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Measuring Transport Difficulty of Data Dissemination inLarge-Scale Online Social Networks:

An Interest-Driven Case

Cheng Wang, Zhenzhen Zhang, Jieren Zhou, Yuan He, Jipeng Cui, Changjun JiangDepartment of Computer Science and Technology

Tongji University, Shanghai, Chinachengwang@tongji.edu.cn

ABSTRACTIn this paper, we aim to model the formation of data dissem-ination in online social networks (OSNs), and measure thetransport difficulty of generated data traffic. We focus on ausual type of interest-driven social sessions in OSNs, calledSocial-InterestCast, under which a user will autonomouslydetermine whether to view the content from his followeesdepending on his interest. It is challenging to figure out theformation mechanism of such a Social-InterestCast, since itinvolves multiple interrelated factors such as users’ socialrelationships, users’ interests, and content semantics. Wepropose a four-layered system model, consisting of physi-cal layer, social layer, content layer, and session layer. Bythis model we successfully obtain the geographical distribu-tion of Social-InterestCast sessions, serving as the precondi-tion for quantifying data transport difficulty. We define thefundamental limit oftransport loadas a new metric, calledtransport complexity, i.e., theminimum requiredtransportload for an OSN over a given carrier network. Specifically,we derive the transport complexity for Social-InterestCastsessions in a large-scale OSN over the carrier network withoptimal communication architecture. The results can actas the common lower bounds on transport load for Social-InterestCast over any carrier networks. To the best of ourknowledge, this is the first work to measure the transport dif-ficulty for data dissemination in OSNs by modeling sessionpatterns with the interest-driven characteristics.

KeywordsOnline social networks, data dissemination, transport load,fundamental limits, scaling behavior

1. INTRODUCTIONAs social networking services are becoming increasingly

popular, online social networks (OSNs) play a growing rolein individuals’ daily lives. The user population of OSNs hasgrown drastically in recent years. According to the researchproceeded by Statista1 demonstrated that the number of so-cial network users worldwide in 2014 had reached 1.87 bil-1One of the largest statistics portals, http://www.statista.com/

lion and estimated that there will be around 2.72 billion so-cial network users around the globe in 2019 [1]. In addition,people are spending more and more time on OSNs. For ex-ample, in January 2015, GlobalWebIndex2 showed that theaverage user spends 1.72 hours per day on social platforms,which represents about 28 percent of all online activity [2].What’s more, various kinds of social applications are con-stantly emerging, which are rendering an increasing userpopulation of OSNs and providing users with more typesof content to choose from, e.g., audios, videos, and pictures.In addition, OSNs are covering a wider range around theworld. All of the factors involved above are resulting in theheavy load imposed on the carrier communication networkof OSNs. Furthermore, the load in the OSNs will increasecontinually with the expansion of OSNs.

Over a long period of time, this growth will give rise tothe limitation of Internet’s bandwidth, so practically mea-suring the load imposed by OSNs enjoys a crucial mean-ing. To measure such a load, i.e., the load imposed on thecarrier communication network by the OSN, we first intro-duce a metric calledtransport load. It is defined as theproduct of two key factors: data generating rate at usersand transport distance of messages. The former depends onthe format of message (e.g., text, picture, and video, etc.)and the performance request in terms of quality of service(e.g., throughput, latency, and deliver ratio, etc.). The lat-ter is directly determined by the geographical distributionof dissemination sessions and the communication architec-ture of network. Both parameters are critical for measuringthe transport capacity of networks. To be specific, it is de-fined as the product of bits and the transport distance overwhich the data is successfully transported from the sourceto the intended destinations. In this work, we further de-fine the fundamental limit of such transport load astransportcomplexity, i.e., theminimum requiredtransport load for anOSN over a given carrier communication network. We notethat, unlike some classical performance metrics, e.g., net-work capacity, the transport complexity is a metric to definethe fundamental transport difficulty of a specific data com-

2A market research firm running world’s largest market researchstudy on the digital consumer, https://www.globalwebindex.net/

Transport Load

Transport Capability

How wella network can do

for a given application

of data transportation?

of A Network

Transport Capacity

Maximum AchievableTransport Throughput

Transport Difficulty

How difficultfor a given network

to make an application

of data transportation?

of An Application

Transport Complexity

Minimum Required

Figure 1: Transport Complexity vs. Transport Capacity.They are the typical and intuitive metrics of the trans-port difficulty of an application and the transport capa-bility of a network, respectively. The transport capabil-ity is the capability of a network in terms of data trans-portation, while the transport difficulty is the inherentdifficulty of an application in terms of data transporta-tion.

munication applications instead of the transport capabilityof a given network for specific applications. Regarding theessential difference between transport capacity of a networkand transport complexity of a data transport application, wegive an explicit explanation with the help of the illustrationin Figure 1.

For computing the transport load, the first thing is to combrightly the procedure of a data dissemination in OSNs. Fromour perspective, the session can be divided into two suc-cessive phases:Passive Phaseand Initiative Phase. In thepassive phase, a source user holding a message sends thedigest of this message to all his followers. The followerspassively receive this digest. This process of such a dis-semination just acts like abroadcastin all the followers ofthis source user. We call this processSocial-BroadCast, andsessions generated in this process are straightforward calledSocial-BroadCast sessions. In the initiative phase, accordingto the interest, a user will autonomously determine whetherto download the complete message based on the digest of amessage from who he follows, [40]. We define such a dis-semination processSocial-InterestCastand sessions gener-ated in this process areSocial-InterestCast sessions. In ourwork, the Social-InterestCast is defined by taking into ac-count specific user interests and their effects on session gen-eration. We give an intuitive explanation as follows: Thereexists usually auser-message mappingbetween the user setand message set, [43]. That is, when a user broadcasts amessage to others, only the users whose interests are con-sistent with the topic of the message can be the potentialdestinations. For example, if a user broadcasts a video mes-sage characterized by several words (a digest or a title) inhis Facebook, only the followers who are interested in thismessage will open the video file. It is convincing that this

behavior occurs due to an underlying user-message map-ping. In fact, the transport load in the former process, i.e.,the transport load for the Social-BroadCast has been inves-tigated in [37], then our focus of this work is to model theformation of a Social-InterestCast. Accordingly, we proposea four-layered model consisting of the physical layer, sociallayer, content layer, and session layer, as illustrated in Fig-ure 2. Compared with the three-layered model in [37], thismodel introduces acontent layerwhere the relationship linksare defined as the semantic similarity among the messages.We take an example shown in Figure 2 to explain this pro-cedure: A user delivers a “Message 1”. Then, all his fourfollowers can receive the glance (or say abstract) of “Mes-sage 1”. Finally, only two followers are filtered through thecontent layer to act as the valid destinations due to their in-terests to “Message 1”.

After the preparations made above, we compute the trans-port load. Recall its definition, we first need to investigatethe complex geographic characteristics of data dissemina-tion sessions in OSNs, i.e., the spacial distribution of traf-fic sessions (the location distribution of sources and destina-tions).

We adopt the following three steps (as shown in Figure 3)to get the spacial distribution of Social-InterestCast sessionsdepending on users’ geographical distribution: Firstly, we model the spatial distribution of social rela-

tionships, i.e., the geographical distribution of followers, byinvestigating the correlations between user’s social relation-ship formation and geographical distribution. We adopt thepopulation-based modelin [37] for the advantages in realis-tic and analytical aspects. Secondly, we build the user-content interest mapping

by matching the topics of content and interests of users. Un-der the Social-InterestCast, for measuring the transport com-plexity, it is important to estimate how many followers of asource user will be interested in a given message and make adecision to view it. We divide this problem into two cases interms of the dependency among followers’ decisions and theattractivity difference of message content. To be specific,when the decisions of a source user’s followers are non-independent, Matthew Effect [30] should be considered dueto the preferential attachment of some messages that have at-tracted more followers; when the attractivity of messages isof significant difference, the preferential attachment of somepopular messages comes on the stage. Furthermore, com-bining with the experimental results based on Foursquareusers’ dataset [11], we get that: Under both cases above, fora source user, sayvk, we reasonably assume that the numberof destinations of a session fromvk follows a Zipf’s distri-bution whose parameters depend on the degree of nodevk. Thirdly, we model the spatial distribution of traffic ses-

sions, i.e., the geographical distribution of session sourcesand destinations, by combining the effects of both users’ so-cial relationships on the social layer and the user-contentin-terest mapping across the social and content layers.

Physical

Layer

Social

Layer

Session

Layer

Content

Layer

Physical

Proximity

Social

Relationship

Data

Flow

Interest

Mapping

Content

Correlation

1

3

2

Figure 2: Four-layered Architecture and Social-InterestCast.

Session Layer

Content Layer

Social Layer

Physical Layer

Step 2

Geographical Distributionof Social Relationships

User ContentInterest Mapping

Geographical Distribution

of Traffic Sessions

Step 3Step 1

Figure 3: Three Steps for Modeling The Generation ofTraffic Sessions.

