1

Impact of Social Relation and Group Size inMulticast Ad Hoc NetworksYi Qin, Riheng Jia, Jinbei Zhang, Weijie Wu, Xinbing Wang

Abstract—This paper investigates the multicast capacity ofstatic wireless social networks. We adopt the two-layer networkmodel, which includes the social layer and the networking layer.In the social layer, the social group size of each source nodeis modeled as power-law distribution. Moreover, the rank-basedmodel is utilized to describe the relation between source anddestinations in the networking layer. Based on the two-layernetwork model, the probability density function (PDF) of thedestination positions is analyzed and verified by numerical sim-ulation, which is different from the traditional ad hoc networks.According to the PDF, the bound of the network capacity isderived, and we propose a Euclidean minimum spanning treebased transmission scheme, which is proved to achieve the orderof capacity bound for most cases. Finally, the capacity of socialnetworks is compared with the traditional multicast ad hocnetworks, which indicates that the capacity scaling performsbetter in social networks than traditional ones. To our bestknowledge, this is the first work of analyzing the impact onthe capacity of social relation and group size in multicast ad hocnetworks for the rank-based model.

I. INTRODUCTION

Network scaling law was firstly studied by Gupta andKumar [1], followed by many works about the networkcapacity and throughput [2]- [4]. However, their models fellshort in well characterizing real social networks, and theyhad to consider simplified models instead. For example, inmost of these works, the destinations associated with eachuser were drawn according to a uniform distribution, which isunrealistic. To give another example, it was also an unrealisticassumption that the number of friends was known aprioriin traditional multicast ad hoc networks [2]- [4]. Therefore,many researchers turned to study social characteristics suchas the way people selected friends (destinations) [5] [6] andthe number of these friends [7] [8]. It was only recently thatstudying social phenomena had become a hot topic in adhoc networks [9], peer to peer networks [10] and some othernetworks [11]- [14].

To study the social phenomenon, two characteristics ofsocial networks are considered in our paper, which are thesocial group size and the social relation. Firstly, we introducethe group size which describes the number of friends for eachnode. In [7] [8] [15], the authors analyzed the probability thatan arbitrary node had q friends 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-lawdistribution, which generally matched the fact.

The authors are with the Department of Electronic Engineering ShanghaiJiao Tong University, China

Email: qinyi 33, jiariheng, abelchina, weijiewu, xwang8@sjtu.edu.cn

The second characteristic of social networks is the socialrelation which has been studied for more than thirty years[9], [16]- [19]. In 1979, the authors in [16] focused on about500 persons in a network. Further research about 130 millionpersons of Myspace was demonstrated in [17]. In their works,the social network was modeled as a graph to describe thegeographical and social relations among users. Moreover, thesocial relation in these papers reflected how users select friendsin the network, and it was usually determined by the factorssuch as friendship, common interest or alliance, which wasdifferent from the traditional ad hoc networks. From the abovestudies of the experiments about how people selected friends,some feasible social relation models were proposed such asdistance-based model and rank-based model [19]. The scalinglaws of these two models were analyzed in [9] and [18],respectively. In [9], the probability that node j was a friend ofnode i was proportional to d(i, j)−α, where d(i, j) representedthe distance between i and j. Instead, in [18], the authorsillustrated that the probability was proportional to Rank−αi (j),where Ranki(j) was the rank of j with respect to i.

The two social characteristics above were both observedthrough statistics from online social networks. However, theycan also be introduced into wireless networks because thatthey are independent from the wireless nature of the networks.Moreover, it is worthwhile to study wireless social networksdue to the two facts: 1) The study of wireless social networks ispromising for the reason that the social wireless networks, suchas MSN wireless client, facebook wireless client, have gained alarge popularity nowadays. Furthermore, in cellular networks,as the number of wireless terminals increases, one base-station needs to service more and more terminals. However,the number of simultaneously serviced terminals is restricteddue to the limited up-link resource. This will reduce theper-terminal throughput when there are too many terminals.Therefore, wireless multi-hop transmission can be adopted toimprove the per-terminal throughput of social networks. 2) Themulti-hop user relaying technology has become an importanttopic and been well studied. It can be demonstrated from [20],which investigated coordination among users through wirelesschannel (such as multi-hop), that the user relaying technologyhelped to improve the capacity of wireless system.

Additionally, the nodes in the network need to transmit theirpackets to all of their friends by multicasting in this paper,which is also a common requirement. For example, a user offacebook may want to send a message to all of his friends.

The capacity of traditional multicast ad hoc networks wasintensively studied in [2]- [4]. However, the traditional resultscannot be directly applied to social networks, and therefore

2

we present this paper to improve and generalize the theoryof ad hoc network capacity. In particular, the main differ-ence between the social networks and the traditional ad hocnetworks is the Probability Distribution Function (PDF) ofdestination positions. Therefore, a challenging question ariseswhen considering the two social characteristics:• How do the two characteristics of static social networks

jointly impact the capacity of network, and what is thecapacity achieving scheme?

To answer this question, we study the two-layer static socialnetwork model, which includes social layer and networkinglayer. In the social layer, the social group size of each sourcenode was modeled as power-law distribution model. Moreover,the rank-based model was utilized to describe the relationbetween source and destinations in the networking layer. Fora node with q friends, the PDF of its friends’ positionsare derived in our paper. The simulations of the PDF arealso illustrated, which verify the theoretical results. We thenanalyze the bound of network capacity and propose a capacityachieving scheme based on the Euclidean Minimum SpanningTree.

The main contributions of this paper are summarized asfollows:• The relation between rank and geographical position of an

arbitrary node is demonstrated for rank-based static socialnetwork model. If there are n nodes in the network whichis of size 1× 1, the result shows that x = Θ

ÈRankn

1

with probability 1 when n goes to infinity, where x isthe distance between source and destination. Moreover,we also derive the PDF of the destination positions for agiven source, which is verified by the simulations.

• We analyze the capacity bound of the social networkmodel based on the PDF of destination positions andEuclidean Minimum Spanning Tree. Moreover, a corre-sponding transmission scheme is proposed to achieve thecapacity bound in most cases.

• Finally, we compare the capacity of social networkswith the traditional ad hoc networks, and the resultsindicate that the capacity scaling performs better in socialnetworks than traditional ad hoc networks.

The rest of this paper is organized as follows. In SectionII, we introduce the network model and definitions. The PDFof destination positions and the capacity bound of the socialnetwork model are derived in Section III, and the theoreticalresults of the PDF is verified by numerical simulations in Sec-tion IV. In Section V, a capacity achieving scheme is proposedto demonstrate that the capacity is achievable in most cases. InSection VI, we compare the capacity of social networks withtraditional ad hoc networks. Finally, we conclude in SectionVII.

1We use standard asymptotic notations in our paper. Consider t-wo nonnegative function f(·) and g(·): (1) f(n) = o(g(n))means limn→∞ f(n)/g(n) = 0. (2) f(n) = O(g(n)) mean-s limn→∞ f(n)/g(n) < ∞. (3) f(n) = ω(g(n)) mean-s limn→∞ f(n)/g(n) = ∞. (4) f(n) = Ω(g(n)) meanslimn→∞ f(n)/g(n) > 0. (5) f(n) = Θ(g(n)) means f(n) = O(g(n))and g(n) = O(f(n)). (6) f(n) = Θ(g(n)) means that there exists two con-stants a and b satisfy f(n) = O(g(n) loga n) and f(n) = Ω(g(n) logb n)

II. NETWORK MODEL AND DEFINITIONS

In this section, the two-layer model of static social networksis proposed based on group size and social relation, whichare the main differences between social network model andtraditional ad hoc network model. Moreover, we also list somenetwork performance metrics and notations for our followinganalysis.

A. The Social Network Model

In this paper, we adopt the two-layer network model, whichincludes

1) Social layer: This layer captures the social relationamong individuals, which is not related with the networktopology.

2) Networking layer: This layer reflects the network topol-ogy based on the node positions.

In the social layer, each node i selects some friends as itsdestinations, which form a social group. Moreover, the size ofthe social group, denoted as qi, is proved to satisfy the power-law degree distribution in [9]. In particular, the experimentsof three online social networking services, each with morethan 10 million users, indicate that Pqi = q follows thedistribution in (1)

Pqi = q =1

Gnqβ, (1)

where β ≥ 0 is a constant and Gn =Pnq=1

1qβ

.In the networking layer, we consider the network which is

a unit square, and n static nodes (users) are uniformly andrandomly distributed in it. The well-known protocol model asin [1] is employed as the interference model in our network.When node i wants to transmit a packet to node j, thetransmission is considered to be successful if the followinginequality is satisfied

‖Xi −Xj‖ ≤ r(n), (2)

where Xi represents node i’s location, and r(n) is the max-imum transmission range of each node, which is a functionof n. ‖ · ‖ here is the Euclidean distance. Moreover, any othernode k who wants to transmit packet at the same time mustsatisfy the inequality as

‖Xk −Xj‖ ≥ (1 + ∆)r(n), (3)

where ∆ > 0 is a constant factor depending on the acceptableSignal to Interference Noise Ratio (SINR) of the network.Furthermore, the bandwidth of the network is finite andconstant. In this model, the transmission range r(n) is assumed

as r(n) =

√logn√n

[1], which guarantees the connectivity ofthe network. In addition, each node transmits packet to itsdestinations through multi-hop.

It is important to study the node mapping from the sociallayer to the networking layer. We adopt the rank-based modelin [18] and [19]. In this model, for a given node i, when itselects one friend from all the nodes, we denote that node j isselected with probability Pi→j . Based on social experiments[19], the friends of a user are more likely to be close to him.

