Learning the Strength of the Factors InfluencingUser Behavior in Online Social Networks

Bo Hu, Mohsen Jamali, and Martin EsterSchool of Computing Science

Simon Fraser UniversityBurnaby, Canada, V5A 1S6Telephone: 778-782-3111

Email: {boh, mohsen jamali, ester}@sfu.ca

Abstract—As social networking is moving into the web, thestudy and exploitation of social correlation has emerged as ahot research topic. Most of these work consider binary socialrelations, called “friendships”. However, online users tend toestablish many friendships of varying degree of strength, e.g.,relatives, friends, co-workers, and acquaintances. We arguethat, due to their different degree of strength, different friendrelationships will have greatly varying degrees of correlation andshould be distinguished. Besides, social correlation is not theonly factor driving user behavior. In this paper, we address theproblem of learning the strength of the social correlation, user,item, and sparsity factors in online social networks. We proposea probabilistic model, Factor Weight Model, for learning thesestrengths which maximize the joint probability of the observeduser behavior, i.e., actions on items. Different from existingmethods, our model considers not only social correlation, butit also considers the other factors affecting user behavior. Wehave conducted experiments on four real life data sets fromEpinions, Flixster, Flickr, and Digg. Our experiments prove thesuperiority of our model over a state-of-the-art method in termsof action prediction. We also analyze the contributions of thevarious factors for the prediction performance.

I. INTRODUCTION

Millions of people now use social networking websites toenjoy online interaction with friends and meeting new people.Social network sites, such as Flixster, Flickr, and Digg, areattracting an increasing number of users, many of whom haveintegrated these sites into their daily practices. Meanwhile, therapid development of online social networking is increasinglyattracting the attention of academic and industry researchersand motivating the study of prominent phenomenons, such ashomophily [30], i.e., similar users are more likely to connectto each other, and social influence [10], i.e., friends tend toinfluence each others preferences and behavior. Thus, a lotof research has investigated patterns of user interests anduser actions in social networks for viral marketing [9], [24],recommender systems [2], [18], and trust propagation [14],[15].

Most of the current work considers only binary socialrelations (friends or not). However, the real cases of onlinesocial networks are much more complicated. Due to the swiftand low cost of creating social relations online, people tend toestablish a large number of online friendships with differentsocial groups, such as friends, relatives, co-workers, or even

strangers, with different degrees of friendships strength [13],[28]. This means that valuable information may be lost if onlyone strength of friendship is considered.

Several works [12], [23], [26], [27], [28] have modeledthe tie strength between pairs of users as hidden parametersand developed methods to learn them from observed userbehavior. However, none of them considers all factors affectingthe dynamics of online social networks. As discussed in theliterature [1], [19], the factors driving user behavior include:(1) homophily and social influence; (2) user; (3) item; and(4) sparsity factors. Since in this paper we do not focuson distinguishing homophily from social influence effects[20], [29], we use the term “social correlation” to representthe combined effect of homophily and social influence. Forexample, if a friend of the user has gone to the “The SocialNetwork” movie, she may watch it as well (social correlation);and the user could watch the movie because of her ownpreference for drama movies (user); the movie may be sopopular that she cannot resist it (item); or users do nothave enough time to check on every movie (sparsity). Thesparsity factor significantly reduces the amount of availabledata and should not be neglected when modeling online socialnetworks.

In this paper, we address the problem of learning thestrengths of social correlation, user, item, and sparsity in thecontext of an online social network. We propose a probabilisticmodel for learning these strengths by maximizing the jointprobability of the observed user behavior.

The major contributions of this paper are as follows:

1) We introduce a more comprehensive user behaviormodel, which considers all major factors impacting userbehavior: social correlation, user, item, and sparsity.

2) The proposed model also considers the strengths of dif-ferent factors, which is crucial in online social networks.

