution of destination Theorem 1: If i is a source node with q i destinations, the location of each destination follows the distribution in polar coordinates when 0≤ α <1, and when α =1, and when α >1.

24. 24 The distribution of destinations The position distribution   Proof of Theorem 1   We first analyze the CDF of destinations Based on the CDF, the PDF can be expressed as (0≤ α <1 as an example) where is the regularized incomplete beta function, is a constant. Further analysis of the expression above proves Theorem 1.

25. 25 The distribution of destinations The position distribution   Verification of Theorem 1   We use Surface Fitting to verify the PDF of destinations. The equation of surface fitting method and the surface fitting parameters can be expressed as follows The surface fitting parameters The differences between the parameters of surface fitting function and theoretical function are no more than 7.5% except for poly-logarithmic factors.

26. 26 The distribution of destinations The position distribution   Verification of Theorem 1   Figures of Surface Fitting α=0.5 α=1 α=1.2

27. 27 Outline Introduction System model and main idea The capacity upper-bound of multicast social networks   The distribution of destinations   Build the EMST   Calculate the capacity upper bound The capacity achieving scheme Discussion Conclusion and future directions

28. 28 Build the EMST The total length of EMST   The theorem of EMST with node distribution unrelated with n [12]   Lemma 1: Let f ( x ) denote the PDF of the related nodes in the network, where x is the position vector. Then, for large number of nodes n and the network dimension d > 1, if f ( x ) is independent from n, the total length for the EMST satisfies with probability 1, where c ( d ) is constant. However, the distribution in this paper is related with n. [12] M. Steele, “Growth rates of euclidean minimal spanning trees with power weighted edges,” in The Annals of Probability, vol. 16, no. 4, pp. 1767-1787, 1988.

29. 29 Build the EMST The total length of EMST   The theorem of EMST with node distribution related with n   Lemma 2: Let f n ( x ) denote the PDF of the related nodes in the network, where x is the position vector. Then, for large number of nodes n and the network dimension d >1, if the following two conditions are satisfied Condition 1: There exists a function satisfying when n goes to infinity, where is the range of total network, is the partition of separating into many non-overlapping parts (i.e., and ), is an indicative function and. Moreover, each part is a d -dimensional hypercube, and the number of its adjacent parts is limited by a constant. Condition 2: with probability 1. the total length for the EMST satisfies with probability 1, where c ( d ) is constant.

30. 30 Build the EMST The total length of EMST   Proof of Lemma 2   We prove the Lemma 2 based on the following lemma given in [12]. Lemma 7: (M. Steele, [12]) In the network with size [0,1] d, we denote the EMSTs of nodes with distribution and as EMST f and EMST g, respectively, where is a continuous distribution and is a blocked distribution. The total length of EMST f can be expressed as when the following two requirements are satisfied, where c ( d ) is a constant. Requirement 1: The total lengths of EMST f and EMST g satisfy where B is a constant, and are the sets of the node positions of EMST f and EMST g, respectively. Requirement 2: The total length of EMST g can be expressed as

31. The total length upper-bound of EMST of the nodes in each part which belonging to 31 Build the EMST The total length of EMST   Proof of Lemma 2   It has been proved in [12] that the Requirement 1 in Lemma 7 is satisfied when the Condition 1 in Lemma 2 holds. Therefore, it only remains to verify Requirement 2.   The upper-bound of It has been proved in [12] that if there are n nodes uniformly distributed in, the total length of the EMST of nodes in is with probability 1 when n goes to infinity, where is a constant. Defining Then the upper-bound can be expressed as The cost needed to unite all of the parts The total length of EMST of the nodes in each part which belonging to

32. 32 Build the EMST The total length of EMST   Proof of Lemma 2   The upper-bound of It is proved in the paper that M is upper-bounded by Thus, the upper-bound of can be calculated as where is a constant.

33. 33 Build the EMST The total length of EMST   Proof of Lemma 2   The lower-bound of Let D i denote the set of edges e in EMST g such that both two endpoints of e are in. Let V i denote the set of nodes in which are jointed by an edge in EMST g with one another endpoint out of. Thus, the edge in D i and the EMST of V i connect all the nodes in. According to [12], for any, the following relation holds when. The total length of EMST of the nodes connecting with the nodes outside The length of EMST in The total length of edges with two endpoints in

34. 34 Build the EMST The total length of EMST   Proof of Lemma 2   The lower-bound of In this paper, we propose the way to build a graph G, which connects all the nodes in the network, satisfying and where. Considering the fact that we can further calculate the lower-bound as where are constants.

35. 35 Build the EMST The total length of EMST   Proof of Lemma 2   According to the upper-bound and lower-bound, it can be obtained that Consequently, based on Lemma 7, the Lemma 2 is proved.