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Denoting P as the set consisting all the nodes and d(i, j) =‖Xi−Xj‖ as the distance between i and j, the rank of j withrespect to i is defined as

Ranki(j) = |k ∈ P : d(i, k) < d(i, j)|, (4)

where | · | represented the number of elements in the set.According to [18] and [19], Pi→j satisfies

Pi→j ∝1

Rankαi (j), (5)

where α ≥ 0 is a constant. When α = 0, the network is thetraditional ad hoc network. (5) shows the fact that a node ismore likely to select a friend that nearby. Furthermore, thismodel focuses on relative location instead of geographic lo-cation. We define Hn =

Pni=1

1iα . Therefore, by normalizing,

(5) can be represented as

Pi→j =1

HnRankαi (j). (6)

In [19], the authors verify the (6) by online data. It should benoticed that we consider the case that one node selects multipledestinations instead of one. Therefore, the distribution of thedestinations will be analyzed based on both social group sizeand (6).

B. Network Performance Metrics and Notations

We give the definitions of throughput and capacity in thissubsection. Moreover, some notations are also demonstrated.

Definition of throughput: For a given scheme, we define thethroughput as the maximum achievable transmission rate. In ttime slots, any i’s destination (denoted as k) receives Mi(k, t)packets and the number of i’s friends is qi. The long termper-node throughput of this multicast session is defined by

λi(n) as λi(n) = lim inft→∞

1tqi

qiPk=1

Mi(k, t). Then the average

throughput of this scheme is defined by T (n)

T (n) = lim infn→∞

1

n

nXi=1

λi(n). (7)

Definition of capacity: In this social network, C(n) is saidto be the asymptotic per-node social multicast capacity withorder Θ(T0(n)), where T0(n) satisfies (8).

T (n) : limn→∞

T (n)

T0(n)=∞ = ∅,

T (n) : limn→∞

T (n)

T0(n)> 0 6= ∅,

(8)

for any feasible T (n).Furthermore, we list some important parameters in Table I

that will be used in later analysis, proofs and discussions.

III. CAPACITY ANALYSIS FOR SOCIAL NETWORKS

In this section, firstly, the relation between rank and ge-ographical position of an arbitrary node is analyzed. Basedon such relation, we derive the PDF of the destinations, andfinally obtain the bound of capacity.

The links between the nodes belonging to one multicastsession are considered, which can organize a spanning tree.

TABLE I: Important Definitions of Symbols and Notations

Symbol Definitionn Number of usersr(n) The maximum transmission rangeqi Number of destinations (friends) of user iα Parameter of rank-based model

βParameter of power-law distribution ofgroup size

Ranki(j) The rank of j with respect to i

fn(x, θ)The probability density function of destina-tion positions

‖EMST‖ The total length of the EMSTC(n) The capacity of the network

Nq(n)The minimum total number of hops for amulticast tree with q + 1 nodes

Rq

The capacity ratio of social networks andtraditional ad hoc networks when each nodehas q friends

Thus, in order to minimize the length of transmission path, theEuclidean Minimum Spanning Tree (EMST) is investigated.The total length for this EMST is proved in [21] to satisfy thefollowing lemma.

Lemma 1 Let f(x) denote the PDF of the related nodes inthe network, where x is the position vector. Then, for largenumber of nodes n and the network dimension d > 1, if f(x)is independent from n, the total length for the EMST satisfies

limn→∞

n−d−1d ‖EMST‖ = c(d)

ZRdf(x)

d−1d dx, (9)

with probability 1, where c(d) is constant.

The proof of this lemma is demonstrated in [21], and d = 2in our paper. However, when f(x) is related with n, Lemma1 does not hold. As a result, we denote the PDF as fn(x) inthis paper and give the following lemma which indicates thebound of the total length of the EMST for infinite n.

Lemma 2 If fn(x) in Lemma 1 is related with infinite n andthe following two conditions are satisfied,

• Condition 1: There exists a function gn(x) =Pξi∈Ψ γi1ξi satisfyingZ

Ψ|gn(x)− fn(x)|dx→ 0, (10)

when n goes to infinity, where Ψ is the range of totalnetwork, ξi(i = 1, 2, · · · ) is the partition of Ψ separatingΨ into many non-overlapping parts (i.e., ξi∩ξj = ∅,∀i 6=j and

Sξi = Ψ), 1ξi is an indicative function andR

ξign(x)dx =

Rξifn(x)dx. Moreover, each part ξi is a

d-dimensional hypercube, and the number of its adjacentparts is limited by a constant ζ.

• Condition 2: limn→∞

nRξign(x)dx = Ω(1) with probabil-

ity 1.

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One than has that with probability 1

limn→∞

n−d−1d ‖EMST‖R

Rd fn(x)d−1d dx

= c(d), (11)

where c(d) is constant.

Proof: The proof can be found in the Appendix.Moreover, according to the sum of p-series, the Gn in (1)

and Hn in (6) can be represented as

Hn =

nXi=1

1

iα=

8<:Θn1−α 0 ≤ α < 1,

Θ (logn) α = 1,

Θ(1) α > 1,

(12)

and

Gn =

nXq=1

1

qβ=

8<:Θn1−β 0 ≤ β < 1,

Θ (logn) β = 1,

Θ(1) β > 1.

(13)

To calculate the PDF of user i’s qi friends, the relationbetween rank and geographical position is important. We willshow this relation in Lemma 4 which is supported by Lemma3. It should be noticed that the impact of boundary effect onscaling laws can be ignored in the proofs of this paper, whichcan also be found in other works of scaling laws [2].

Lemma 3 We denote two constants a and b where 0 < a < 1and b > 1. For any X ∈ 1, 2, · · · , n, there must be

n!

X!(n−X)!

aX

n

X 1− aX

n

n−X< c1(ae1−a)X , (14)

where e is the base of the natural logarithm and c1 is constant,when n is large enough. Moreover, if bX ≤ n, there must be

n!

X!(n−X)!

bX

n

X 1− bX

n

n−X< c2(be1−b)X , (15)

where c2 is constant and n is large enough.

Proof: Firstly, the left-hand side of (14) satisfies

g(X) =n!

X!(n−X)!

aX

n

X 1− aX

n

n−X(a)

<c3ne

n√n

Xe

X √Xn−Xe

n−X √n−X

aXXX

nX(n− aX)n−X

nn−X

=c3aXn− aXn−X

n−XÉ 1

X+

1

n−X

<2c3aXn− aXn−X

n−X,

(16)

where c3 > 1 is constant, and (a) holds due to the fact that

limn→∞

ne

n √2πn

n!= 1. (17)

Moreover, for any c > 0, there must be 1 < (1 + c)1c < e.

Hence, g(X) satisfies

g(X) < 2c3aX

1 +

(1− a)X

n−X

n−X(1−a)X

(1−a)X

< 2c3(ae1−a)X .

(18)

Therefore, if we define c1 = 2c3, equation (14) is proved.Furthermore, equation (15) can be proved in the similar way.

Lemma 4 Node i is the source node. If the rank of node j to iis Ranki(j) = X , the distance between node i and j satisfiesx = Θ

ÈXn

with probability 1, when n→∞.

Proof: The probability that the distance between node iand j does not follow x = Θ

ÈXn

is

1− Px = Θ

ÉX

n

< P

x ≤

ÉaX

πn

+ P

x ≥

ÉbX

πn

,

(19)

where 0 < a < (2e)−1 and b > 2e. The Px ≤

ÈaXn

satisfies

Px ≤

ÉaX

πn

=

nXk=X

P(There are k nodes within

ÉaX

πnaway from node i)

=

nXk=X

n!

k!(n− k)!(aX

n)k(1− aX

n)n−k.

(20)

Denoting g(k) = n!k!(n−k)! (

aXn )k(1 − aX

n )n−k, based onLemma 3, there must be g(X) < c1(ae1−a)X . Furthermore,the ratio of g(k) and g(k + 1) can be represented as

g(k)

g(k + 1)=

n!k!(n−k)!

(aXn

)k(1− aXn

)n−k

n!(k+1)!(n−k−1)!

(aXn

)k+1(1− aXn

)n−k−1>k + 1

aX.

(21)

Thus, for any k > X , g(k) satisfies

g(k) <aX

kg(k − 1) < · · · < (aX)k−XX!

k!g(X). (22)

Therefore, Px ≤

ÈaXn

in (19) can be upper-bounded as

Px ≤

ÉaX

πn

<

nXk=X

(aX)k−XX!

k!g(X)

(a)

< c3

nXk=X

(aX)k−XXe

X √X

ke

k√k

g(X)

= c3

nXk=X

aXe

k

k 1

ae

XÉX

kg(X)

< c3g(X)

nXk=X

(ae)k−X < 2c3g(X),

(23)

(a) holds due to (17). For Px ≥

ÈbXπn

, we denote g∗(k) =

n!k!(n−k)! (

bXn )k(1 − bX

n )n−k and Px ≥

ÈbXπn

< 2c3g

∗(k)

can be proved in the similar way as above.

Hence, the probability that the distance between node i andj does not satisfy x = Θ

ÈXn

can be represented as

1− Px = Θ

ÉX

n

< 2c3c1(ae1−a)X + 2c3c2(be1−b)X .

(24)

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It should be noticed that c1(ae1−a)X and c2(be1−b)X arethe monotonic functions of a and b with lower-bound 0,respectively. For any given constant c > 0, there must exist aand b satisfying c3c1(ae1−a)X+c3c2(be1−b)X < c. Therefore,based on the definition of o(·), the following equation can beobtained.

1− Px = Θ

rX

n

= o(1), (25)

which means that the x = ΘÈ

Xn

with probability 1 when

n goes to infinity.