3) We present an extensive experimental evaluation on fourreal data sets (Epinions1, Flixster2, Flickr3, and Digg4),demonstrating the accuracy of our proposed model and

1http://www.epinions.com2http://www.flixster.com3http://www.flickr.com4http://www.digg.com

2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining

978-0-7695-4799-2/12 $26.00 © 2012 IEEE

DOI 10.1109/ASONAM.2012.67

368

2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining

978-0-7695-4799-2/12 $26.00 © 2012 IEEE

DOI 10.1109/ASONAM.2012.67

368

providing interesting insights into the most importanteffects in real life social networks.

The rest of the paper is organized as follows. We presentour proposed probabilistic model in Section II and the methodfor parameter learning in Section III. In Section IV, weexperimentally evaluate and compare our model with othermodels. In Section V, we survey related work. We concludethe paper in Section VI.

II. THE FACTOR WEIGHT MODEL

In this section, we first explain some definitions and intu-itions behind our model, and then we introduce our researchproblem and describe our probabilistic model.

A. Preliminaries

In this subsection, we introduce the basic definitions and ourresearch problem. We are given users (henceforth, interchange-able with “individuals”) and items (henceforth, interchange-able with “products”, “movies”, and “photos” etc.), and friendrelationships (henceforth, interchangeable with “neighbors”).For the sake of clarity, we reserve special indexing lettersfor distinguishing users from items: 𝑢 and 𝑣 for users and𝑖 for items. We use 𝑡 for time or date. In addition, we reservesuperscripts for time and subscripts for users and items.

In this work, a social network is represented as a directedgraph 𝐺 = (𝑈,𝐸), where 𝑈 denotes the set of users, anda directed edge (𝑢, 𝑣) ∈ 𝐸, 𝑢 ∈ 𝑈, 𝑣 ∈ 𝑈 from user 𝑢to user 𝑣 represents the fact that 𝑢 has added 𝑣 as her“friend”. We assume an ordered set of discrete timestamps𝑇 = {1, 2, ...𝑡𝑒𝑛𝑑}, where 𝑡𝑒𝑛𝑑 is the last timestamp. We use𝑡𝑢,𝑣 ∈ 𝑇 to denote the time of creating the edge from 𝑢 to 𝑣.We define a snapshot of a social network as containing all thenodes and edges that have been created up to time 𝑡.

Definition 1 (Social network snapshot): A social networksnapshot is defined as 𝐺𝑡 = (𝑈 𝑡, 𝐸𝑡), where 𝑈 𝑡 is the setof users at time 𝑡 ∈ 𝑇 and 𝐸𝑡 is the set of links betweenusers that have been created at or before time 𝑡.

Note that most available data sets, such as Epinions, Flixster,Flickr, and Digg in our experiments, contain only creations ofnodes and edges and no deletions, but our model can easilybe applied to both cases.

In this paper, we consider a sequence of social networksnapshots [𝐺1, 𝐺2, ..., 𝐺𝑡𝑒𝑛𝑑 ] together with user actions. In ourcontext, user actions are performed on items, and we assumethat only one action on one item. We associate a user with aset of actions, i.e., the set of items on which she has performedan action at some point of time. We define the action moreformally as follows:

Definition 2 (Action): Given a set of items 𝐼 , each user 𝑢 ∈𝑈 is associated with a set of actions, denoted as

𝐼𝑢 = {𝑖1, 𝑖2, ..., 𝑖𝑀}with 𝐼𝑢 ⊆ 𝐼 . We use 𝑡𝑢,𝑖 ∈ 𝑇 to denote the time of the actionof user 𝑢 on item 𝑖.

Interaction is a common way to maintain friend relation-ships. Typically, users are willing to spend some time to

communicate with other users, especially friends. In the Flixterdata set, for example, users’ interaction is based on actions onthe same movies. We define this interaction formally as itemadoption similar to the notation of [8].