Based on these three steps, we can obtain the geographi-cal distribution of traffic sessions, and thereby compute thebound on the aggregate transport distance over which thedata is successfully transported from the source to the in-tended destinations. Under a realistic assumption that ev-ery source sustains a data generating rate of constant or-der, our result demonstrates that the transport complexityforSocial-InterestCast in an OSN over a carrier network withan optimal communication architecture varies in the range[Θ(n),Θ(n2)

]depending on the clustering exponents of re-

lationship degree, relationship formation and disseminationpattern, wheren is the user number of the OSN.

The aforementioned results for Social-InterestCast clarifythe differences from those for data dissemination in conven-tional communication networks, and can serve as a valuablemetric to measure network performance and the difficulty ofdata dissemination in large-scale networks. Furthermore,ifa specific carrier communication network is introduced, ourresults on transport distances can also play an important role

in analyzing some system performances, e.g., network ca-pacity and latency.

To the best of our knowledge, this is the first work to mea-sure the transport difficulty for data dissemination in OSNsfor modeling session patterns by taking into account the interest-driven characteristic and users’ behaviour in real-world OSNs.

The rest of this paper is organized as follows. Firstly, weshow the related work in Section 2, and give the metric oftransport difficulty in Section 3. In Section 4, we proposeour system model of Social-InterestCast. In Section 5, wederive the transport complexity for the Social-InterestCast.Finally, we draw a conclusion and make a discussion on ourfuture work in Section 6.

2. RELATED WORKOnline social networks (OSNs) provide a platform for hun-

dreds of millions of the Internet users worldwide to produceand consume content. Users in OSNs have the access tothe unprecedented large-scale information repository [21].Moreover, OSNs play an important role in the informationdiffusion by increasing the spread of novel information anddiverse viewpoints, and have shown their power in manysituations [10]. There are some representative topics thathave been extensively studied in the research community ofOSNs, such as detecting popular topics [29], digging po-tential popularity of contents [7], modeling information dif-fusion [23, 31], identifying influential spreaders, leaders orfollowers [33], presenting influence mechanisms [26], max-imizing the spread of an information epidemic [24], predict-ing the properties/signs of links [42] and the missing prefer-ence of a user [27], estimating the proximity of social net-works [34], and exploring security issues [36], and so on.However, most existing work mainly focused on the infor-mation diffusion scheme in overlay relationship networks ofusers in social networking sites/services (SNSs), [41].

Meanwhile, as SNSs become increasingly popular for in-formation exchange, the traffic generated by social appli-cations rapidly expands [4]. A report of Shareaholic [5]showed that, between November 2012 and November 2013,social media referral traffic from the top five social mediasites increased by111%while search traffic from the top fivesearch engines had decreased by6%. Therefore, besides theanalysis of information diffusion schemes in overlay socialnetworks [10,24,29,31], and the gain of social relationshipsin terms of the information dissemination [17, 39], an in-depth understanding of the impact of increasing traffic gen-erated by OSNs on carrier communication networks, e.g.,the Internet, is convincingly necessary for evaluating currentOSNs systems, optimizing network architectures and the de-ployment of servers for OSNs, and even designing futureOSNs [16]. To address this issue, we need to propose prac-tical modeling and effective analytic methods for contentdistribution of OSNs implemented in carrier communicationnetworks, since OSNs change both information propagationschemes and traffic session patterns in communication net-

works due to the involvement of overlay social relationships,users’ preferences and decisions, [20]. Accordingly, in thispaper, we aim at modeling content distribution [8] in OSNs,and measuring its transport difficulty imposed on the carriercommunication networks of OSNs.

The most relevant work investigating the load imposed onthe carrier communication network is [37], where the met-ric called transport load(or traffic load ) was proposed toquantify such a load. In [37], only a theoretical lower boundon transport load was derived without sufficiently clarifyingthe role of this result in measuring the transport difficultyof data dissemination for specific online social networkingapplications. We state that transport difficulty for a specificapplication should be an intrinsic property of this applica-tion when the carrier network is given. Consequently, themetric for transport difficulty should be a fundamental limitof a certain metric on transport burden. From such a per-spective, we define the fundamental limit oftransport loadas a new metric calledtransport complexity, i.e., themini-mum requiredtransport load for an OSN over a given car-rier network. Besides this, comparing with [37], we makesome significant improvements in this work: In [37], the au-thors proposed a three-layered model to formulate data dis-semination sessions for social applications in OSNs. Thesession generation is simply modeled as Social-BroadCast,where the source broadcasts messages to all of its followers,and all followers have to be the passive destinations. Appar-ently, they neglected the important features of session gen-eration in OSNs, i.e., the fact that the generation of trafficsessions depends on both users’ social relationships and theuser-content interest mappings. Accordingly, in this work,we focus on an interest-driven session pattern, called Social-InteretCast. To model its formation, we introduce acontentlayer as an interest-based filter to build interest links fromusers to messages, and propose afour-layered system modelas illustrated in Figure 2.

3. METRIC OF TRANSPORT DIFFICULTYConsidering an online social network (OSN), denoted by

N, consisting ofn users, we denote the set of all users byU = uini=1. Let a subsetS = uS,kns

k=1 ⊆ U denotethe set of all sources, where|S| = ns. Denote a data dis-semination session from a sourceuS,k by an ordered pairDS,k =< uS,k,DS,k >, whereDS,k is the set of all desti-nations ofuS,k.

In this work, we intend to investigate the transport diffi-culty for data dissemination in OSNs, i.e., the load on thecarrier communication networks generated by a specific so-cial applications. To quantify such a load, we introduce ametric from [37], calledtransport load, which depends ontwo factors:data requested rateanddata transport distance.

Data Requested Rate:Data requested rate is determinedby QoS (quality of service) of the application. For data dis-semination applications in OSNs, the QoS of data transportapplications is usually set up according tothe generating

rate of content at source users, i.e., the so-calleddata arrivalrate. Moreover, data requested rate is generally defined as acertain portion of data arrival rate. In other words, a portionof data arriving at the source user is requested to be success-fully disseminated. Therefore, we reasonably assume thatthe data requested rate has the same order as the data ar-rival rate.

The temporal behavior of messages arriving at a user in anOSN has been addressed by analyzing some real-life OSNs,[12,32]. For example, Perera et al. [32] developed a softwarearchitecture that uses a Twitter application program interface(API) to collect the tweets sent to specific users. They indi-cated that the arrival process of new tweets to a user can bemodeled as a Poisson Process. In this paper, we just take itas an empirical argument for assuming the data arrival for auser as a data source follows a Poisson Process, [32]. In ourwork, for each sessionDS,k =< uS,k,DS,k >, we simplyset thedata requested rateto be a portion of the data arrivalrate. Then, we can denote the data requested rate by a vectorΛS = (λS,1, λS,2, · · · , λS,ns

), whereλS,k is the rate of aPoisson Process at useruS,k (for k = 1, 2, · · · , ns).

In practice, the data arrival rate is dispensable on the scaleof the specific OSN, i.e., the value ofn, although the dataarrival rate depends on many factors, such as the specificform and quality of social services. Combining the facts thatthedata requested rateis a certain portion of the data arrivalrate, we can make a reasonable and practical assumption thatλS,k = Θ(1) for k = 1, 2, · · · , ns.

In this work, we aim to analyze the fundamental limits onthe transport load for data dissemination in OSNs accordingto the network size. Therefore, it is appropriate to note atthis point that the specific distribution of data requested ratehas no impact on the results (in order sense) as long as itholds thatλS,k = Θ(1) for k = 1, 2, · · · , ns. This is whywe do not make an intensive study of the specific distributionof data requested rates.

Data Transport Distance: The data transport distanceis comprehensively determined by traffic session pattern ofthe application, communication network architecture, andtransmission schemes. For a given transmission scheme in agiven OSNN, saySN, define a vector

DS(SN) = (dS,1(SN), dS,2(SN), · · · , dS,ns(SN)) ,

wheredS,k(SN) represents thetransport distanceover whichthe message for sessionDS,k is successfully transported fromthe sourceuS,k to all destinations.

Transport Load and Transport Complexity: In the OSNN, given a specific carrier communication network, definethe transport load for a dissemination session, sayDS,k, as

LN,S(DS,k) = λS,k · dS,k(SN).

Furthermore, the aggregated transport load for disseminationsessions from all sources inS can be defined as

LN,S = minSN∈S ΛS ∗DS(SN), (1)

whereS is the set of all feasible transmission schemes, and∗ is an inner product.

Based on the definition of transport load, we further intro-duce thefeasible transport load.

DEFINITION 1 (FEASIBLE TRANSPORTLOAD). For asocial data dissemination with a set of social sessionsDS =DS,knk=1, we say that the transport loadLN,S is feasibleifand only if there exists an appropriate transmission schemewith a communication deployment, denoted bySN, such thatit holds thatDS(SN)∗ΛS ≤ LN,S ensuring that the networkthroughput ofΛS is achievable.