To derive the fn(x) in Lemma 2, we first show the distri-bution of rank of i’s friends when it has qi friends in the nextLemma. Afterwards, we derive the PDF in Theorem 1 basedon the relation between rank and geographical position.

Lemma 5 If i is a source node with qi destinations, and j isone of them. The probability that Ranki(j) = X satisfies

PRanki(j) = X = Θ

1

qi + n1−αXα

, (26)

when 0 ≤ α < 1, and

PRanki(j) = X = Θ

1

qi +XC1(qi, n)

, (27)

when α = 1 and C1(qi, n) is given in (45), and

PRanki(j) = X = Θ

1

qi + q1−αi Xα

, (28)

when α > 1.

Proof: We sort the nodes based on the rank respectsto i as v1, · · · , vn−1, where Ranki(vX) = X . The set ofdestinations of i is denoted as Qi, where |Qi| = qi. Therefore,the probability that Ranki(j) = X is

PRanki(j) = X = Pj = vX |vX ∈ QiPvX ∈ Qi,(29)

which is shown in [9]. In particular, Pj = vX |vX ∈ Qi = 1qi

and PvX ∈ Qi can be represented as

PvX ∈ Qi =

1HnXα

P1≤i1<···<iqi−1≤n,ih 6=X

qi−1Qh=1

1HniαhP

1≤i1<···<iqi≤n

qiQh=1

1Hniαh

.

(30)

We define σX,qi−1 =P

1≤i1<···<iqi−1≤n,ih 6=X

qi−1Qh=1

1iαh

and hence

X1≤i1<···<iqi≤n

qiYh=1

1

iαh

=X

1≤i1<···<iqi≤n,∃h:ih=X

qiYh=1

1

iαh+

X1≤i1<···<iqi≤n,ih 6=X

qiYh=1

1

iαh

=1

XασX,qi−1 + σX,qi .

(31)

Denoting Cσ(X, qi, n) as

Cσ(X, qi, n) = qiσX,qiσX,qi−1

, (32)

and thus PRanki(j) = X can be transformed as

PRanki(j) = X =1

qi + Cσ(X, qi, n)Xα. (33)

Then, we prove that the order of C(X, qi, n) is independentfrom X when C(X, qi, n)Xα = ω(qi) (i.e., for any constantc4 there must be C(X, qi, n)Xα > c4qi). We arbitrarily selecttwo integers Xa and Xb. Based on (32) and C(X, qi, n)Xα >c4qi we can obtain

σX1X2,qi

+σX1X2,qi−1

Xα2

> c4

σX1X2,qi−1

Xα1

+σX1X2,qi−2

Xα2 X

α1

,

σX1X2,qi

+σX1X2,qi−1

Xα1

> c4

σX1X2,qi−1

Xα2

+σX1X2,qi−2

Xα2 X

α1

.

(34)

If c4 > 1, σX1X2,qican be lower-bounded as

σX1X2,qi

>(c4 − 1)σ

X1X2,qi−1

2Xα2

+(c4 − 1)σ

X1X2,qi−1

2Xα1

. (35)

According to the Newton’s inequality in [22] that for anyconstant c5 > 0, σX1X2,qi−1 can be lower-bounded as

σX1X2,qi−1

>c5σX1X2,qi

σX1X2,qi−2

σX1X2,qi−1

. (36)

Therefore, the relation between σX1X2,qi−1 andσX1X2,qi−2

Xα2can be derived as

σX1X2,qi−1

>c5σX1X2,qi

σX1X2,qi−2

σX1X2,qi−1

=c5Xα

2

σX1X2,qiσX1X2,qi−2

σX1X2,qi−1

Xα2

>c5(c4 − 1)σX1X2,qi−2

Xα2

.

(37)

Thus, we calculate the bound of C(X1, qi, n) as follows

C(X1, qi, n) = qiσX1X2,qi

+σX2X2,qi−1

Xα2

σX1X2,qi−1 +σX1X2,qi−2

Xα2

>qiσX1X2,qi

1c5(c4−1)

+ 1σX1X2,qi−1

,

C(X1, qi, n) <

1

c5(c4−1)+ 1qiσX1X2,qi

σX1X2,qi−1

.

(38)

Therefore, it is obvious that

C(X1, qi, n) = Θ(qiσX1X2,qi

σX1X2,qi−1

). (39)

Similarly,

C(X2, qi, n) = Θ

qiσX1X2,qi

σX1X2,qi−1

= Θ(C(X1, qi, n)), (40)

which means that the order of C(X, qi, n) is independentfrom X when C(X, qi, n)Xα = ω(qi). Hence, we denoteC(X, qi, n) as cXC(qi, n) where cX may be related to X, qi.

6

According to equation (38), the upper-bound and lower-boundof cX are

1

c5(c4−1) + 12

and

1c5(c4−1) + 1

−2, respectively.

Since c4 > 1 and c5 > 0 are arbitrary constants, we obtainthat cX = 1. Thus, (33) can be rewritten as

PRanki(j) = X =1

qi + C(qi, n)Xα, (41)

Now we analyze PRanki(j) = X for three cases.Case 1: 0 ≤ α < 1. In this case, because of (29) and (30),the sum of (41) is 1, which means that

nXX=1

1

qi + C(qi, n)Xα=

c6q1−αα

i

C1α (qi, n)

+c6n

1−α

C(qi, n)= 1, (42)

where 1/2 < c6 < 1 is constant. Therefore,

c6n1−α < C(qi, n) < 2c6n

1−α, (43)

which shows that C(qi, n) = c′Xn1−α, where c′X is related to

cX and 1/2 < c′X < 2. Hence, (33) can be rewritten as

PRanki(j) = X =1

qi + c′Xn1−αXα

= Θ

1

qi + n1−αXα

.

(44)Case 2: α = 1. We derive C(qi, n) and PRanki(j) = Xin the same way as in case 1. It can be obtained that

C(qi, n) = C1(qi, n) =

Θ(log n

qi) limn→∞

nqi> 1,

Θ(1) limn→∞nqi

= 1,(45)

and

PRanki(j)=X= 1

qi + c′′XXC1(qi, n)= Θ

1

qi +XC1(qi, n)

,

(46)where 1/2 < c′′X < 2.

Case 3: α > 1. C(qi, n) and PRanki(j) = X can alsobe obtained in the similar way, and the results are C(qi, n) =Θ(q1−α

i ) and

PRanki(j) = X =1

qi + c′′′Xq1−αi Xα

= Θ

1

qi + q1−αi Xα

,

(47)where 1/2 < c′′′X < 2.

Based on Lemma 4 and 5, the PDF of destinations isobtained in the following theorem in polar coordinates.

Theorem 1 If node i is a source, and it selects qi destinationsbased on (6), the location of each destination follows thedistribution

fn(x, θ) = Θ

n

qi + nx2α

, (48)

when 0 ≤ α < 1, and

fn(x, θ) = Θ

n

qi + nx2C1(qi, n)

, (49)

when α = 1, and

fn(x, θ) = Θ

n

qi + q1−αi nαx2α

, (50)

when α > 1.

Proof: Firstly, we analyze the PDF for the case 0 ≤ α <1. For node i and its destination j, denoting the CumulativeDistribution Functions (CDF) of d(i, j) as Fi(x), it can be

represented as

Fi(x) = Pd(i, j) < x

=

nXX=1

Pd(i, j) < x|Ranki(j) = XPRanki(j) = X.

(51)

Hence, the PDF can be expressed as

fn(x, θ) =dFi(x)

2πxdx

=

nXR=1

dPd(i, j) < x|Ranki(j) = k2πxdx

· PRanki(j) = R

=

nXX=1

dnP

k=X

n!k!(n−k)!

(πx2)k(1− πx2)n−k

2π(qi + c′Xn1−αXα)xdx

(a)=

nXX=1

dIπx2(X,n−X+1)

2π(qi + c′Xn1−αXα)xdx

,

(52)

where Ix(a, b) is the regularized incomplete beta functionand (a) holds due to equation (8.17.5) in [23]. If qi >(2eπ)αnx2α, the PDF is as follow

fn(x, θ)

=

nXX=1

n!(X−1)!(n−X)!

(πx2)X−1(1− πx2)n−X

2πqi + 2πc′Xn1−αXα

<n

2πqi

X0XX=1

(n− 1)!

(X − 1)!(n−X)!(πx2)X−1(1− πx2)n−X+

nαnX

X=X0+1

(n− 1)!

2πc′X(X − 1)!(n−X)!Xα(πx2)X−1(1− πx2)n−X

=n

2πqi

X0XX=1

(n− 1)!

(X − 1)!(n−X)!

aX0

n

X−1 1− aX0

n

n−X+

nαnX

X=X0+1

(n− 1)!

2πc′X(X − 1)!(n−X)!Xα

aX0

n

X−1 1− aX0

n

n−X(a)

<n

2πqi+c1(ae1−a)X0+1nα

2πc′XXα0

<n

πqi,

(53)

where X0 =

q

1αi

πn1−αα

, a = nx2

X0< (2e)−1 and c′X is the

lower-bound of c′X , which equals to 1/2. It should be noticedthat the fn(x, θ) is uniform for different directions, and thusthere is no θ in the expression of fn(x, θ). (a) in (53) holdsdue to the proof of Lemma 4 which shows that

nXX=X0+1

(n− 1)!aX0n

X−1 1− aX0

n

n−X(X − 1)!(n−X)!