Definition 3 (Item adoption): For a pair of friends 𝑢 and𝑣, (𝑣, 𝑢) ∈ 𝐸, and an item 𝑖, there is an item adoption 𝑖𝑢,𝑣from user 𝑢 to user 𝑣, if the following two conditions hold:(1) 𝑡𝑢,𝑖 < 𝑡𝑣,𝑖, i.e., user 𝑢 performs the action on item 𝑖 aheadof user 𝑣; (2) 𝑡𝑣,𝑢 < 𝑡𝑢,𝑖 and 𝑡𝑣,𝑢 < 𝑡𝑣,𝑖, i.e., the time 𝑡𝑣,𝑢 ofestablishing the friend relationship is ahead of both actions byuser 𝑢 and user 𝑣. Furthermore, the set of all items adoptedby 𝑣 from 𝑢 is defined as follows:

𝐼𝑢,𝑣 = {𝑖∣𝑖 ∈ 𝐼𝑢∧𝑖 ∈ 𝐼𝑣∧𝑡𝑢,𝑖 < 𝑡𝑣,𝑖∧𝑡𝑣,𝑢 < 𝑡𝑢,𝑖∧𝑡𝑣,𝑢 < 𝑡𝑣,𝑖}Notice that based on the definition of item adoption, there

are two types of actions. Actions that are adopted from friendsare called “social actions”, i.e., users follow their friends,while other actions are not adopted from friends, which wecall “individual actions”. Social actions could be attributed tosocial correlation. An individual action is the result of the userand item effects, which is well known in the recommendersystems literature, where latent factor models [16], [18], [19],especially matrix factorization techniques, have been success-ful in explaining the user actions by characterizing both theuser and item factors. For movies, for example, each userfactor measures how much the user likes movies based on usercharacteristics, such as demographic attributes and preferencesfor movie genres, while the item factors may measure howmuch the movie attracts users on different dimensions, suchas movie genre or the depth of character development.

Based on the above definitions, we introduce the factorstrength prediction problem in online social networks asfollows:

Problem 1 (Factor strength prediction): Given a sequenceof social network snapshots [𝐺1, 𝐺2, ..., 𝐺𝑡𝑒𝑛𝑑 ], sets of actions𝐼𝑢 for all users 𝑢 ∈ 𝑈 and a series of timestamps 𝑇 , the goalof the factor strength prediction problem is to output sets ofstrength of the social correlation, users, items, and sparsityfactors.

B. The Probabilistic Model

In order to infer the strength of social correlation, user, item,and sparsity factors, we plan to model the user states.

For every timestamp 𝑡 ∈ 𝑇 and every item 𝑖 ∈ 𝐼 , thereare two states for a user 𝑢: active, i.e., the user has alreadyperformed the action on the item, and inactive, i.e., the user hasnot performed the action. Therefore, we use a binary randomvariable 𝑥𝑡

𝑢,𝑖 ∈ {0, 1} to represent a single user state, where𝑥𝑡𝑢,𝑖 = 1 indicates that user 𝑢 has performed an action on item

𝑖 by time 𝑡, and 𝑥𝑡𝑢,𝑖 = 0 means that user 𝑢 has not performed

an action on item 𝑖 by time 𝑡. We also assume that once auser 𝑢 performed an action on a particular item 𝑖 at time 𝑡,she stays active on this item up to the end, so that all randomvariables in the set {𝑥𝑡+1

𝑢,𝑖 , ..., 𝑥𝑡𝑒𝑛𝑑𝑢,𝑖 } are equal to one. In our

data sets, all user states are observed. Our goal is to computethe probability of these observed states and to obtain a joint

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probability of distribution of the 𝑥𝑡𝑢,𝑖 for all 𝑢 ∈ 𝑈 , 𝑖 ∈ 𝐼 ,

and 𝑡 ∈ 𝑇 . To that purpose, we introduce the Factor Weight(𝐹𝑊 ) Model.

Now we define the notations needed for our 𝐹𝑊 model,which are listed in Table I. 𝑁 𝑡

𝑢 is the set of friends of user 𝑢by time 𝑡. We assume that user actions are driven by socialcorrelation, user, item, and sparsity factors. In the following,we use term “weight” instead of the term “strength”. Let 𝑤𝑢,𝑣

denote the social correlation weight from user 𝑢 to user 𝑣, 𝜃𝑢denote the user weight of user 𝑢, 𝜃𝑖 denote the item weight ofitem 𝑖, and 𝜌 denote the sparsity weight. In addition, we use𝑋𝑁𝑡

𝑢,𝑖to represent the set of all 𝑥𝑡

𝑣,𝑖 for 𝑣 ∈ 𝑁 𝑡𝑢.