Based on the definition of feasible transport load, we fi-nally define thetransport complexity.

DEFINITION 2 (TRANSPORTCOMPLEXITY). We say thatthe transport complexityof the class of random social datadisseminationsDS is of orderΘ(f(n)) bit-meters per sec-ond if there are deterministic constantsc1 > 0 andc2 < ∞such that: There exists a communication architectureN andcorresponding transmission schemes such that

limn−→∞

Pr(LN,S = c1 · f(n) is feasible) = 1,

and for any possible communication architectures and trans-mission schemes, it holds that:

lim infn−→∞

Pr(LN,S = c2 · f(n) is feasible) < 1.

4. MODEL OF SOCIAL-INTERESTCASTFor each session, the geographical distribution of the source

and destination(s) plays a key role in generating the transportload. So it is critical to analyze the correlation between thespatial distribution of sessions and geographical distributionof users in online social networks (OSNs).

To address this issue, we propose a four-layered model,consisting of the physical layer, social layer, content layer,and session layer, as illustrated in Figure 2.

4.1 Four-Layered System Model

4.1.1 Physical Layer- Physical Network Deployment

The deployment of the so-called physical network can bedivided into two parts. The first is the geographical distri-bution of social users. The second is the communicationarchitecture of the carrier network.

Geographical Distribution of Social Users: We con-sider the random network consisting of a random numberN (with E(N) = n)3 users who are randomly distributedover a square region of areaS := n, whereE[N ] = n. Toavoid border effects, we consider wraparound conditions atthe network edges, i.e., the network area is assumed to be thesurface of a two-dimensional TorusO. To simplify the de-scription, we assume that the number of nodes is exactlyn,3Throughout the paper, letE[X] denote the mean of a random vari-ableX.

and denote the set of nodes byV = vknk=1, without chang-ing our results in order sense. We make a compromise in thegenerality and practicality of the geographical distribution ofsocial users, in order to concentrate on clarifying the impactsof users’ interest on the session formation. Specifically, wefollow the setting where all users are distributed according toa homogeneous Poisson point process, taking no account ofthe inhomogeneous property of the uneven population dis-tribution in real-life OSNs. The derived results are expectedto serve as the basis for investigating more realistic scenar-ios under the more practical but complex deployment mod-els, such as the Clustering Random Model (CRM) accordingto the shotnoise Cox process [6] and Multi-center GaussianModel (MGM) in [15].

Communication Architecture of Carrier Network: Foronline social networking services, we state that the carriernetwork is indeed the mobile Internet. This means that thereis hardly a uniform communication architecture, e.g., cen-tralized or distributed network architecture, practically char-acterizing the architecture of a real-world carrier network forOSNs. This is also the reason why we derive the transportcomplexity for Social-InterestCast sessions over the carriernetwork with optimal communication architecture. The re-sults can serve as the common lower bounds on transportcomplexity for Social-InterestCast sessions over any carriernetworks.

4.1.2 Social Layer- Social Relationship

To model the geographical distribution of relationships,we introduce thepopulation-distance-basedmodelfrom [37].In [37], Wang et al. provided a numerical evaluation for thissocial model based on a Brightkite users’ dataset [18]. Be-fore deciding to adopt this model, we complement an eval-uation based on another dataset from [22], i.e., a Gowallausers’ dataset. For the completeness, we include the evalua-tions based on these datasets in Appendix B.

Let D(u, r) denote the disk centered at a nodeu with ra-dius r in the deployment regionO, and letN(u, r) denotethe number of nodes contained inD(u, r). Then, for any twonodes, sayu andv, we can define thepopulation-distancefrom u to v asN(u, |u − v|), where|u − v| denotes theEuclidean distance between nodeu and nodev.

For completeness, we include the population-distance-basedmodel as follows: For a nodevk ∈ V , we construct its re-lationship setFk of qk (qk ≥ 1) follower nodes by the fol-lowing procedures: For a particular nodevk ∈ V , denotethe number of followers byqk, we assume that it follows aZipf’s distribution [28], i.e.,

Pr(qk = l) =

(∑n−1

j=1j−γ

)−1

· l−γ . (2)

Note that we accordingly give a numerical evaluation basedon Gowalla dataset [22] for the Zipf’s degree distribution inAppendix B.2.

Then, to model the geographical distribution of social re-

lationships, we make the position of nodevk as thereferencepointand chooseqk pointsindependently on the torus regionO according to a probability distribution with the followingdensity function:

fvk(X) = Φk(S, β) · (E[N(vk, |X − vk|)] + 1)−β

, (3)

where the random variableX := (x, y) denotes the positionof a selected point in the deployment region, andβ ∈ [0,∞)represents the clustering exponent of relationship formation;the coefficientΦk(S, β) > 0 depends onβ andS (the areaof deployment region), and it satisfies:

Φk(S, β) ·∫

O(E[N(vk, |X − vk|)] + 1)

−βdX = 1. (4)

Further, we determine the nearest followers for specificusers. LetAk = pki

qki=1 denote the set of theseqk points.Let vki

be the nearest node topki, for 1 ≤ i ≤ qk (ties

are broken randomly). Denote the set of theseqk nodes byFk = vki

qki=1. We call pointpkitheanchor pointof vki

,and define a setPk := vk ∪ Ak.

In this model, we can observe that the degree distributionabove depends on the specific network size, i.e., the num-ber of usersn. Note that the correlation between the users’degree distribution and geographical distribution shouldnotbe neglected, although we simplify it in this work. We willentirely address this issue in the future work. Throughoutthis paper, we useP(γ, β) to denote the population-distance-based model. Here,γ andβ denote the clustering exponentsof relationship degree and relationship formation, respec-tively. For their detailed meanings, please refer to Eq.(2)and Eq.(3), respectively.

4.1.3 Content Layer- User-Content Interest Mapping

In this work, we focus on the case where a user only viewthe messages within his interest. Then, modeling the map-ping between users and content is the first critical step forthe final generation of traffic sessions. The basic idea forthis issue is to build the interest links from users to mes-sages according to the similarity of users’ interests to mes-sage semantics. Specifically, we dig the user’s interest byabstracting the topics of all his posted messages.

4.1.4 Session Layer- Data Traffic Session

When observing users’ behaviour in real-life OSNs, thereexists a significant feature of traffic session in OSNs, i.e.,theusers’ intrinsic interest and subjective choice are often indis-pensable for forming the sessions, [25]. For all the commontraffic patterns in OSN, we focus on a typical data dissem-ination, called SocialCast, which can be divided into twosuccessive phases:Passive Phaseand Initiative Phase. Inthe passive phase, a source user holding a message sendsself-appointedly the digest of this message to all his fol-lowers. The followers passively receive this digest. Thisprocess just acts like a broadcast in all the followers of thissource user. This process is calledSocial-BroadCastin [37].In the initiative phase, according to the interest, a user will

autonomously determine whether to download the completemessage based on the digest of a message from who he fol-lows. We define such an interest-driven dissemination pro-cessSocial-InterestCast. The Social-InterestCast is definedby taking into account specific user interests and their ef-fects on session generation. Our focus in this work is Social-InterestCast.

4.2 Social-InterestCastIn real-life OSNs, there exists naturally a common phe-

nomenon: When a user broadcasts messages to others, thenumber of potential destinations is depending on underlyingrelationships among users (both the source and destinations),[25]. For a delivered message, to explore the characteristicsof source user and final destinations, we begin with studyingrelationships between users and messages. Then, it is com-mon that if a user broadcasts a message characterized by sev-eral words (abstract/digest) in his Facebook, only the userswhose interests are consistent with the topic of the messagecan be the final destinations. We say that this behavior oc-curs due to an underlyinguser-message mappingbetweenuser set and message set. For such type of session, we callit Social-InterestCast, which is different from Unicast in [9]and Social-BroadCast in [37]. In the following, we firstly digthe distribution of destinations of Social-InterestCast ses-sions based on a real-life dataset of Foursquare [11].

4.2.1 Analyzing Social-InterestCast Formation

In this part, we aim to analyze the distribution of destina-tion number for Social-InterestCast by exploring the sessionformation mechanism over a real-life Foursquare dataset [11].Foursquare was created in 2009. It is a location-based socialnetworking service provider where users share their loca-tions via “checking-in” function. The dataset contains104, 478check-in tips generated by31, 544users in Los Angeles (LA).It provides the follower lists and check-in messages of allusers.