< c1(ae1−a)X0+1,

(54)

when a < (2e)−1. In the same way, it can also be proved that

fn(x, θ) >n

2πqi− c1(ae1−a)X0+1nα

πc′XXα0

>n

8πqi, (55)

7

where c′X is the upper-bound of c′X , which equals to 2.Hence, fn(x, θ) = Θ( nqi ) is proved when qi = ω(nx2α).Moreover, if qi = O(nx2α), after some similar mathematicalmanipulations, the fn(x, θ) can be represented as

fn(x, θ) = Θx−2α . (56)

Consequently, the PDF for 0 ≤ α < 1 is

fn(x, θ) = Θ

max

§x−2α,

n

qi

ª= Θ

n

qi + nx2α

, (57)

Moreover, the fn(x, θ) for α ≥ 1 can be obtained in thesame way, and the results are shown in (49) and (50).

According to Lemma 2 and Theorem 1, we analyze theexpression of the ‖EMST‖ in the following theorem.

Theorem 2 When α = 1, the expectation of the bound‖EMST‖ in (11) satisfies

‖EMST‖ =

8<: ΘÈ

qiC1(qi,n)

qi = ω(log n),

O

qilogn

qi = O(log n).

(58)

Furthermore, when 0 ≤ α < 1, the expectation of the bound‖EMST‖ in (11) satisfies

‖EMST‖ = Θ (√qi) . (59)

Finally, when α > 1, the expectation of the bound ‖EMST‖in (11) satisfies

‖EMST‖ =

8>>>>><>>>>>:

Oqαi n

1−α qi = O(nα−1α ), 1 < α ≤ 3

2 ,

Θqα2i n

1−α2

qi = ω(n

α−1α ), 1 < α ≤ 3

2 ,

Oq

α2α−2

i n−12

qi = O(n

α−1α ), 3

2 < α < 2,

Θqα2i n

1−α2

qi = ω(n

α−1α ), 3

2 < α < 2,

Θqi√n

α ≥ 2.

(60)

Proof: Case 1: α = 1. Based on Lemma 2 and Theorem1, the ‖EMST‖ can be derived for different value of qi. Ifqi = ω(log n), we prove that the ‖EMST‖ can be calculatedby (11) first. The network can be separated into some smallsquare regions as in Lemma 2, and we assume that the sidelength of the region ξ, which is d away from the source, isr, where d > r. The number of destinations in this region islower-bounded asZξfn(x, θ)dξ >

qi2

Z rd

0

Z d+ r2

d− r2xfn(x, θ)dxdθ =

c8qir2

d2C1(qi, n),

(61)

where d ≥È

qinC1(qi,n) , and 1

8π < c8 <1

2π is constant. It isnecessary to guarantee that (61) is no less than 1, and therefore,r ≥ d

qC1(qi,n)c8qi

. Since a small r is needed for gn(x, θ) whichsatisfies Condition 1 in Lemma 2, r can be represented as

r = d

ÊC1(qi, n)

c8qi. (62)

Moreover, d > r because qi = ω(log n) = ω(C1(qi, n)). Sinced > r and r ∝ dα, there are at most 2α + 1 adjacent parts at

each side of ξi. Considering the fact that ξi has only 4 sides,the ζ in the Theorem 2 is no greater than 2α+2 + 4, whichis a constant. Thus the Conditions in Lemma 2 are satisfied.Similarly, the Conditions in Lemma 2 also hold in the case0 ≤ α < 1 and α > 1, and we do not show it in the followingproof for brevity.

We define gn(x, θ) = 1r2

Rξ xfn(x, θ)dxdθ when (x, θ) is

in ξ, where ξ is a square region with side length r. If d ≥Èqi

nC1(qi,n) , the difference between gn(x, θ) and fn(x, θ) inξ can be derived asZξ

|fn(x, θ)− gn(x, θ)|dξ <Z r

d

0

Z d+ r2

d− r2

|fn(x, θ)− gn(x, θ)|xdxdθ

=c9r

3

d3C1(qi, n),

(63)

where 18π < c9 <

1π is constant. Thus, according to (53) and

(55), the condition 1 of Lemma 2 can be proved as followZΨ

|fn(x, θ)− gn(x, θ)|dξ

=

Z 2π

0

Z 1Èqi

nC1(qi,n)

|fn(x, θ)− gn(x, θ)|xdxdθ

+

Z 2π

0

Z È qinC1(qi,n)

0

|fn(x, θ)− gn(x, θ)|xdxdθ

(a)

<Xd

2πd

r

Z rd

0

Z d+ r2

d− r2

|fn(x, θ)− gn(x, θ)|xdxdθ

+

Z 2π

0

Z È qinC1(qi,n)

0

c1(ae1−a)X0+1(c′X + c′X)n

2πc′Xc′Xqi

xdxdθ

<Xd

2πc9r2

d2C1(qi, n)+c1(ae1−a)X0+1(c′X + c′X)

c′Xc′X

,

(64)

(a) holds because there are less than 2πdr regions with side

length r and d from the source. Moreover, considering therelation between d and r in (62) and the maximum value ofd is 1

2 , (64) can be bounded byXd

2πc9r2

d2C1(qi, n)+c1(ae1−a)X0+1(c′X + c′X)

c′Xc′X

<4πc9 lognpc8qiC1(qi, n)

+c1(ae1−a)X0+1(c′X + c′X)

c′Xc′X

=o(1).

(65)

Therefore, the two conditions in Lemma 2 are satisfied. Wedenote that x† =

Èqi

nC1(qi,n) and the ‖EMST‖ is as

‖EMST‖ <c10c(d)√qi

Z 2π

0

Z 1

0

x

Én

qi + nC1(qi, n)x2dxdθ

<2c10c(d)π√qi

"Z x†

0

x√n

√qidx+

Z 1

x†

É1

C1(qi, n)dx

#

<2c10c(d)π

Éqi

C1(qi, n),

(66)

8

where c10 is constant according to the proof of Theorem 1.Similarly, (67) can also be proved.

‖EMST‖ >c10c(d)π

2

Éqi

C1(qi, n), (67)

where c10 is constant according to Theorem 1. Thus,

‖EMST‖ = ΘÈ

qiC1(qi,n)

.

If qi = Θ(1), (11) cannot be applied. However, the‖EMST‖ is upper-bounded by the sum distance between iand its qi friends and lower-bounded by the distance betweeni and one of its friends. Therefore, the order of ‖EMST‖ canbe expressed as

‖EMST‖ = Θ

1

logn

. (68)

If qi = ω(1) and qi = O(log n), it is obvious that thetwo conditions in Lemma 2 cannot be satisfied simultaneouslybased on (61) to (65). Therefore, (11) cannot be adopted inthis case, and we give the upper-bound of ‖EMST‖ as follow

‖EMST‖ = O

qi

log n

, (69)

which is the sum length of qi direct links. Consequently,equation (58) is proved.

Case 2: 0 ≤ α < 1. We also derive the ‖EMST‖ based onqi. The network is separated into some small square regionsas in Lemma 2. We assume that the side length of the regionξ, which is d away from the source, is r, and d > r. Similarto the proof of case 1, the two conditions in Lemma 2 are

satisfied for the region x <qin

12α . Therefore, we only focus

on the case x >qin

12α . The number of destinations in the

region with side length r and d >qin

12α is lower-bounded

byqi2

Z rd

0

Z d+ r2

d− r2xfn(x, θ)dxdθ =

c11qir2

d2α, (70)

where 18π < c11 <

12π . Since it is necessary that c11qir

2

d2α ≥ 1,the r can be bounded as r ≥ dα√

c11qi. Thus, r is selected as

r =dα√c11qi

. (71)

Moreover, d > r, i.e., d > (c11qi)1

2α−2 . We define gn(x, θ) =1r2

Rξ xfn(x, θ)dxdθ, where ξ is a square region with side

length r. Thus, if (c11qi)1

2α−2 <qin

12α ,

RΨ |fn(x, θ) −

gn(x, θ)|dξ = o(1) can be proved in the similar way as in (63)to (65), and therefore the fn(x, θ) satisfies two conditions inLemma 2.

If (c11qi)1

2α−2 >qin

12α , we will propose another PDF

f ′n(x, θ), which satisfies the two conditions and its correspond-ing ‖EMST‖ is the same as fn(x, θ) in order sense. The num-

ber of destinations within the ringqin

12α< d < (c11qi)

12α−2

is upper-bounded as

qi

Z 2π

0

Z (c11√qi)

1α−1

( qin )12α

xfn(x, θ)dxdθ <π

c11(1− α). (72)

Hence, the total length of links in this region is smaller thanπ(c11)

2−αα−1 q

12α−2i

1−α . Thus, we define the new PDF as

f ′n(x, θ)

=

8>>><>>>:fn(x, θ) x ≤

qin

12α ,R 2π

0

R (c11qi)1

2α−2

( qin )12α

xfn(x,θ)dxdθ

π(c11qi)1

α−1−π( qin )1α

qin

12α < x < (c11qi)

12α−2 ,

fn(x, θ) x ≥ (c11qi)1

2α−2 .(73)

The difference of ‖EMST‖ between fn(x, θ) and f ′n(x, θ)

is smaller than π(c11)2−αα−1 q

12α−2i

1−α . Moreover, similar to theproof from (63) to (65), there exists a gn(x, θ) satisfyingthe two conditions in Lemma 2 with respect to f ′n(x, θ).Hence, the ‖EMST‖ of f ′n(x, θ) can be derived by Lemma

2. Furthermore, since ‖EMST‖ > 1 >2π(c11)

2−αα−1 q

12α−2i

1−α , theorder of ‖EMST‖ of fn(x, θ) and f ′n(x, θ) are the same.Consequently, ‖EMST‖ for fn(x, θ) is the same as f ′n(x, θ)in order sense for the case 0 < α < 1. Similar to (66) and(67), the result of ‖EMST‖ can be represented as in (59).