TABLE INOTATIONS IN THE FW MODEL

𝑥𝑡𝑢,𝑖 binary random variable: whether user 𝑢 has performed an

action on item 𝑖 by time 𝑡𝑁𝑡

𝑢 set of friends of user 𝑢 by time 𝑡𝑋𝑁𝑡

𝑢,𝑖 set of random variables 𝑥𝑡𝑣,𝑖 for all friends 𝑣 ∈ 𝑁𝑡

𝑢 of user𝑢

𝑤𝑢,𝑣 social correlation weight from user 𝑢 to user 𝑣𝑊𝑁𝑡

𝑢set of 𝑤𝑢,𝑣 for all 𝑣 ∈ 𝑁𝑡

𝑢

𝜃𝑢 user weight of user 𝑢𝜃𝑖 item weight of item 𝑖𝜌 sparsity weight of online social network

To model the probability of the states of each user at eachtimestamp, we assume the following generative process. Forevery user 𝑢, every item 𝑖, at every time 𝑡, the following twosteps are performed: (1) with probability 1 − 𝜌, user 𝑢 isexposed to item 𝑖; (2) if user 𝑢 is exposed to the item 𝑖, user𝑢 performs the action on item 𝑖 with a probability dependingon the combination of the weights of user, item, and socialcorrelation factors. The intuition behind the model is that theonline users cannot perform actions on all the items, becausethere are far too many items. Therefore, we attribute most ofthe absent actions to the sparsity weight 𝜌.

Now we introduce our 𝐹𝑊 model for user, item, socialcorrelation, and sparsity weight prediction. Given a socialnetwork snapshot 𝐺𝑡 and a set of actions up to time 𝑡 − 1,we model the conditional probability of the user 𝑢 state onan item 𝑖 by time 𝑡 in Equation 1 and 2. We make a Markovassumption that the state of user 𝑢 by time 𝑡 is conditionallyindependent of all the previous states given the state of allusers at time 𝑡− 1.

𝑝(𝑥𝑡𝑢,𝑖 = 1∣𝑥𝑡−1

𝑢,𝑖 ,𝑊𝑁𝑡−1𝑢

, 𝑋𝑁𝑡−1𝑢 ,𝑖, 𝜃𝑢, 𝜃𝑖, 𝜌) =⎧⎨

⎩(1− 𝜌)

×𝑝(𝑥𝑡𝑢,𝑖 = 1∣𝑊𝑁𝑡−1

𝑢, 𝑋𝑁𝑡−1

𝑢 ,𝑖, 𝜃𝑢, 𝜃𝑖) if 𝑥𝑡−1𝑢,𝑖 = 0

1 if 𝑥𝑡−1𝑢,𝑖 = 1

(1)

𝑝(𝑥𝑡𝑢,𝑖 = 1∣𝑥𝑡−1

𝑢,𝑖 ,𝑊𝑁𝑡−1𝑢

, 𝑋𝑁𝑡−1𝑢 ,𝑖, 𝜃𝑢, 𝜃𝑖, 𝜌) =

1− 𝑝(𝑥𝑡𝑢,𝑖 = 1∣𝑥𝑡−1

𝑢,𝑖 ,𝑊𝑁𝑡−1𝑢

, 𝑋𝑁𝑡−1𝑢 ,𝑖, 𝜃𝑢, 𝜃𝑖, 𝜌) (2)

Note that after a user performs an action on a given item, theprobability of performing the same action afterwards is always

one, and in order to model user states at timestamp 1 ∈ 𝑇 , weassume a special timestamp 0, which is ahead of timestamp1, for which we set all user state random variables 𝑥 to 0.