In real-life OSNs, the users’intrinsic interestandsubjec-tive choiceare usually indispensable for forming the trafficsessions. Under such a common type of sessions, a user willautonomously determine whether to download the contentfrom whom he follows depending on his interest, [14]. Weapply theLatent Dirichlet Allocation(LDA) [13] to extracttopics from the check-in messages generated by users. LDAassumes that words of each document are drawn from a mix-ture of topics. We define all messages generated by a user,sayv, as auser documentcv. Then, we apply LDA model toextract the topic distributionθcv , serving as user’s interests.Figure 4 depicts the LDA model. With the user’s interestdistribution θcv , we introduceJensen-Shannon divergenceJSD(P ||Q) [19] to measure the similarity between users’interestθcv and newly arrived messageθcnew

.In our evaluation, we assume that only when the user’s in-

terestθcv is similar to the message’s topic distributionθcnew,

userv will further view the newly arrived messagecnew. In

Corpus

Word

User Interest/

Message TopicDocument

z

φ

θ wλ

Figure 4: and are the super parameters of LDAmod-

el, is the topic distribution of user document, i.e., user

interest or message topic, is the topic assignment of

and is a word of the message in message set.

! " # $ % & ' ( ) ! " # $

)

#))

'))

!"#$%%&"%'("#)*+,"&#-.#/0"12.21#/"332-43

!"#$%&'"(#)*#+",-./0-.)/,#)*#1#2",,.)/

Figure 5: Destinations Distribution with Relationship

Degree = 22

larity between users’ interest and newly arrived message

new

In our evaluation, we assume that only when the user’s in-

terest is similar to the message’s topic distributionnew

user will further view the newly arrived message new. In

other words, when the value of JSD ||new

is smaller

than that of a pre-defined threshold , the user is selected

as a final destination of the message new

As in Figure 5 illustrated, one of the experimental result-

s based on Foursquare dataset [11] shows the distribution of

session destinations initiated from users who have 22 follow-

ers. In Figure 5, the -axis denotes the number of destina-

tions at a certain degree 22, the -axis denotes the number

of cases where destination number equals . We can see

that decreases rapidly at first and then gently with the in-

creasing of

In order to provide more insights, we include the distribu-

tions of destination number for different relationship degree

with the threshold = 6 in Figure 6. Here, Figure 6

only gives the results with degree from to 32, since user-

-1

5000

degree=3

5000

degree=4

5000

degree=5

2000

4000

degree=6

5000

degree=7

2000

degree=8

2000

degree=9

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ggre

gate

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ber

of S

pecific

Sessio

ns

10

1000

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degree=10

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1000

2000

degree=11

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degree=12

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degree=13

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degree=14

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degree=15

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degree=16

10 15

500

1000

degree=17

10

500

1000

degree=18

10 15

500

degree=19

10 15

500

degree=20

10 20

500

degree=21

10 15

500

degree=22

10 20

500

degree=23

10 15

500

degree=24

10 20

500

degree=25

10 20

200

400

degree=26

10 20

500

degree=27

10 20

200

400

degree=28

10 20

200

degree=29

10 20

200

degree=30

10 20

500

degree=31

10 20

100

200

300

degree=32

The Number of Destinations of A Session

Figure 6: Destinations Distribution in Foursquare.

s with larger relationship degree cannot show convincingly

the statistical characteristics due to the small number of such

users.

The experimental result in each figure provides an insight

that the distribution of destination number for each Social

InterestCast session appears a long-tailed feature. As an em-

pirical argument for this result, Li et al. [24] has provided

a numerical validation based on Renren dataset for such a

long-tailed distribution.

4.2.2 Modeling Social-InterestCast

Based on the observations above, we assume that the dis-

tribution of session destinations for Social-InterestCast fol-

lows approximately a Zipf’s distribution. To be specific, fo

a source node , based on the dependency among follower-

s’ decisions and the attractivity difference of message con

tent, we make a reasonable assumption that the number of

destinations follows a Zipf’s distribution whose parameters

depend on the relationship degree of i.e.

Pr( ) ==1

(5)

where is the number of final destinations of a session ini-

tiated by , and [0 is the exponent of data dis-

semination. Here, for problem simplification, we first study

the special case where for every

In the following Section 5, we mainly study this type of

data disseminations and give the corresponding results for

transport complexity.

5. TRANSPORT COMPLEXITY FOR

SOCIAL-INTERESTCAST

In this section, we aim to derive the transport complexity

for Social-InterestCast.

5.1 Social-InterestCast Sessions

Figure 4: λ is the hyperparameter of LDA model,φ is thetopic-word distribution, θ is the topic distribution of userdocument, i.e., user interest,z denotes the latent topic indocumentcv, andw is an observed word.if a user sends a message characterized by several

words (an abstract) in his Facebook, only the users whose

of the message can be

We say that this behavior occurs due

to a underlying ge mapping

For this data dissemination, we call it

, which is a more common session pattern than

in [29] and Social-BroadCast in [7]. In the following,

to model the internal mathematical rule in this, we first provide

validations as follows.

1) Validations Based on Foursquare Dataset: In this sec-

we give the validation based on Foursquare dataset [9]

to find out the distribution of session destinations for Social-

to Foursquare: Foursquare was created in

It is a location-based social networking service provider

by ”checking-in” function,

by

31 in Los Angeles (LA) and it was collected using

It provides each user with the friend lists and

by users.

of Social-InterestCast Session Destinations:

In Foursquare dataset [9], each of user has personal interests,

to facilitate the comprehension of users’ interests, we first

A) [41] to extract

xt generated by users. For LDA,

it is assumed that words of each document are arised from a

of topics. What’s more, LDA allows each document

to share multiple topics with different proportions, and each

words with the joint influence of word proportions

To be specific, in our validation,

we first integrate each user document

is an arbitrary random user) with the check-in tips generated

by users. Then we apply LDA model to extract the topics

as user’s interests.

With the user’s interest distribution , we introduce

Kullback-Leibler Divergence to measure the similarities of

KL distance

is defined as follows:

KL , θ ) = ) log ) log

is the Ith in the dataset,

is the ith xt generated by user

of user xt

vely. Here, we point out that the smaller KL , θ

of

In the validation, considering the practical observation that

xt which they are interested in,

we assume that only when the user’s interest distribution is

to that of the context , user

xt . And when the value of KL , θ is smaller

of the threshold , the user is chose as a final

of the context

we declare that since most users only farther

a few messages, threshold is chose as

! " # $ % & ' ( ) ! " # $

)

#))

'))

!"#$%%&"%'("#)*+,"&#-.#/0"12.21#/"332-43

!"#$%&'"(#)*#+",-./0-.)/,#)*#1#2",,.)/

4. Destinations Distribution with = 22

5. Destinations Distribution of Foursquare Messages.

on Foursquare dataset [9], one of the experimental

is showed in Fig.4.

In Fig.4, the friendship degree is 22of destinations at certain degree, of

to , and

of

For the completeness of validations, we include some other

experimental results based on the threshold = 6 in Fig.5.

we want to say that due to the number of users with

degree is few, which cannot show the general

of destinations distribution, So Fig.5 only gives

degree from to 42to all cases above, the experimental result in each

is approximately in long tail, which basically validate

of destinations for each Social-InterestCast

ws a Zipf’s distribution.

2) Modeling for Social-InterestCast: ve validations

of destinations for Social-InterestCast

is approximately in Zipf’s distribution. To be specific, for

a source node , based on the dependency among friends’

of message content, we

e a reasonable assumption that the number of destinations

Figure 5: Destinations Distribution with RelationshipDegreeqk = 22.

other words, when the value ofJSD(θcv ||θcnew) is smaller

than that of a pre-defined thresholdδ, the userv is selectedas a final destination of the messagecnew .

As illustrated in Figure 5, one of the experimental resultsbased on Foursquare dataset [11] shows the distribution ofsession destinations initiated from users who have22 fol-lowers. In Figure 5, theX-axis denotes the number of des-tinations at a certain degree22, theY -axis denotes the num-ber of cases where destination number equalsX . We cansee thatY decreases rapidly at first and then gently with theincreasing ofX .

To provide more insights, Figure 6 illustrates the distri-butions of destination number for different relationship de-greeqk with the thresholdδ = 6.5. Since users with largerrelationship degree cannot show convincingly the statisticalcharacteristics due to the small number of such users (thesmall sample size), we have removed those users whose de-gree is larger than32 to obtain a more representative result.In Figure 6, the experimental result in each subfigure showsthat the distribution of destination number for each Social-

Figure 6: Destinations Distribution in Foursquare.

InterestCast session appears a long-tailed feature.Note that the long-tailed property in the distribution of

destination number is just derived under our hypotheticalformation mechanism of Social-InterestCast using LDA model.As an important empirical argument, Li et al. [25] has pro-vided a numerical validation based on RenRen dataset forsuch a long-tailed distribution by counting the ratio of thenumber of viewed videos and that of the received videosfrom friends.

4.2.2 Modeling Social-InterestCast

Based on the result in Section 4.2.1, we assume that thedistribution of session destinations for Social-InterestCastfollows approximately a Zipf’s distribution. Specifically, fora source nodevk, based on the dependency among follow-ers’ decisions and the attractivity difference of message con-tent, we make a reasonable assumption that the number ofdestinations follows a Zipf’s distribution whose parametersdepend on the relationship degree ofvk, i.e.,

Pr(dk = d|qk = l) =

(∑l

m=1m−ϕk

)−1

· d−ϕk , (5)

wheredk is the number of final destinations of a session ini-tiated byvk, andϕk ∈ [0,∞) is the exponent of data dis-semination. Here, for problem simplification, we first studythe special case whereϕk ≡ ϕ for everyvk.