Case 3: α > 1. We try to find when the two conditions inLemma 2 are satisfied first. The network is also separated intosome small square regions as above. Therefore, the number ofdestinations in one region ξ and d >

Èqin is lower-bounded

as

qi

Z rd

0

Z d+ r2

d− r2

xfn(x, θ)dxdθ >c12q

αi r

2

d2αnα−1, (74)

where 18π < c12 <

12π . We select r = n

α−12 dα

√c12q

α2i

to ensure that

(74) equals to 1. Moreover, d > r which indicates that d <(c12)

12α−2 q

α2α−2i√

n. We define gn(x, θ) = 1

r2

Rξ xfn(x, θ)dxdθ

when (x, θ) is in ξ. Thus, if qi > nα−1α , it can be proved in

the similar way as in (63) to (65) thatZΨ

|fn(x, θ)− gn(x, θ)|dξ = o(1). (75)

Hence, when qi > nα−1α , the two conditions in Lemma 2 are

satisfied. and the ‖EMST‖ can be calculated by (11).

However, if qi < nα−1α , the two conditions in Lemma

2 are no longer satisfied. In order to analyze the bound of‖EMST‖, we divide the network into two groups. In thegroup 1, the nodes in the circle centered at the source with

radius qα

2α−2i√n

are considered, and the nodes out of it arenot considered. In the group 2, other nodes are considered.The PDFs of them are denoted as f (1)

n (x, θ) and f(2)n (x, θ),

respectively. The f (1)n (x, θ) can be derived as

f (1)n (x, θ) =

fn(x, θ)

R 2π

0

R q

α2α−2i√n

0 xfn(x, θ)dxdθ

= c13fn(x, θ),(76)

for x <q

α2α−2i√n

, and f1(x, θ) = 0 for x >q

α2α−2i√n

, wherec13 → 1 when n → ∞. According to (63) to (65), it can be

9

found that the nodes in group 1 satisfy condition 1 and 2 inLemma 2. Thus, the ‖EMST‖ for group 1 can be calculatedbased on Lemma 2. Then we derive f (2)

n (x, θ) as

f (2)n (x, θ) =

fn(x, θ)R 2π

0

R 1

q

α2α−2i√n

xfn(x, θ)dxdθ>qifn(x, θ)

c14,

(77)

for x >q

α2α−2i√n

, and f2(x, θ) = 0 for x <q

α2α−2i√n

, wherec14 > 0 is constant according to Theorem 1. The number ofdestinations in group 2 is

qi

Z 2π

0

Z 1

q

α2α−2i√n

xfn(x, θ)dxdθ < c14. (78)

And the average distance from these destinations to the sourceis

Υqi <c14

Z 2π

0

Z 1

q

α2α−2i√n

x2f2(x, θ)dxdθ. (79)

Since the total ‖EMST‖ for this case is upper-bounded bythe sum of ‖EMST‖ of the two groups which are calculatedindependently, the upper bound of the total ‖EMST‖ can berepresented as

‖EMST‖ < c(d)√qi

Z 2π

0

Z qα

2α−2√n

0

x

Èf

(1)n (x, θ)dxdθ + Υqi ,

(80)

when qi = Onα−1α

. Moreover, the total ‖EMST‖ must be

greater than the ‖EMST‖ of nodes within the range of√qi√n

from the center. Therefore, the ‖EMST‖ is lower-boundedby

‖EMST‖ > c(d)√qi

Z 2π

0

Z √qi√n

0

Ìx2fn(x, θ)R 2π

0

R √qi√n

0 xfn(x, θ)dxdθ

dxdθ.

(81)

Consequently, according to (80) and (81), the ‖EMST‖ issummarized in (60). Moreover, since the capacity ratio ofsocial networks and traditional ad hoc networks is greater thanΘ(1) when α > 1, which is shown in Section VI, we use Θ(·)instead of Θ(·) here and ignore the poly-logarithmic factors.

Denoting the expectation of the minimum total number ofhops for a multicast tree is Nq(n). If the packet is transmittedthrough the path which is close to the line of multicasttree within Θ

Èlognn

, the order of the total length of the

path equals to ‖EMST‖ in order sense. However, anotherimportant case must be considered, which is that one nodecan transmit to multiple nodes simultaneously if the distancebetween them are smaller than r(n). Thus, these hops canbe treated as one, and we denote this as ‘hop reduction’.For example, if there are k0 destinations distributed in asmall region with side length less than r(n) uniformly andindependently, the total length of the EMST among them isr(n)√k0 which consists at least k0 hops. But the total hop

for them is 1 since the source can transmit to other nodes

simultaneously. In our network model, the number of nodesin a region with side length r(n) is Θ(log n). Thus, the relationbetween Nq(n) and ‖EMST‖ is

Nq(n) = O

‖EMST‖r(n)

, and Nq(n) = Ω

‖EMST‖r(n) logn

,

(82)due to the ‘hop reduction’. To obtain the order of Nq(n), wegive the following lemma.

Lemma 6 The expectation of the minimum total number ofhops for a multicast tree satisfies

Nq(n) =

8<: Θ‖EMST‖r(n)

q = o

n

logn log logn

,

Θ‖EMST‖r(n)

q = Ω

n

logn log logn

,

(83)

where α = 1, and

Nq(n) =

8<: Θ‖EMST‖r(n)

q = o

n

logn

,

Θ‖EMST‖r(n)

q = Ω

n

logn

,

(84)

where 0 ≤ α < 1, and

Nq(n) = Θ

‖EMST‖r(n)

, (85)

where α > 1.

Proof: Firstly, we prove the lemma under the conditionthat α = 1. Considering a square region with side lengthr(n), which is kr(n) away from the source, when the totalnumber of destinations is q, the expectation of the number ofdestinations in this region can be upper-bounded as

Dk < q

Z 1k

0

Z (k+1)p

lognn

kp

lognn

fn(x, θ)dxdθ. (86)

If k >È

qC1(q,n) logn , Dk satisfies

Dk < 2q

Z 1k

0

Z (k+1)p

lognn

kp

lognn

c15

xC1(q, n)dxdθ <

2c15q

k2C1(q, n),

(87)

where c15 is constant according to Theorem 1. If k <Èq

C1(q,n) logn , Dk can be derived as

Dk < 2q

Z 1k

0

Z (k+1)p

lognn

kp

lognn

nx

qdxdθ < 2 logn. (88)

Notice that if there is o(1) destination in one region withside length r(n) in average, there is no ‘hop reduction’.When k >

q2c15qC1(q,n) , which means 2c15qi

k2C1(q,n) < 1, there

is no ‘hop reduction’. Moreover, if k <È

qC1(q,n) logn , the

number of ‘hop reduction’ is smaller than√

log n in eachregion with side length r(n). Finally, the total number of ‘hopreduction’ is the sum of the two parts, i.e., k <

Èq

C1(q,n) logn

andÈ

qC1(q,n) logn < k < min

§q2c15qC1(q,n) ,

√n

2√

logn

ª, where

√n

2√

lognis the maximum value of k. Thus, according to (87),

10

the total number of ‘hop reduction’ is upper-bounded by

π√

lognq

C1(q, n) logn+

min

nÈ2c15q

C1(q,n),√n

2√

logn

oX

k=p

qC1(q,n) logn

4c15πq

kC1(q, n)

<

( 2c15πq log 2c15 lognC1(q,n)

q ≤ C1(q,n)n8c15 logn

,

2c15πq lognC1(q,n)

2q

C1(q,n)q > C1(q,n)n

8c15 logn.

(89)

Furthermore, when q < nlogn log logn , the total number of hops

of the EMST is lower-bounded as

‖EMST‖r(n)

>c10c(d)π

2

Éqn

C1(q, n) logn

>4c15πq log 2c15 logn

C1(q, n),

(90)

when n goes to infinite. Hence, the ‘hop reduction’ can be ig-nored in this condition in order sense. Consequently, accordingto (82), the expectation of the minimum total number of hopsfor a multicast tree is as in (83) when α = 1.

Similarly, the expectation of the minimum total number ofhops for the case 0 ≤ α < 1 and α > 1 can be derived, andthe results are demonstrated in (84) and (85), respectively.

According to Lemma 6, the capacity bound of this socialnetwork model can be derived in the following theorem.

Theorem 3 The expectation of the capacity of this socialnetwork model is bounded by

C(n) =

8><>:Θ

1n

0 ≤ β ≤ 1,

Θnβ−2

1 < β ≤ 3

2,

Θ

1√n logn

β > 3

2,

(91)

when 0 ≤ α < 1, and

C(n) =

8>>>>>><>>>>>>:

Θn−1

0 ≤ β ≤ 1,

Θnβ−2

1 < β ≤ 3

2,

Ω

log

β− 32 n√n

32< β < 2,

Ω √

logn√n log logn

β = 2,

ΘÈ

lognn

β > 2,

(92)

when α = 1, and

C(n) = Θ

1

E‖EMST‖√n

, (93)

when α > 1, and E‖EMST‖ =nPq=1

‖EMST‖qβGn

. We do not

show the detailed expressions for the sake of conciseness.

Proof: From Lemma 6, we know that the total numberof hops is Nq(n) for each multicast session. Therefore, theexpectation of the total number of hops is ENq(n) =Pnq=1

Nq(n)qβGn

. Furthermore, there are n multicast sessions inthe network. Since the transmission range is limited by r(n),there are at most 1

r2(n) active nodes in one time slot in average.Thus, the capacity of the network is bounded by

1

ENq(n)nr2(n)=

1

ENq(n) logn. (94)

The capacity bound can be calculated based on Theorem 2,and the results are presented in (91), (92) and (93).