We assume that a user is influenced by each friend accordingto the social correlation weight similar to the Linear ThresholdModel [17]. At each timestamp, the neighbor has a chance totrigger the user to perform the action, and for each user, thesocial correlation weight of active neighbors is summed up.In addition to take into account the user and item factors,we add the user and item weights. In conclusion, we define𝑝(𝑥𝑡

𝑢,𝑖 = 1∣𝑊𝑁𝑡−1𝑢

, 𝑋𝑁𝑡−1𝑢 ,𝑖, 𝜃𝑢, 𝜃𝑖) as follows:

𝑝(𝑥𝑡𝑢,𝑖 = 1∣𝑊𝑁𝑡−1

𝑢, 𝑋𝑁𝑡−1

𝑢 ,𝑖, 𝜃𝑢, 𝜃𝑖)

=1

1 + 𝑒−(∑

𝑣∈𝑁𝑡−1𝑢

(𝑤𝑣,𝑢𝑥𝑡−1𝑣,𝑖 )+𝜃𝑢+𝜃𝑖

) (3)

where we use a sigmoid function (non-linear) for two reasons.The first one is to simplify the parameter learning. The secondone is that we observe the social correlation weight grows upto a number of friends but then it stabilizes, which meanswhen the number of active friends meets a threshold, addinganother active friend cannot make contribution any more. Thesigmoid function fits this observation very well based on ourexperiment results. To simplify the presentation, we replace∑

𝑣∈𝑁𝑡−1𝑢

(𝑤𝑣,𝑢𝑥𝑡−1𝑣,𝑖 ) with

∑𝑊 .

We assume that user 𝑢 can influence user 𝑣 if and only if𝑣 has added 𝑢 as her friend and define the social correlationweights only on pairs of users that are “friends” at the endof the time window 𝑡𝑒𝑛𝑑. Note that social correlation is notsymmetric, and the social correlation weight 𝑤𝑢,𝑣 has ingeneral a different value than 𝑤𝑣,𝑢.

We plug Equation 3 into Equation 1, and we obtain thefollowing Equation 4 for the case 𝑥𝑡−1

𝑢,𝑖 = 0.

𝑝(𝑥𝑡𝑢,𝑖 = 1∣𝑊𝑁𝑡−1

𝑢, 𝑋𝑁𝑡−1

𝑢 ,𝑖, 𝜃𝑢, 𝜃𝑖, 𝜌) =(1− 𝜌

)×

(𝑔(∑

𝑊 +𝜃𝑢 + 𝜃𝑖))

(4)

where 𝑔(𝑥) = 11+𝑒−𝑥 is the sigmoid function. Furthermore,

we obtain 𝑝(𝑥𝑡𝑢,𝑖 = 0∣𝑊𝑁𝑡−1

𝑢, 𝑋𝑁𝑡−1

𝑢 ,𝑖, 𝜃𝑢, 𝜃𝑖, 𝜌) for the case𝑥𝑡−1𝑢,𝑖 = 0 as follows:

𝑝(𝑥𝑡𝑢,𝑖 = 0∣𝑊𝑁𝑡−1

𝑢 ,𝑢, 𝑋𝑁𝑡−1𝑢 ,𝑖, 𝜃𝑢, 𝜃𝑖, 𝜌) =

𝜌+(1− 𝜌

)×

(1− 𝑔

(∑𝑊 +𝜃𝑢 + 𝜃𝑖

))(5)

In order to make the learning problem tractable, we assumethat all the weights are static, i.e., do not change in the courseof the time as also implemented in [12], [27]. However, onemight argue that the degree of correlation of an item adoptiondecreases over time. We also consider such a temporal versionof our model. Intuitively, the social correlation from user 𝑢 touser 𝑣 is at its peak at the time of action by user 𝑢. Afterthat time, the correlation is decreasing and finally disappearsafter a certain amount of time. We assume that the peak timecoincides with the time 𝑡𝑢,𝑖 of the friends’ action. Hence, we

replace the term 𝑤𝑣,𝑢 in Equation 4 by 𝑤𝑣,𝑢×𝑒−𝑡−𝑡𝑢,𝑖

𝜏 , where

370370

𝜏 is the decaying parameter. In our experiments, we tune 𝜏 inorder to obtain the best results.