In the following Section 5, we mainly study this type ofdata disseminations and give the corresponding results fortransport complexity.

5. TRANSPORT COMPLEXITY FORSOCIAL-INTERESTCAST

In this section, we aim to derive the transport complexityfor Social-InterestCast.

5.1 Social-InterestCast SessionsDenote a Social-InterestCast session by an ordered pair

DIk =< vk, Ik >, wherevk is the source and each element

vkiin Ik = vki

dk

i=1 is the nearest node to the correspond-ing pki

in AIk = pki

dk

i=1, the random variabledk denotesthe number of potential destinations for sessionDI

k, i.e., thefollowers of vk who are interested in the message fromvkin this session. We call pointpki

the anchor pointof vki,

and define a setP Ik := vk ∪ AI

k. Then, we can get thefollowing lemma.

LEMMA 1. For a Social-InterestCast sessionDIk, when

dk = ω(1), with probability1, it holds that

|EMST(AIk)| = Θ(|EMST(P I

k)|) = Θ(LIP(β, dk)),

whereEMST(·) denotes the Euclidean minimum spanningtree over a set, and

LIP(β, dk) =

Θ(√

dk), β > 2;

Θ(√

dk · logn), β = 2;

Θ(√

dk · n1− β

2

), 1 < β < 2;

Θ(√

dk ·√

nlogn

), β = 1;

Θ(√

dk ·√n), 0 ≤ β < 1.

(6)

PROOF. By Lemma 4 in Appendix A, with probability1,it holds that

|EMST(AIk)| = Θ(

√dk ·

∫

O

√fX0

(X)dX),

where

fX0(X) =

Θ((|X −X0|2 + 1)−β

), β > 1;

Θ((

logn · (|X −X0|2 + 1))−1), β = 1;

Θ(nβ−1 · (|X −X0|2 + 1)−β

), 0 ≤ β < 1.

Next, we compute∫O√fX0

(X)dX . According to the valueof β, we have:

(1) When0 ≤ β < 1,∫

O

√fX0

(X)dX = Θ

(n

β−1

2 ·∫

O

dX

(|X −X0|2 + 1)β

2

)

= Θ(n

β−1

2 · n1− β

2

)= Θ(

√n).

(2) Whenβ = 1,∫

O

√fX0

(X)dX = Θ

(1√logn

·∫

O

dX

(|X −X0|2 + 1)1

2

)

= Θ(

1√logn

· √n)= Θ

(√n

log n

).

(3) Whenβ > 1,∫

O

√fX0

(X)dX = Θ

(∫

O

dX

(|X −X0|2 + 1)β2

).

Especially,

when1 < β < 2, Θ

(∫O

dX

(|X−X0|2+1)β2

)= Θ

(n1−β

2

);

whenβ = 2, Θ(∫

OdX

(|X−X0|2+1)

)=Θ(logn);

whenβ > 2, Θ

(∫O

dX

(|X−X0|2+1)β2

)=Θ(1).

Then, by summarizing the derived results above, we canobtain that

|EMST(AIk)| = Θ(LI

P(β, dk)),

whereLIP(β, dk) is defined in Eq.(6).

Let L denote the smallest distance from the points inAIk

to pointX0. We can get that

|EMST(AIk)| ≤ |EMST(P I

k)| ≤ |EMST(AIk)|+ L,

Furthermore, according to the fact thatL = O(|EMST(AIk)|)

with probability1, we get|EMST(P Ik)| = Θ(|EMST(AI

k)|),which finally completes the proof.

5.2 Main Results on Transport Complexity

5.2.1 Bounds on Aggregated Transport Distance

The bound on the transport complexity depends on thevalue of

∑n

k=1 |EMST(P Ik)|, which we compute in the fol-

lowing theorem.

THEOREM 1. For all Social-InterestCast sessionsDIknk=1

with the Zipf ’s distribution as defined in Eq.(5), with highprobability, the order of

∑n

k=1 |EMST(P Ik)| holds as pre-

sented in Table 1.

PROOF. LetTl denote the number of users withl destina-tions. Then, by laws of larger numbers (LLN, Lemma 5 inAppendix A), we have

Tl = n · Pr(qk = l) = n ·(∑n−1

j=1j−γ

)−1

· (l)−γ .

For all sessionsDIknk=1, we define two sets:

K1 := k|qk = Θ(1),K∞ := k|qk = ω(1).Then, it follows that

∑n

k=1|EMST(P I

k)| = Σ1 + Σ∞, (7)

where

Σ1 =∑

k∈K1

|EMST(P Ik)|, Σ∞ =

∑

k∈K∞

|EMST(P Ik)|.

We first address the part ofΣ1. Since forqk = Θ(1), itholds that|EMST(P I

k)| = Θ(|X−vk|), we then haveΣ1 =∑k∈K1 |X − vk|. Fork ∈ K1, we define a sequence of ran-

dom variablesφ1k := |X− vk|/

√n, which have finite mean-

ing as follows:E[φ1k] = E[|X−vk|]/

√n,whereE[|X−vk|]

is given in Lemma 5 of [37]. Then, according to Lemma 4,with probability1, it holds that

Σ1 = Θ(√

n ·∑

k∈K1φ1k

),

and∑

k∈K1 φ1k/|K1| = Θ(E [|X − vk|/

√n]), where|K1|

denotes the cardinality of setK1. Therefore,∑

k∈K1φ1k = Θ

(|K1| · E

[|X − vk|/

√n])

.

Table 1: The Order of∑n

k=1 |EMST(P Ik)|

ϕ \ β β > 2 β = 2 1 < β < 2 β = 1 0 ≤ β < 1

ϕ > 32 Θ(n) , γ ≥ 0; Θ(n · logn) , γ ≥ 0; Θ(n2−β

2 ) , γ ≥ 0; Θ(n3

2 /√logn) , γ ≥ 0; Θ(n

3

2 ) , γ ≥ 0.

ϕ = 32

Θ(n),γ > 1;

Θ(n · logn),0 ≤ γ ≤ 1.

Θ(n · logn),γ > 1;

Θ(n · (log n)2),0 ≤ γ ≤ 1.

Θ(n2−β2 ),

γ > 1;

Θ(n2−β

2 · logn),0 ≤ γ ≤ 1.

Θ(n3

2 /√log n),

γ > 1;

Θ(n3

2 · √logn),0 ≤ γ ≤ 1.

Θ(n3

2 ),γ > 1;

Θ(n3

2 · logn),0 ≤ γ ≤ 1.

1 < ϕ < 32

Θ(n),γ > 5

2 − ϕ;Θ(n · logn),

γ = 52 − ϕ;

Θ(n7

2−γ−ϕ),1 < γ < 5

2 − ϕ;

Θ(n5

2−ϕ/ logn),γ = 1;

Θ(n5

2−ϕ),0 ≤ γ < 1.

Θ(n · logn),γ > 5

2 − ϕ;Θ(n · (log n)2),

γ = 52 − ϕ;

Θ(n7

2−γ−ϕ · log n),1 < γ < 5

2 − ϕ;

Θ(n5

2−ϕ),γ = 1;

Θ(n5

2−ϕ · logn),0 ≤ γ < 1.

Θ(n2−β

2 ),γ > 5

2 − ϕ;

Θ(n2−β

2 · logn),γ = 5

2 − ϕ;

Θ(n9

2−γ−ϕ−β

2 ),1 < γ < 5

2 − ϕ;

Θ(n7

2−ϕ−β

2 / logn),γ = 1;

Θ(n7

2−ϕ−β

2 ),0 ≤ γ < 1.

Θ(n3

2 /√log n),

γ > 52 − ϕ;

Θ(n3

2 ·√logn),

γ = 52 − ϕ;

Θ(n4−γ−ϕ/√logn),

1 < γ < 52 − ϕ;

Θ(n3−ϕ/(logn)3

2 ),γ = 1;

Θ(n3−ϕ/√logn),

0 ≤ γ < 1.

Θ(n3

2 ),γ > 5

2 − ϕ;

Θ(n3

2 · logn),γ = 5

2 − ϕ;Θ(n4−γ−ϕ),

1 < γ < 52 − ϕ;

Θ(n3−ϕ/ logn),γ = 1;

Θ(n3−ϕ),0 ≤ γ < 1.

ϕ = 1

Θ(n),γ ≥ 3

2 ;

Θ(n5

2−γ/ logn),1 < γ < 3

2 ;

Θ(n3

2 /(logn)2),γ = 1;

Θ(n3

2 / logn),0 ≤ γ < 1.

Θ(n · logn),γ ≥ 3

2 ;

Θ(n5

2−γ),1 < γ < 3

2 ;

Θ(n3

2 / logn),γ = 1;

Θ(n3

2 ),0 ≤ γ < 1.