IV. SIMULATIONS

In this section, we will give simulations about the desti-nation PDFs for the source which has q destinations. In oursimulations, the number of nodes n is from 1×104 to 1×105,and the number of destinations is q = n

45 . The distance range

is 0 ≤ x ≤ 1. The simulation results are averaged over 10000realizations. For each realization, firstly, n nodes are randomlyand uniformly distributed in the network, and we assumethat the source is at the center of the network. Afterwards,q destinations are selected based on the rank-based model in(6). Finally, the statistical PDF of these nodes is obtained andcompared with the theoretical result.

The statistical PDF is obtained according to the simulationresults. On the other hand, the corresponding theoretical PDFis fn(x) = 2πxfn(x, θ), which can be expressed as

fn(x) =

8>>><>>>:Θ

n15 x

1+n15 x2α

0 ≤ α < 1

Θ

n

15 x

1+n15 x2 logn

α = 1

Θ

n15 x

1+nα5 x2α

α > 1

(95)

In our simulation, we use Surface Fitting to obtain thesimulation results. Therefore, based on (95), we give theequation of surface fitting method as follows

fn(x) =

8><>:a1n

b1xc3

1+d1ne1xf10 ≤ α < 1

a2nb2xc1

1+d2ne2xf2 logg2 nα = 1

a3nb3xc2

1+d3ne3xf3α > 1

(96)

where ai to gi are the surface fitting parameters. In oursimulations, we only focus on bi, ci, ei, fi and gi to showthe order of the PDF.

The simulation results are illustrated in Figure 1 for thecase 0 ≤ α < 1, and α is assumed to be 0.5 here. Moreover,

00.2

0.40.6

0.8

1

0

2

4

6

8

10

x 104

Distance x

n

PD

F

Fig. 1: The comparison of theoretical PDF and statistical PDFfor α = 0.5.

the corresponding parameters are given in Table II. R-square inTable II is the coefficient of determination. If R-square is closeto 1, it means that the fitting is accurate. From the simulation

11

TABLE II: Surface fitting parameters

Parameters α = 0.5 α = 1 α = 1.2ai 2.841 2.164 2.421bi 0.1886 0.2002 0.1948ci 1.07 1.057 1.075di 1.658 0.5765 2.109ei 0.2098 0.1983 0.2546fi 1.066 2.056 2.473gi - 0.8885 -

R− square 0.9996 0.9998 0.9992

results we can find that the differences between the parametersof surface fitting function and theoretical function are no morethan 7%, which indicates that our theoretical PDF is very closeto the simulation one in order sense.

We also give the simulations for the case α = 1 and α > 1in Figure 2 and 3, respectively. The corresponding parametersare listed in Table II. For the case α > 1, we assume α = 1.2.

0

0.2

0.4

0.6

0.8

1 0

2

4

6

8

10

x 104

n

Distance x

PD

F

Fig. 2: The comparison of theoretical PDF and statistical PDFfor α = 1.

00.2

0.40.6

0.81

0

2

4

6

8

10

x 104

n

Diatance x

PD

F

Fig. 3: The comparison of theoretical PDF and statistical PDFfor α = 1.2.

From Figure 2 and 3, we can find that the theoretical PDFsare very close to the simulation ones in order sense. In TableII, when α = 1, the differences between the parameters ofsurface fitting function and theoretical function are no morethan 5.7% except for g2. The difference of g2 is 11.2% becauseit is the power of log n, which is not very large even whenn = 1×105. Therefore, the surface fitting parameter g2 is notaccurate enough. However, this difference will not seriouslyinfluence the PDF because log n is quite small comparing

with n when n is large enough. For the case α = 1.2, thedifferences between the parameters of surface fitting functionand theoretical function are no more than 7.5%. Therefore,from Figure 1-3 and Table II, it is indicated that the theoreticalPDF can describe the real PDF of destinations in order sensefor all cases of α.

V. CAPACITY ACHIEVING SCHEME

A social network transmission scheme will be proposed inthis section, and we will analyze the corresponding throughput.In this scheme, TDMA medium access control scheme basedon the protocol model is employed. The network is dividedinto many cells with side-length R0r(n) where R0 is constant.Therefore, M2-TDMA can be employed, which allows eachM2 adjacent cells to be active with a round-robin fashion,where M is constant. Moreover, considering the protocol mod-el, M must satisfy MR0r(n) ≥ (1 + ∆)r(n), i.e., M ≥ 1+∆

R0.

The EMST is established for the social networks and theEMST-based scheme is as follows.

1) Initially, the network is separated into cells, and TDMAis employed as described above.

2) For each arbitrary source i, an EMST is built among theqi destinations and node i. This tree is the basic EMST.

3) Considering the limited transmission range r(n), eachdirect link in basic EMST can be constructed by sendingpackets hop by hop along the cells where link linesacross. When a node is allowed to be active in this timeslot, it will select the oldest packet in its buffer andtransmit it to the next node according to the route ofEMST.

To analyze the throughput of the EMST-based scheme, wepropose the following theorem.

Theorem 4 The throughput of EMST-based scheme which isproposed above is of the same order as the capacity given inTheorem 3 when 0 ≤ α < 2 or 0 ≤ β ≤ 1. Moreover, forthe case α ≥ 2 and β > 1, there is a gap nη between them,where η > 0 is an arbitrary small constant.

Proof: We define the node set Si consisting the nodeswhich are one of the relays or destinations for one multicastsession with source i. Considering an arbitrary cell s, thenumber of multicast sessions that invoke s can be representedas

N(s) =nXi=1

1s is invoked by Si, (97)

where the 1s is invoked by Si are i.i.d. Bernoullian randomvariables. Thus, according to the definition of throughput, thistheorem is equivalent to

Pr

¨\s

¦N(s) < cn‖EMST‖r(n)

©«→ 1, (98)

when 0 ≤ α < 2, and

Pr

¨\s

¦N(s) < cn1+η‖EMST‖r(n)

©«→ 1, (99)

when α ≥ 2 and c > 0 is constant.

12

In order to prove (98) and (99), we consider a region whichis defined as

Ri = x|The minimum distance from x to theEMSTi is smaller than 2R0r(n).

(100)

Therefore, all the invoked cells by Si is covered by Ri, andits average size is no greater than 3R0‖EMSTi‖r(n). Sincethe s is randomly selected, mean value of 1s is invoked by Si ism

(i)1 < min3R0‖EMSTi‖r(n), 1 = m

(i)2 .

Since there are total n EMSTs, N(s) is upper-bounded byn. Thus, throughput is lower-bounded by 1

n , which indicatesthat the capacity bound in Theorem 4 can be achieved when0 ≤ β ≤ 1.

For the case β > 1, q represents the mean of qi, whichsatisfies q = Θ(maxn2−β , 1). Firstly, we prove the theoremfor the case 0 < α < 1. Considering two multicast sessionswith source i and j, the sets of their friends are defined asFi and Fj , respectively. Therefore, the correlation between1s is invoked by Si and 1s is invoked by Sj is related with the totallength of links among nodes in set Fi ∪ Fj . For the case 0 <α < 1, the number of nodes k in Fi∪Fj can be upper-boundedas

|Fi ∪ Fj | <nX

X=1

q2PRanki(k) = XPRankj(k) = X

<c16q qn

1−αα

= o(q),

(101)

where c16 is constant according to Lemma 5. Furthermore,the probability that a node belongs to Fi ∪ Fj is small if itis far from i and j due to Theorem 1. However, the linksconnected to it in the EMSTi and EMSTj are long. Thus,the total length of links among nodes in set Fi∪Fj is smallerthan ‖EMSTi‖ and ‖EMSTj‖ in order sense. Hence, thecorrelation between 1s is invoked by Si and 1s is invoked by Sjcan be ignored. Therefore, denoting N(s) as another sum ofi.i.d. Bernoullian random variables with mean m

(i)2 , we can

find that N(s) is statistically larger than N(s). Thus, basedon Chernoff bounds, we can obtain the following relation

PrN(s) > 2E[N(s)] < PrN(s) > 2E[N(s)]< (e)−nm2/3,

(102)

where m2 is the mean value of m(i)2 which equals to

min3R0‖EMST‖r(n), 1. Then we consider all the cells,since m2 > 3R0r(n),

Pr

¨\s

¦N(s) < 6R0n‖EMST‖r(n)

©«≥1−

Xs

PrN(s) > 2E[N(s)]

≥1− ne−3R0

√n logn/3 → 1 as n→∞.

(103)

Therefore, the throughput lower-bound of the proposed schemeis Θ

1

n‖EMST‖r(n)

based on the definition of throughput.

Moreover, considering the capacity upper-bound in Theorem3, which is the same as the lower-bound in order sense, theproposed scheme achieves the capacity upper-bound.

For the case α = 1, the number of nodes in Fi ∪Fj can be

upper-bounded as

|Fi ∪ Fj | <nX

X=1

q2PRanki(k) = XPRanki(k) = X

<c17q

C1(q, n)= o(q),

(104)

where c17 is constant according to Lemma 5. Therefore, it issimilar to the case 0 < α < 1 that the correlation between1s is invoked by Si and 1s is invoked by Sj can be ignored, andthe proposed scheme can be proved to achieve the capacityupper-bound in order sense.

For the case α > 1, we consider two sources i and j, and thedistance between them satisfies d(i, j) > q

1−2ε2 n

2ε−12 , where

0 < ε < 1/3 is a small constant. Then, the nodes in thenetwork can be divided into 3 parts.

• Part 1: Defined as set P1 = k|Ranki(k) < q1−εnε.• Part 2: Defined as set P2 = k|Rankj(k) < q1−εnε.• Part 3: Defined as set P3 = k|Ranki(k) >q1−εnε and Rankj(k) > q1−εnε.

