Impact of Social Relation and Group Size in Multicast Ad Hoc Networks Yi Qin, Riheng Jia, Jinbei...
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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
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
Slide 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
Slide 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.
Slide 4
Previous Works & Motivation Social networks The history of
social networks[1] 4 In the late 1890s, Ferdinand Tnnies and mile
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] Tnnies, 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.
Slide 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.
Slide 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.
Slide 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)
Slide 8
Previous Works & Motivation The multicast uses 8
Slide 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.
Slide 10
Previous Works & Motivation Main challenge and contribution
10
Slide 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
Slide 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
Slide 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.
Slide 14
System model The two-layer model 14 Social layer For user i,
the probability that it has q i friends satisfies where >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.
Slide 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 >0 is a constant and H n is defined as This
model is verified by the study of large scale social networks
[10].
Slide 16
System model The network model 16 Static networks Total number
of nodes : n Multicast Protocol model Transmission range:
Slide 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.
Slide 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
Slide 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
Slide 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 1.
Slide 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
Slide 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.
Slide 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 1.
Slide 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
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.
Slide 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.
Slide 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
Slide 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
Slide 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.
Slide 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
Slide 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.
Slide 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.
Slide 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 1.
Slide 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
Slide 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 < 1. The capacity upper-bound can be obtained
based on the total length of EMST and the theory in [13].
Slide 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 =
1, and when > 1, and. We do not show the detailed expressions
for the sake of conciseness.
Slide 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
Slide 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
Slide 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 1, there is a gap n between them, where
>0 is an arbitrary small constant.
Slide 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 1, where c is a
constant. The detailed proof can be found in the paper.
Slide 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
Slide 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.
Slide 46
CaseCapacity ratio 01 46 Discussion The capacity ratio can be
expressed as follows
Slide 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.
Slide 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
Slide 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?