Θ(n2−β2 ),

γ ≥ 32 ;

Θ(n7

2−γ−β

2 / logn),1 < γ < 3

2 ;

Θ(n5

2−β

2 /(logn)2),γ = 1;

Θ(n5

2−β

2 / logn),0 ≤ γ < 1.

Θ(n3

2 /√log n),

γ ≥ 32 ;

Θ(n3−γ/(logn)3

2 ),1 < γ < 3

2 ;

Θ(n2/(logn)5

2 ),γ = 1;

Θ(n2/(logn)3

2 ),0 ≤ γ < 1.

Θ(n3

2 ),γ ≥ 3

2 ;Θ(n3−γ/ logn),

1 < γ < 32 ;

Θ(n2/(logn)2),γ = 1;

Θ(n2/ logn),0 ≤ γ < 1.

0 ≤ ϕ < 1

Θ(n),γ > 3

2 ;Θ(n · logn),

γ = 32 ;

Θ(n5

2−γ),1 < γ < 3

2 ;

Θ(n3

2 / logn),γ = 1;

Θ(n3

2 ),0 ≤ γ < 1.

Θ(n · logn),γ > 3

2 ;Θ(n · (log n)2),

γ = 32 ;

Θ(n5

2−γ · logn),1 < γ < 3

2 ;

Θ(n3

2 ),γ = 1;

Θ(n3

2 · logn),0 ≤ γ < 1.

Θ(n2−β

2 ),γ > 3

2 ;

Θ(n2−β2 · logn),

γ = 32 ;

Θ(n7

2−γ−β

2 ),1 < γ < 3

2 ;

Θ(n5

2−β

2 / logn),γ = 1;

Θ(n5

2−β

2 ),0 ≤ γ < 1.

Θ(n3

2 /√log n),

γ > 32 ;

Θ(n3

2 · √logn),γ = 3

2 ;Θ(n3−γ/

√logn),

1 < γ < 32 ;

Θ(n2/(logn)3

2 ),γ = 1;

Θ(n2/√logn),

0 ≤ γ < 1.

Θ(n3

2 ),γ > 3

2 ;

Θ(n3

2 · logn),γ = 3

2 ;Θ(n3−γ),

1 < γ < 32 ;

Θ(n2/ logn),γ = 1;

Θ(n2),0 ≤ γ < 1.

Then, by Lemma 5 in Appendix A, with probability1, itholds that

Σ1 = Θ(|K1| ·E[|X − vk|]

). (8)

Next, we considerΣ∞. By introducing anchor points, allrandom variables|EMST(P I

k)| with k ∈ K∞ are indepen-dent. For users with follower number inK∞, we define twosets:

K∞1 = k|dk = Θ(1),K∞

∞ = k|dk = ω (1).

Then, it follows that

Σ∞ = Σ∞1 + Σ∞

∞, (9)

where

Σ∞1 =

∑

k∈K∞

1

|EMST(P Ik

)|, Σ∞

∞ =∑

k∈K∞

∞

|EMST(P Ik

)|.

We first consider theK∞1 . For dk = Θ(1), the order of

Σ∞1 is lower than that ofΣ1. For the final summation, the

specific value ofΣ∞1 is relatively infinitesimal.

Next, we considerK∞∞. According to Lemma 1, with

probability1, it holds that

|EMST(P Ik

)| = Θ

(LIP(β, dk)

),

whereLIP(β, dk) is defined in Eq.(6). And using LLN (Lemma

5), with probability1, it follows that

Σ∞∞ =

∑n−1

l=2

∑l

d=1Tl · Pr(dk = d|qk = l) · LI

P (β, d) .

(10)Combining with Eq.(7), Eq.(8), Eq.(9) and Eq.(10), we com-plete the proof of Theorem 1.

5.2.2 Tight Bounds on Transport Complexity

Firstly, we give the main result (Theorem 2), i.e., the scal-ing laws of transport complexity, and prove them tight boundson the transport complexity by deriving lower bounds (Lemma2) and computing upper bounds (Lemma 3), respectively.

THEOREM 2. LetCIN

denote the transport complexity forSocial-InterestCast sessions in a large-scale OSN over thecarrier network with optimal communication architecture.Then, it holds that

CIN = Θ(G (β, γ, ϕ)) ,

whereG (β, γ, ϕ) depends on the clustering exponents of re-lationship degreeγ, relationship formationβ and dissemi-nation patternϕ, and the value is presented in Table 3 inAppendix C.

Note that the results in Table 3 synthetically depend onthree parameters, i.e.,γ, β, andϕ. This contributes to theinformative and comprehensive form of results. For cardingthe flow and improving the readability, we move the detailedresults to Appendix C (Table 3). We will provide an intuitiveexplanation and discussion on the results in Section 5.2.3.

In the following proofs, we letLIN

denote the transportcomplexity for all data dissemination sessions in OSNN.

Lower Bounds on Transport Complexity: The follow-ing Lemma 2 demonstrates a lower bound on transport com-plexity for OSNN.

LEMMA 2. For the Social-InterestCast with the Zipf ’sdistribution as defined in Eq.(5), it holds that

LIN = Ω(G (β, γ, ϕ)) ,

whereG (β, γ, ϕ) is presented in Table 3 in Appendix C.

PROOF. Since

n∑

k=1

dk = Θ

(n−1∑

l=1

l∑

d=1

Tl · Pr(dk = d|qk = l) · d),

we can get that∑n

k=1 dk = W (γ, ϕ) , whereW (γ, ϕ) isdescribed in Table 2. Additionally, for allvk ∈ V ,

Table 2: The Number of All Destinations,W (γ, ϕ)ϕ W (γ, ϕ)

ϕ > 2 Θ(n) , γ ≥ 0.

ϕ = 2

Θ(n), γ > 1;

Θ(n · logn), 0 ≤ γ ≤ 1.

1 < ϕ < 2

Θ(n), γ > 3− ϕ;

Θ(n · logn), γ = 3− ϕ;

Θ(n4−γ−ϕ), 1 < γ < 3− ϕ;

Θ(n3−ϕ/ logn), γ = 1;

Θ(n3−ϕ), 0 ≤ γ < 1.

ϕ = 1

Θ(n), γ ≥ 2;

Θ(n3−γ/ logn), 1 < γ < 2;

Θ(n2/(logn)2), γ = 1;

Θ(n2/ logn), 0 ≤ γ < 1.

0 ≤ ϕ < 1

Θ(n), γ > 2;

Θ(n · logn), γ = 2;

Θ(n3−γ), 1 < γ < 2;

Θ(n2/ logn), γ = 1;

Θ(n2), 0 ≤ γ < 1.

E[|vki− pki

|] = Θ

(∫ √n

0

x · e−π·x2

dx

).

That is,E[|vki− pki

|] = Θ (1) . Therefore, according toLemma 5, with probability1, it follows that

∑n

k=1

∑dk

i=1|vki

−pki| = Θ

(∑n

k=1dk

)= Θ(W (γ, ϕ)) .

(11)

γ1 3− ϕ

γ = 1 :

1 < γ < 3− ϕ :

γ ≥ 3− ϕ :

•

0 ≤ γ < 1 :

0

G(β, γ, ϕ)

Θ(n3−ϕ)

Θ(n3−ϕ/ log n)

Θ(2− β

2)

•

Θ(n4−γ−ϕ)

1 < β < 2, 1 < ϕ < 3

2

e 7: The Order of Lower Bounds on The Derived

Transport Load, β, γ, ϕ , under < β <

ϕ <

6. CONCLUSION AND FUTURE WORK

To measure the transport difficulty for data dissemination

in online social networks (OSNs), we have defined a new

. To model the formation

of the interest-driven social session, i.e., Social-InterestCast,

we have proposed a four-layered architecture to model the

dissemination in OSNs, including the physical layer,

, content layer, and session layer. By analyz-

vances among these four layers, we have ob-

of dis-

in OSNs. We have presented the den-

of general social relationship distribution and

of bounds on transport load for large-scale

we have derived the tight lower bound

on transport complexity of Social-InterestCast.

work still remains to be done. Main issues are as

In our work, although we have assumed that users are

it is also applicable to the mobile scenario where each

moves within a bounded distance from exact one

, in our numerical evaluation, we have

of each mobile user as his stat-

ic location. However, mobile users in real-life scenario are

by more than one home point rather than

exact one home point. Therefore, it is necessary to further

e into account a more realistic model for users distribu-

in the physical deployment layer, such as Multi-center

in [14].

We have provided only the explicit result for the model

of users. This

of real-

or the advantages of the introduced γ, β

& social-based information dissemi-

a subsequent traffic session initiated from a

is usually triggered by the previous session from an-

However, we have focused exclusively on the

val model without considering the correlations of

transport complexity to wire-

overestimate the transport

in OSNs. To be specific, in

to multiple destinations can be

by a simple wireless broadcast, while our met-

overly accumulate the transport distance of such a dis-

Even so, our results are still reasonable based

on the explanation as follows: For a data dissemination, the

of the last hop is relatively infinitesimal to the ag-

gate distance of this dissemination. What’s more, for the

of scaling laws issue, we only care about the order

of transport load imposed on the carrier communication net-

works of OSNs. Anyway, it is a significant work to seek for

a more accurate and practical metric to measure the transport

of data dissemination in OSNs.