Since the nodes are uniformly and randomly distributed inthe network, it is obvious that Ranki(k) > q1−εnε when k ∈P1 and Rankj(k) > q1−εnε when k ∈ P2 with probability1. Therefore, the number of nodes in Fi ∪ Fj can be upper-bounded by

|Fi ∪ Fj |= |Fi ∪ Fj ∪ P1|+ |Fi ∪ Fj ∪ P2|+ |Fi ∪ Fj ∪ P3|

<

q1−εnεXX=1

q2PRanki(k) = XPRankj(k) = q1−εnε

+

q1−εnεXX=1

q2PRanki(k) = q1−εnεPRankj(k) = X

+

nXX=q1−εnε+1

q2PRanki(k) = XPRankj(k) = X

<c18q qn

εα+ c19q

qn

εα+ c20q

qn

(2α−1)ε

=o(q),

(105)

where c18, c19, c20 are constant according to Lemma 5.Therefore, the correlation between 1s is invoked by Si and

1s is invoked by Sj can be ignored if d(i, j) > q1−2ε

2 n2ε−1

2 .According to this result, we divide all of the nodes intoq1−3εn3ε groups with the same nodes number, and the distancebetween nodes in the same group is larger than q

1−2ε2 n

2ε−12 .

Owing to the uniform and independent distribution of nodes,this division exists with probability 1 when n goes to infinity.Hence, we define Gu as Gu =

Pi is in group u

1s is invoked by Si

and Gu as the corresponding sum of i.i.d. Bernoullian randomvariables with mean m

(i)3 = min3R0n

η‖EMSTi‖r(n), 1,where η ≥ 0 is a small constant.

Considering the case that there are some links belonging toboth EMSTi and EMSTj , s is more likely to be invokedby Sj if it is invoked by Si. On the other hand, if thecommon links of EMSTi and EMSTj do not invoke s,1s is invoked by Si and 1s is invoked by Sj are independent fromeach other. Therefore, the correlation coefficient between

13

1s is invoked by Si and 1s is invoked by Sj is non-negative. Thus,the correlation coefficients between different Gu are also non-negative. Consequently, we can obtain the following relation

PrN(s) > 2E[N(s)] =Pr¦X

Gu > 2E[N(s)]©

<Pr

maxuGu >

2E[N(s)]

q1−3εn3ε

=1− Pr

¨\u

Gu <2E[N(s)]

q1−3εn3ε

«

<Xu

Pr

Gu <

2E[N(s)]

q1−3εn3ε

<q1−3εn3ε(e)

−nq

1−3εm3/3,

(106)

According to (81),

m3 ≥ min3R0

Èlog n

qn

minα,22

nη2 , 1, (107)

and therefore

Pr

¨\s

¦N(s) < 6R0n

1+η‖EMST‖r(n)©«

≥1−Xs

PrN(s) > 2E[N(s)]

≥1− q1−3εn1+3εe−nq

1−3εmin3R0

√logn( qn )

minα,22 n

η2 ,1/3

=1− q1−3εn1+3εe−ncβ(1−3ε)

min3R0

√lognn

cβcαnη2 ,1/3,

(108)

where β > 1, cβ = minβ − 1, 1 and cα = minα,22 . We

select ε = 2−α12 and η = 0 when 1 < α < 2, and ε = η

12 andη > 0 when α ≥ 2, and therefore the value of (108) becomes1 when n goes infinity. Thus, the proposed scheme achievesthe capacity upper-bound when 1 < α < 2, and there is a gapnη between them when α ≥ 2. Consequently, the theorem isproved.

VI. DISCUSSIONS

In this session, we discuss about the difference of capacitybetween our social networks and traditional multicast ad hocnetworks. The authors in [2] show that the capacity of thenetworks with uniformly and randomly selected destinationsis Ctraditional(n) = Θ

1

Nq(n)nr2(n)

, where Nq(n) is the

average total number of hops of a multicast session, and r(n)is the transmission range. Therefore, the difference of capacityis only caused by the difference of Nq(n). Moreover, [2] showsthat

Nq(n) =

¨Θ‖EMST‖r(n)

q = o(r−2(n)),

Θr−2(n)

q = Ω(r−2(n)),

(109)

where q is the number of destinations. The similar results canbe found in [3] and [4]. Since the number of destinations iscertain before transmission in their networks, we compare thecapacity of our network with theirs under the assumption thatthe number of destinations is certain. The capacity ratio isdefined as

Rq =C

(q)social(n)

C(q)traditional(n)

, (110)

where C(q)social(n) is the capacity of social networks when des-

tination number is q, and C(q)traditional(n) is the corresponding

capacity for traditional ad hoc networks in [2]- [4].Comparison for 0 ≤ α < 1. In this case, ‖EMST‖ = Θ(

√q)

in both of our networks (59) and traditional multicast ad hocnetworks [2]. Therefore, the capacity ratio is

Rq =

(Θ(1) q = O

n

logn

,

Θ(1) q = ω

nlogn

.

(111)

From the expression of PDF of friends in Theorem 1, it can befound that the concentration degree of friends is determinedby α. In this case, the value of α is small, and thus the PDF issimilar to the uniform distribution in order sense. Therefore,the order of ‖EMST‖ in social network is the same as thetraditional ad hoc networks.Comparison when α = 1. In this case, ‖EMST‖ = Θ(

√q)

in traditional multicast ad hoc networks. However, in our socialnetworks, according to Theorem 2, the ‖EMST‖ is smaller.Thus, the capacity ratio is

Rq =

8><>:Ω

logn√q

q = O(logn),

Θ√

logn

q = ω(logn), q = o

nlogn log logn

,

Θ(1) q = Ω

nlogn log logn

.

(112)When α = 1, the majority of nodes are centralized withina small region close to the source. Due to the concentration,the average distance between nodes is decreased comparingwith the uniform distribution. Furthermore, the ‖EMST‖ isdecreased, and the capacity is increased.Comparison for α > 1. In this case, we use Θ(·) insteadof Θ(·) to make the result brief. Firstly, ‖EMST‖ = Θ(

√q)

in traditional multicast ad hoc networks. Based on the corre-sponding ‖EMST‖ in (60), the capacity ratio is

Rq =

8>>>>>><>>>>>>:

Ωq

12−αnα−1

qi = O(n

α−1α ), 1 < α ≤ 3

2,

Θ

nq

α−12

qi = ω(n

α−1α ), 1 < α ≤ 3

2,

Ωq

12−2α√n

qi = O(nα−1α ), 3

2< α < 2,

Θ

nq

α−12

qi = ω(n

α−1α ), 3

2< α < 2,

ΘÈ

nq

α ≥ 2.

(113)

The capacity ratio is larger than other two cases because thatthe α is sufficiently large and the destinations are centralized.Thus, most of the destinations are distributed around thesource, which causes large Rq .

Figure 4 illustrates the variances of Rq versus α whenq = Θ

n

23

in order sense. In Figure 4, when α = 0, the

network is the traditional ad hoc network. When 0 < α < 1,the distribution of destination is similar to uniformly distribu-tion, and therefore the corresponding capacity ratio is Θ(1).When α = 1, the destinations distribute closer to the sourcethan that of case 0 ≤ α < 1, which results in a smaller‖EMST‖. Therefore, the capacity ratio becomes Θ(log n)which is greater than Θ(1). Moreover, when 1 < α < 2, theconcentration degree of destinations becomes higher when αis larger. Hence, the capacity ratio increases with α. However,

14

when α ≥ 2, since the density of destinations is limitedby the node density of all the n nodes in the network, theconcentration degree of destinations does not increase with αin this case. Thus, the corresponding capacity ratio does notchange with α here. In Figure 4, we ignore poly-logarithmicfactors when α > 1.

a

1 20

qR 2

3( )q n= Q

1log n

1.5

112n

16n

Fig. 4: The variances of Rq versus α. When α = 0, thenetwork is the traditional ad hoc network

VII. CONCLUSION

In this paper, we introduce the two-layer network model,which includes social layer and networking layer. In the sociallayer, the social group size of each source node is modeledas power-law distribution model. Moreover, the rank-basedmodel is utilized to describe the relation between source anddestinations in the networking layer. In this model, the PDF ofdestinations is derived and verified by numerical simulations.Based on the PDF, we analyze the capacity bound of thenetworks and propose an EMST-based scheme to achieve thecapacity in most cases. Finally, the social network capacity iscompared with the traditional ad hoc networks.

VIII. ACKNOWLEDGEMENT

This work is supported by NSF China (No.61325012,61271219, 61221001, 61428205);China Ministry of Educa-tion Doctor Program(No.20130073110025); Shanghai BasicResearch Key Project (12JC1405200, 11JC1405100);ShanghaiInternational Cooperation Project: (No. 13510711300).

APPENDIX

We prove the Lemma 2 based on the following lemma. Theproof of Lemma 7 for the case fn(x) related with n is similarto the proof of Theorem 3 in [21], and we do not show it heredue to the space limitation.

Lemma 7 In the network with size [0, 1]d, we denote the EM-STs of nodes with distribution fn(x) and gn(x) as EMSTfand EMSTg , respectively, where fn(x) is a continuous dis-tribution and gn(x) is a blocked distribution. The total lengthof EMSTf can be expressed as

limn→∞

‖EMSTf‖n−d−1dR

Rd fn(x)d−1d dx

= c(d), (114)

when the following two requirements are satisfied, where c(d)is a constant.

• Requirement 1: The total lengths of EMSTf andEMATg satisfy

limn→∞

‖EMSTf‖ − ‖EMSTg‖

|i : xfi 6= xgi |d−1d

≤ B, (115)

where B is a constant, xfi and xgi are the sets of thenode positions of EMSTf and EMATg , respectively.