36. 36 Build the EMST The total length of EMST   Based on Lemma 2, the total length of EMST for the social network can be calculated as Theorem 2: If i is a source node with q i destinations, the total length of EMST for social network satisfies when 0≤ α <1, and when α =1, and when α >1.

37. 37 Outline Introduction System model and main idea The capacity upper-bound of multicast social networks   The distribution of destinations   Build the EMST   Calculate the capacity upper bound The capacity achieving scheme Discussion Conclusion and future direction s

38. 38 Calculate the capacity upper bound The capacity upper-bound [13] Z. Wang, H. Sadjadpour, J. Garcia-Luna-Aceves, “A Unifying Perspective on the Capacity of Wireless ad hoc Networks,” in Proc. of IEEE INFOCOM 2008, Phoenix, AZ, USA, Apr. 2008. Theorem 2 (part 1):The expectation of the capacity of this social network model is bounded by when 0 ≤ α < 1.   The capacity upper-bound can be obtained based on the total length of EMST and the theory in [13].

39. 39 Calculate the capacity upper bound The capacity upper-bound   The capacity upper-bound can be obtained based on the total length of EMST and the theory in [13]. Theorem 2 (part 2):The expectation of the capacity of this social network model is bounded by when α = 1, and when α > 1, and. We do not show the detailed expressions for the sake of conciseness.

40. 40 Outline Introduction System model and main idea The capacity upper-bound of multicast social networks The capacity achieving scheme Discussion Conclusion and future directions

41. 41 The capacity achieving scheme The EMST-based scheme   Initially, the network is separated into cells, and M 2 -TDMA is employed.   For each arbitrary source i, an EMST is built among the q i destinations and node i. This tree is the basic EMST.   Considering the limited transmission range r ( n ), each direct link in basic EMST can be constructed by sending packets hop by hop along the cells where link lines across. When a node is allowed to be active in this time slot, it will select the oldest packet in its buffer and transmit it to the next node according to the route of EMST.   The EMST based scheme

42. 42 The capacity achieving scheme The performance of EMST-based scheme   By considering the bottleneck user of the network, the throughput can be derived. Theorem 4: The throughput of EMST-based scheme which is proposed above is of the same order as the capacity given in Theorem 3 when 0≤ α 1, there is a gap n η between them, where η>0 is an arbitrary small constant.

43. 43 The capacity achieving scheme The performance of EMST-based scheme   Proof of Theorem 4   The network is divided into multiple cells with side-length r ( n ). We define the node set S i consisting the nodes which are one of the relays or destinations for one multicast session with source i. Considering an arbitrary cell s, the number of multicast sessions that invoke s can be represented as where the 1 {s is invoked by S i } are i.i.d. Bernoullian random variables. Thus, according to the definition of throughput, this theorem is equivalent to when 0≤ α <2 or 0 ≤ β ≤ 1, and for the case α ≥2 and β>1, where c is a constant. The detailed proof can be found in the paper.

44. 44 Outline Introduction System model and main idea The capacity upper-bound of multicast social networks The capacity achieving scheme Discussion Conclusion and future directions

45. 45 Discussion   In order to analyze the impact of social features, we compare the capacity of social networks with traditional ad-hoc networks in [13].   Since the number of destinations in [13] is certain before transmission, we compare the capacity of social network with theirs under the assumption that the number of destinations, i.e., q, is certain. The capacity ratio is defined as   It should be noticed that the destinations are uniformly selected in traditional ad-hoc networks.

46. CaseCapacity ratio 0≤α<1 α=1 α>1 46 Discussion   The capacity ratio can be expressed as follows

47. 47 Discussion   An example of capacity ratio The concentration degree of destinations is low The concentration degree of destinations grows The destination density is limited by user density of the network.

48. 48 Outline Introduction System model and main idea The capacity upper-bound of multicast social networks The capacity achieving scheme Discussion Conclusion and future directions

49. 49 Conclusion and future direction   In this paper, we introduce the two-layer network model, which includes social layer and networking layer. In the social layer, the social group size of each source node is modeled as power-law distribution model. Moreover, the rank-based model is utilized to describe the relation between source and destinations in the networking layer.   In this model, the PDF of destinations is derived and verified by numerical simulations. Based on the PDF, we analyze the capacity bound of the networks and propose an EMST-based scheme to achieve the capacity in most cases. Finally, the social network capacity is compared with the traditional ad hoc networks to demonstrate the difference between them.   There are two interesting future directions: What is the impact of social features in mobile network? What will happen if more social features are introduced in the model?

50. 50 Comments ? Questions ? Thank you!