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ware caching and traffic management in

In Proc. IEEE

E. Bakshy, I. Rosenn, C. Marlow, and L. Adamic. The

of social networks in information diffusion. In

Proc. ACM WWW 2012

J. Bao, Y. Zheng, and M. F. Mokbel. Location-based

ware recommendation using sparse

, pages 199–208, 2012.

F. Benevenuto, T. Rodrigues, M. Cha, and V. Almeida.

in online social

In Proc. ACM IMC 2009

Figure 7: The Order of Lower Bounds on The DerivedTransport Complexity, G (β, γ, ϕ), under 1 < β < 2 and1 < ϕ < 3/2.

Combining with Theorem 1, for all Social-InterestCast ses-sionsDI

k, k = 1, 2, ..., n, with high probability, it holds that∑n

k=1|EMST(DI

k)| = Ω(G (β, γ, ϕ)) ,

whereG (β, γ, ϕ) is provided in Table 3 in Appendix C.Then, we complete the proof of Lemma 2.

Upper Bounds on Transport Complexity: Here, we an-alyze the upper bounds on transport complexity for Social-InterestCast sessions over anoptimal carrier network, i.e.,a dedicated carrier communication network for this applica-tion. In such a carrier network, a dedicated link can be builtfor every link inEMST(P I

k), k = 1, 2, · · · , n. Therefore,from Lemma 1 and Theorem 1, for Social-InterestCast ses-sions, the aggregated transport distances of each session andall sessions can respectively reach to the orders as presentedin Eq.(6) and Table 1. Then, we have

LEMMA 3. For the Social-InterestCast with the Zipf ’sdistribution as defined in Eq.(5), it holds that

LIN = O (G (β, γ, ϕ)) ,

whereG (β, γ, ϕ) is presented in Table 3 in Appendix C.

5.2.3 Explanation of Results

Based on the complete results in Table 3 in AppendixC, we can observe that the final results vary in the range[Θ(n),Θ(n2)

], when the parametersγ, β, andϕ have dif-

ferent values in the range[0,+∞).For Table 3, it is too complex to provide a clear insight on

the results. To facilitate the understanding of our results, wechoose a case of results and make them visualized. Figure 7illustrates the results for the case1 < β < 2 and1 < ϕ < 3

2 .Next, we mainly discuss the impacts of the clustering ex-

ponents of relationship degreeγ, relationship formationβand dissemination patternϕ on the transport complexity. Ourresult demonstrates that the transport complexity for Social-InterestCast isnon-increasingwithin the range

[Θ(n),Θ(n2)

]

in terms of the parametersγ, β, andϕ. An intuitive explana-tion for this impact can be provided as follows: Under eachSocial-InterestCast, a larger clustering exponent of relation-ship degreeγ can limit the number of friends of each user

into smaller upper bound with high probability, then leadsto a lower transport complexity; a larger clustering expo-nent of relationship formationβ makes the followers morecloser to each user with high probability, then possibly re-duces the total transport distance of each Social-InterestCastsession, finally also leads to a lower transport complexity;alarger clustering exponent of dissemination patternϕ leadsto a smaller probability that the source chooses larger num-ber of destinations from its followers, which leads to a lowertransport load.

6. CONCLUSION AND FUTURE WORKTo measure the transport difficulty for data dissemination

in online social networks (OSNs), we have defined a newmetric, called transport complexity. To model the formationof the interest-driven social session, i.e., Social-InterestCast,we have proposed a four-layered architecture to model thedata dissemination in OSNs, including the physical layer, so-cial layer, content layer, and session layer. By analyzing mu-tual relevances among these four layers, we have obtainedthe geographical distribution characteristics of dissemina-tion sessions in OSNs. We have presented the density func-tion of general social relationship distribution and the gen-eral form of bounds on transport load for large-scale OSNs.Furthermore, we have derived the tight bounds on transportcomplexity of Social-InterestCast.

Much work still remains to be done. Main issues are listedas follows: In our work, although we have assumed that users are

static, it is also applicable to the mobile scenario where eachmobile user moves within a bounded distance from exact onehome point. Actually, in our numerical evaluation, we havemade the most visited point of each mobile user as his staticlocation. However, mobile users in real-life scenario are usu-ally constrained by more than one home point rather thanexact one home point. Therefore, it is necessary to furthertake into account a more realistic model for users distribu-tion in the physical deployment layer, such as Multi-centerGaussian Model (MGM) in [15]. We have provided only the explicit result for the model

with homogeneous geographical distribution of users. Thiscannot still highlight sufficiently the characteristics ofreal-life OSNs or the advantages of the population-distance-basedmodel. Under the profile & social-based information dissemi-

nation pattern, a subsequent traffic session initiated fromasource is often triggered by the previous session from an-other source. However, we have focused exclusively on thedata arrival model without considering the correlations ofdata generating processes. When applying the metrictransport complexityto wire-

less broadcast, the definition will overestimate the transportdifficulty for data dissemination in OSNs. To be specific, insome scenarios, data targeted to multiple destinations canbetransmitted by a simple wireless broadcast, while our met-

ric overly accumulate the transport distance of such a dis-semination. Even so, our results are still reasonable basedon the explanation as follows: For a data dissemination, thedistance of the last hop is relatively infinitesimal to the ag-gregate distance of this dissemination. What’s more, for thefeature of scaling laws issue, we only care about the orderof transport load imposed on the carrier communication net-works of OSNs. Anyway, it is a significant work to seek fora more accurate and practical metric to measure the transportdifficulty of data dissemination in OSNs.

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APPENDIX

A. SOME USEFUL LEMMASWe provide some useful lemmas as follows:

LEMMA 4 (MINIMAL SPANNING TREE [35]). LetXi,1 ≤ i < ∞, denote independent random variables with val-ues inRd, d ≥ 2, and letMn denote the cost of a minimalspanning tree of a complete graph with vertex setXini=1,where the cost of an edge(Xi, Xj) is given byΨ((|Xi −Xj |)). Here, |Xi − Xj | denotes the Euclidean distancebetweenXi and Xj and Ψ is a monotone function. Forbounded random variables and0 < σ < d, it holds thatasn → ∞, with probability1, one has

Mn ∼ c1(σ, d) · nd−σd ·

∫

Rd

f(X)d−σd dX,

providedΨ(x) ∼ xσ, wheref(X) is the density of the ab-solutely continuous part of the distribution of theXi.

LEMMA 5 (KOLMOGOROV’ S STRONG LLN [38]). LetXn be an i.i.d. sequence of random variables having fi-nite mean: For∀n,E[Xn] < ∞. Then, a strong law of largenumbers (LLN) applies to the sample mean:

Xna.s.−→ E[Xn],

wherea.s.−→ denotesalmost sure convergence.

B. EVALUATIONS BASED ON GOWALLADATASET AND BRIGHTKITE DATASET

In this section, we provide the evaluations of the adopteddegree distribution model and population-distance-basedmodelusing Gowalla and Brightkite users datasets [18,22].

B.1 Gowalla and Brightkite DatasetsGowalla and Brightkite were both created in 2007. They

were once two location-based social networking service providerswhere users shared their locations by “checking-in” func-tion, [3, 18, 22]. For Gowalla, the relationship network con-sists of196, 591 nodes and950, 327 undirected edges. TheGowalla users’ dataset in [22] collected a total of6, 442, 890checkins of these users over the period from February 2009to October 2010. It provides each user with the incomingand outgoing follower lists as well as the latitude and lon-gitude. For Brightkite, the relationship network consistsof58, 228 nodes and214, 078 directed edges. The Brightkiteusers’ dataset in [18] collected a total of4, 491, 143 check-ins of these users over the period from April 2008 to October2010. It provides each user with the incoming and outgoingfollower lists.

In our evaluations, because of the deficiency of users’ datain Asia and other areas, for Gowalla users’ dataset [22], weextracted52, 161 users who locate in North America to im-prove the accuracy and decrease the computation complex-ity. Figure 8 shows the geographical distribution of users inNorth America.Y andX represent the latitude and longi-tude of users’ locations, respectively.

−130 −120 −110 −100 −90 −80 −70 −60

25

30

35

40

45

50

55

X=Longitude

Y=

Latit

ude

Figure 8: The geographical position of Gowalla Users inNorth America.

B.2 Evaluation of Degree DistributionIn this work, we assume that the number of followers of

a particular nodevk ∈ V , denoted byqk, follows a Zipf’sdistribution [28], i.e.,

Pr(qk = l) =

(∑n−1

j=1j−γ

)−1

· l−γ .

Next, we validate the Zipf’s degree distribution of social re-lationships by investigating the negative linear correlationbetween

Y := Nout(Kout) andX := Kout,

whereKout represents an outgoing degree, andNout(Kout)denotes the number of the users with an outgoing degreeKout.