• Requirement 2: The total length of EMSTg can beexpressed as

limn→∞

‖EMSTg‖n1−ddR

Rd gn(x)d−1d dx

= c(d). (116)

The proof of Lemma 2.5 in [21] can be used to prove thatthe Requirement 1 in Lemma 7 is satisfied when the Condition1 in Lemma 2 holds, and we do not show it here due tothe space limitation. Therefore, it only remains to prove thatRequirement 2 is also satisfied when the two conditions ofLemma 2 hold.

Firstly, we calculate the upper-bound of ‖EMSTg‖. Con-sidering a d-dimensional hypercube network with size S (sidelength S

1d ), it has been proved in [21] that if there are n nodes

uniformly distributed in it, the total length of the correspondingEMST is c0(d)n

d−1d S

1d with probability 1 when n goes to

infinity, where c0(d) is a constant. Hence, defining

X = i : nγi|ξi| = Θ(1), Y = i : nγi|ξi| = ω(1), (117)

the upper-bound of ‖EMSTg‖ can be expressed as

lim supn→∞

‖EMSTg‖M + c0(d)

Pi∈Y

(nγi|ξi|)d−1d |ξi|

1d +

Pi∈X

nγi|ξi|√d|ξi|

1d

≤ 1,

(118)

where |ξi| is the size of ξi, M is the cost needed to unite allof the parts ξi, c0(d)

Pi∈Y

(nγi|ξi|)d−1d |ξi|

1d is the total length

of EMST of the nodes in each part which belonging to Y ,and

Pi∈X

nγi|ξi|√d|ξi|

1d is the upper-bound of it for X . When

two adjacent parts ξi and ξj are connected, the length ofedge between them is no greater than

√d(|ξi|

1d + |ξj |

1d ).

Since there are at most ζ adjacent parts for each ξi, theM in (118) satisfies M ≤ ζ

√dPi∈X∪Y |ξi|

1d when each

part only connects to its adjacent parts. Defining a constant

$ = maxi∈X

§ √d(nγi|ξi|+ζ)

c0(d)(nγi|ξi|)d−1d

ª, we can obtain

limn→∞

‖EMSTg‖Pi∈X∪Y

(nγi|ξi|)d−1d |ξi|

1d

≤ max2, $c0(d) (119)

Afterwards, we analyze the lower-bound of ‖EMSTg‖. LetDi denote the set of edges e in EMSTg such that both twoendpoints of e are in ξi. Let Vi denote the set of nodes in ξiwhich are jointed by an edge in EMSTg with one anotherendpoint out of ξi. Thus, the edges in Di and the EMST of Viconnect all the nodes in ξi. Based on the Lemma 2.2 in [21],

15

for any i ∈ Y , the following relation holds when n→∞

c0(d)(nγi|ξi|)d−1d |ξi|

1dP

e∈di|e|+ c0(d)(|Vi|)

d−1d |ξi|

1d

≤ 1, (120)

where |Vi| is the number of nodes in Vi, |e| is the length ofe, c0(d)(nγi|ξi|)

d−1d |ξi|

1d is the total length of EMST among

nodes in ξi,Pe∈di|e| is the total length of edges with two

endpoints in ξi, and the total length of EMST of the nodesconnecting with the nodes outside ξi can be bounded byc0(d)(|Vi|)

d−1d |ξi|

1d .

In order to bound |Vi|, we consider a d-dimensional hyper-cube ξ

′

i with side length li = 12 |ξi|

1d centered at the center

point of ξi. For any node v′

within ξ′

i , if v′ ∈ Vi, we assume

that it is connected with a node v out of ξi by an edge ev′v ,which is denoted as the replaceable edge. Since ξ

′

i is withinξi and there are ω(1) nodes in ξi − ξ

′

i (since i ∈ Y), anappropriate v∗ satisfying ev∗v < ev′v can be found as arelay with probability 1. Therefore ev′v can be replaced byev′v∗ + ev∗v , and thus ev′v∗ + ev∗v < 3ev′v . After replacingall the replaceable edges of each part (i ∈ Y), the EMSTgbecomes a new graph G. It should be noticed that there maybe some circuits after replacing edges, and thus we denote it asa graph instead of a spanning tree. Since ev′v∗+ev∗v < 3ev′v ,the total length of G satisfies ‖G‖ < 3‖EMSTg‖. Moreover,the relation (120) also holds for the graph G when i ∈ Y , i.e.,

c0(d)(nγi|ξi|)d−1d |ξi|

1dP

e∈d(G)i

|e|+ c0(d)(|V (G)i |)

d−1d |ξi|

1d

≤ 1, (121)

where n→∞, and d(G)i , V (G)

i are similar to those in (120).

Moreover, for any i ∈ X , let Ji denote the sum length ofedge(s) of G within ξi. It should be noted that if part of anedge is within ξi, only the part inside ξi is considered. Sincenodes in ξi are uniformly distributed, Ji|ξi|−

1d is greater than

a constant with probability 1. Thus, for any i ∈ X ,

c0(d)(nγi|ξi|)d−1d |ξi|

1dJ−1

i ≤ $∗, (122)

where $∗ = maxi∈X

§c0(d)(nγi|ξi|)

d−1d |ξi|

1d

Ji

ªis a constant.

Hence, according to (121), (122), the following relation holdsXi∈X∪Y

c0(d)(nγi|ξi|)d−1d |ξi|

1d

≤Xi∈Y

Xe∈d(G)

i

|e|+$∗Xi∈X

Ji +Xi∈Y

c0(d)(|V (G)i |)

d−1d |ξi|

1d

≤ max3, $∗ lim infn→∞

‖EMSTg‖+Xi∈Y

c0(d)(|V (G)i |)

d−1d |ξi|

1d ,

(123)

when n → ∞. Furthermore, based on the edge replacementand the law of large numbers, for any i ∈ Y ,

nγi|ξi| − |V (G)i | ≥ |v : v is within ξ

′i| > 2−d(1−∆0)nγi|ξi|,

(124)

with probability 1 when n → ∞, where 0 < ∆0 < 1 is anyconstant. Hence, (123) can be further transformed as

lim infn→∞

‖EMSTg‖Pi∈X∪Y

(nγi|ξi|)d−1d |ξi|

1d

≥ µc0(d), (125)

where µ =

1−(1−2−d(1−∆0))

d−1d

max3,$∗ < 1 is a constant.

Consequently, based on (119) and (125), it is obvious that

limn→∞

‖EMSTg‖n1−ddR

Rdgn(x)

d−1d dx

= limn→∞

‖EMSTg‖n1−ddP

i∈X∪Y(γi|ξi|)

d−1d |ξi|

1d

= c(d),

(126)

where µc0(d) ≤ c(d) ≤ max2, $c0(d) is a constant. Thus,the Requirement 2 of Lemma 7 is satisfied. According toLemma 7, Lemma 2 is proved.

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Yi Qin received the B.S. and M.S. degrees inelectronic engineering from Shanghai Jiao TongUniversity, Shanghai, China, in 2009 and 2012,respectively. He is currently working toward thePh.D. degree with Shanghai Jiao Tong University,Shanghai, China.

His research interests include D2D communica-tion, D2D discovery, and Ad Hoc network scalinglaws study.

Riheng Jia received his B. E. degree in Electronicsand Information Engineering from Huazhong Uni-versity of Science and Technology, China, in 2012.He is currently pursuing the PHD degree in Elec-tronic Engineering in Shanghai Jiao Tong University.His research of interests are in the area of wirelessnetworks and information-centric networking.

Jinbei Zhang received his B. E. degree in ElectronicEngineering from Xidian University, Xi’an, China,in 2010, and is currently pursuing the Ph.D. degreein electronic engineering at Shanghai Jiao TongUniversity, Shanghai, China. His current researchinterests include network security, capacity scalinglaw and mobility models in wireless networks.

Weijie Wu is an Assistant Professor in School ofElectronic, Information and Electrical Engineering,Shanghai Jiao Tong University. Before that, he was aresearch fellow working with Dr. Richard T.B. Ma inNational University of Singapore, and a postdoctoralfellow working with Prof John C.S. Lui at TheChinese University of Hong Kong. He obtainedhis Ph.D. degree in computer science from TheChinese University of Hong Kong in August 2012,and Bachelor’s degree in electronic & informationscience and technology from Peking University in

July 2008. When he was a Ph.D. student, he spent two months at NationalUniversity of Singapore working as a research intern. His current researchinterests are in computer networks from mathematical modelling, data an-alytics, and economic perspectives. In particular, he is recently interestedin network science (e.g., online social networks, large scale network withdata implications, etc.), network economics (e.g, game theoretic analysis oncommunication networks, pricing and incentive design in network applica-tions, etc.), and network optimization (e.g., resource allocation and pricing incloud computing, information centric network, and P2P systems). His personalinterests include table-tennis, badminton and hiking.

Xinbing Wang received the B.S. degree (with hon-s.) from the Department of Automation, ShanghaiJiaotong University, Shanghai, China, in 1998, andthe M.S. degree from the Department of Comput-er Science and Technology, Tsinghua University,Beijing, China, in 2001. He received the Ph.D.degree, major in the Department of electrical andComputer Engineering, minor in the Departmentof Mathematics, North Carolina State University,Raleigh, in 2006. Currently, he is a professor inthe Department of Electronic Engineering, Shanghai

Jiaotong University, Shanghai, China. Dr. Wang has been an associate editorfor IEEE/ACM Transactions on Networking and IEEE Transactions on MobileComputing, and the member of the Technical Program Committees of severalconferences including ACM MobiCom 2012, ACM MobiHoc 2012-2014,IEEE INFOCOM 2009-2014.