In Gowalla and Brightkite datasets [18, 22], the correla-tion betweenY andX is described as in Figure 9. It showsthat the correlation is approximately to be a line segmentwith negative slope, which basically matches our proposedmodel.

B.3 Evaluation of Population-Distance-BasedModel

We discretize the network areaO into a lattice consistingof 120, 000 points. Each point acts as a candidateanchorpoint. We denote this lattice byOd.

Let d(u, p) denote the distance between useru and a ran-dom position/cellp in Od that serves as a candidateanchorpoint. Let D(u, p) denote the disk centered atu with a ra-diusd(u, p). LetN(u, p) denote the number of nodes in thediskD(u, p). Let vp denote the closest user to the candidateanchor pointp. Furthermore, we define a variable

I(u, vp) = 1 · vp is a follower ofu.We validate the geographical distribution of relationships

by investigating thenegative linear correlationbetweenY

(a) (b)

Figure 9: (a) is the validation for social degree distribution of Gowalla users, while (b) is of Brightkite users from [37].

(a) (b)

Figure 10: (a) is the validation for Population-Distance-Based social probability distribution of Gowalla users, while (b)is of Brightkite users.

andX , whereX denotes a number of nodes contained in acertain disk, and

Y :=1

|E| ·∑

I(u,vp)=1

1 · N(u, p) = N,

with E denoting the set of all social links, respectively.In real-world dataset, the candidateanchor pointslocated

on the sea or in the desert are quite far from their nearestusers, which leads to a high probability of outer sphere ofusers chosen to be a follower ofvk. To get rid of these can-didateanchor pointswhich are apart from the correspondingusers, we set a threshold distancedf to filter such positionsas outliers. In this Gowalla dataset [22],df is set to be200kilometers, which makes the positionsp cover most of theland and filter the ocean area simultaneously.

In the Gowalla and Brightkite datasets [18,22], the corre-lation betweenY andX is described by Figure 10. It shows

that the correlation tendency is approximated very coarselyto a line segment with a negative slope.

The experimental result also basically validates the model,although it does not perfectly match. The main reason ofmismatch possibly lies in the facts as follows: (1) The lo-cations of users in these datasets are actually the positionswhere they check-in, instead of the place where they usu-ally stay. (2) Based on these dataset, more than90% resultsfall within the cases withX > 103. The accumulation ofexperimental errors here leads to the “bloated” tails in theevaluation figures.

C. COMPLETE MAIN RESULTSHere, we summarize the main results of a complete form

in the following table:

Table 3: The Order of Transport Complexity, G (β, γ, ϕ)

ϕ \ β β > 2 β = 2 1 < β < 2 β = 1 0 ≤ β < 1

ϕ > 2 Θ(n), γ ≥ 0 Θ(n · logn), γ ≥ 0 Θ(n2−β

2 ), γ ≥ 0 Θ(n3

2 /√log n), γ ≥ 0 Θ(n

3

2 ), γ ≥ 0

ϕ = 2

Θ(n), γ > 1;

Θ(n · logn), 0 ≤ γ ≤ 1.

Θ(n · logn) , γ ≥ 0. Θ(n2−β2 ), γ ≥ 0. Θ(n

3

2 /√log n) , γ ≥ 0. Θ(n

3

2 ), γ ≥ 0.

32 < ϕ < 2

Θ(n), γ > 3− ϕ;

Θ(n · logn), γ = 3− ϕ;

Θ(n4−γ−ϕ), 1 < γ < 3− ϕ;

Θ(n3−ϕ/ logn), γ = 1;

Θ(n3−ϕ), 0 ≤ γ < 1.

Θ(n · logn), γ ≥ 3− ϕ;

Θ(n4−γ−ϕ), 1 < γ < 3− ϕ;

Θ(n3−ϕ/ logn), γ = 1;

Θ(n3−ϕ), 0 ≤ γ < 1.

Θ(n2−β2 ), γ ≥ 0. Θ(n

3

2 /√log n), γ ≥ 0. Θ(n

3

2 ), γ ≥ 0.

ϕ = 32

Θ(n), γ > 32 ;

Θ(n · logn), γ = 32 ;

Θ(n5

2−γ), 1 < γ < 3

2 ;

Θ(n3

2 / logn), γ = 1;

Θ(n3

2 ), 0 ≤ γ < 1.

Θ(n · logn), γ ≥ 32 ;

Θ(n5

2−γ), 1 < γ < 3

2 ;

Θ(n3

2 / logn), γ = 1;

Θ(n3

2 ), 0 ≤ γ < 1.

Θ(n2− β2 ), γ ≥ 3

2 ;

Θ(n5

2−γ), 1 < γ < 3

2 ;

Θ(n3

2 / logn), γ = 1;

Θ(n3

2 ), 0 ≤ γ < 1.

Θ(n3

2 /√logn), γ > 1;

Θ(n3

2 · √logn), 0 ≤ γ ≤ 1.

Θ(n3

2 ), γ > 1;

Θ(n3

2 · logn), 0 ≤ γ ≤ 1.

1 < ϕ < 32

Θ(n), γ > 3− ϕ;

Θ(n · logn), γ = 3− ϕ;

Θ(n4−γ−ϕ), 1 < γ < 3− ϕ;

Θ(n3−ϕ/ logn), γ = 1;

Θ(n3−ϕ), 0 ≤ γ < 1.

Θ(n · logn), γ ≥ 3− ϕ;

Θ(n4−γ−ϕ), 1 < γ < 3− ϕ;

Θ(n3−ϕ/ logn), γ = 1;

Θ(n3−ϕ), 0 ≤ γ < 1.

Θ(n2− β2 ), γ ≥ 3− ϕ;

Θ(n4−γ−ϕ), 1 < γ < 3− ϕ;

Θ(n3−ϕ/ logn), γ = 1;

Θ(n3−ϕ), 0 ≤ γ < 1.

Θ(n3

2 /√logn), γ > 5

2 − ϕ;

Θ(n3

2 · √logn), γ = 52 − ϕ;

Θ(n4−γ−ϕ), 1 < γ < 52 − ϕ;

Θ(n3−ϕ/ logn), γ = 1;

Θ(n3−ϕ), 0 ≤ γ < 1.

Θ(n3

2 ), γ > 52 − ϕ;

Θ(n3

2 · logn), γ = 52 − ϕ;

Θ(n4−γ−ϕ), 1 < γ < 52 − ϕ;

Θ(n3−ϕ/ logn), γ = 1;

Θ(n3−ϕ), 0 ≤ γ < 1.

ϕ = 1

Θ(n), γ ≥ 2;

Θ(n3−γ/ logn), 1 < γ < 2;

Θ(n2/(logn)2), γ = 1;

Θ(n2/ logn), 0 ≤ γ < 1.

Θ(n · logn), γ ≥ 2;

Θ(n3−γ/ logn), 1 < γ < 2;

Θ(n2/(logn)2), γ = 1;

Θ(n2/ logn), 0 ≤ γ < 1.

Θ(n2− β

2 ), γ > 32 ;

Θ(n3−γ/ logn), 1 < γ ≤ 32 ;

Θ(n2/(logn)2), γ = 1;

Θ(n2/ logn), 0 ≤ γ < 1.

Θ(n3

2 /√logn), γ ≥ 3

2 ;

Θ(n3−γ/ logn), 1 < γ < 32 ;

Θ(n2/(logn)2), γ = 1;

Θ(n2/ logn), 0 ≤ γ < 1.

Θ(n3

2 ), γ ≥ 32 ;

Θ(n3−γ/ logn), 1 < γ < 32 ;

Θ(n2/(logn)2), γ = 1;

Θ(n2/ logn), 0 ≤ γ < 1.

0 ≤ ϕ < 1

Θ(n), γ > 2;

Θ(n · logn), γ = 2;

Θ(n3−γ), 1 < γ < 2;

Θ(n2/ logn), γ = 1;

Θ(n2), 0 ≤ γ < 1.

Θ(n · logn), γ ≥ 2;

Θ(n3−γ), 1 < γ < 2;

Θ(n2/ logn), γ = 1;

Θ(n2), 0 ≤ γ < 1.

Θ(2− β2 ), γ ≥ 2;

Θ(n3−γ), 1 < γ < 2;

Θ(n2/ logn), γ = 1;

Θ(n2), 0 ≤ γ < 1.

Θ(n3

2 /√logn), γ > 3

2 ;

Θ(n3

2 ·√logn), γ = 3

2 ;

Θ(n3−γ), 1 < γ < 32 ;

Θ(n2/ logn), γ = 1;

Θ(n2), 0 ≤ γ < 1.

Θ(n3

2 ), γ > 32 ;

Θ(n3

2 · logn), γ = 32 ;

Θ(n3−γ), 1 < γ < 32 ;

Θ(n2/ logn), γ = 1;

Θ(n2), 0 ≤ γ < 1.