Based on the above equations, we obtain the followinglikelihood function 𝐿 as a joint probability of every observeduser states on every items on every timestamps in Equation 6.

𝐿 =∏𝑢∈𝑈

∏𝑖∈𝐼

∏𝑡∈𝑇

𝑝(𝑥𝑡𝑢,𝑖 = 1∣𝑥𝑡−1

𝑢,𝑖 ,𝑊𝑁𝑡−1𝑢

, 𝑋𝑁𝑡−1𝑢 ,𝑖, 𝜃𝑢,

𝜃𝑖, 𝜌)𝑥𝑡𝑢,𝑖×𝑝(𝑥𝑡

𝑢,𝑖 = 0∣𝑥𝑡−1𝑢,𝑖 ,𝑊𝑁𝑡−1

𝑢, 𝑋𝑁𝑡−1

𝑢 ,𝑖, 𝜃𝑢, 𝜃𝑖, 𝜌)1−𝑥𝑡

𝑢,𝑖

(6)III. PARAMETER LEARNING

In this section, we employ a maximum likelihood estimationmethod to learn the latent parameters {𝑤𝑢,𝑣, 𝜃𝑢, 𝜃𝑖, 𝜌} fromthe observed user states {𝑥𝑡

𝑢,𝑖} of our probabilistic model.The likelihood function is given in Equation 6. To simplifythe learning process, we replace the parameter 𝜌 with valuesin the range between [0, 1] with a sigmoid function 𝑔(𝜌) inorder to make the domain of 𝜌 from −∞ to +∞.

Instead of simply maximizing the log-likelihood function,we minimize the following error function in Equation 7,which is the sum of the negative log-likelihood and quadraticregularization terms [3] on 𝑤𝑣,𝑢, 𝜃𝑢, 𝜃𝑖 and 𝜌.

Φ =∑𝑢∈𝑈

∑𝑖∈𝐼

∑𝑡∈𝑇((

− 𝑥𝑡𝑢,𝑖 × ln

(1− 𝑔(𝜌)

)− 𝑥𝑡

𝑢,𝑖 × ln(𝑔(∑𝑊

+𝜃𝑢 + 𝜃𝑖)))

+

(−(1−𝑥𝑡

𝑢,𝑖)×ln(𝑔(𝜌)+

(1−𝑔(𝜌))×(

1−𝑔(∑𝑊

+𝜃𝑢+𝜃𝑖)))))

+𝜆𝑤

2

∑𝑢∈𝑈

∑𝑣∈𝑁𝑡−1

𝑢 ,𝑡∈𝑇𝑤𝑣,𝑢

2+𝜆𝜃𝑢

2

∑𝑢∈𝑈

𝜃𝑢2+

𝜆𝜃𝑖

2

∑𝑖∈𝐼

𝜃𝑖2+

𝜆𝜌

2𝜌2

(7)

The initial values of all parameters are sampled from normaldistributions with zero mean. We define the update method ofthe parameters 𝜇 ∈ ({𝑤𝑢,𝑣} ∪ {𝜃𝑢} ∪ {𝜃𝑖} ∪ {𝜌}) in Equation8.

𝜇𝑛𝑒𝑤 = 𝜇𝑜𝑙𝑑 − 𝑠𝑡𝑒𝑝× ∂Φ

∂𝜇(8)

where 𝑠𝑡𝑒𝑝 is the learning rate which varies from different ran-dom variables and different data sets. We omit the derivativeequations to save space.

IV. EXPERIMENTS

In this section, we report our experimental results on fourreal life data sets comparing various versions of our 𝐹𝑊model and one of the latest state-of-the-art methods [12].Closely related work are [27], [23], [25], [26], [12]. [27] useuser features to compute the strength of homophily on pairsof friends. If we represent the action as user features, then wecould make this work very similar to [12]. In [23], [25], [26],their models are for the heterogeneous networks, while ourmodel is for the homogeneous networks. In conclusion, wechoose the most related work [12] as our comparison partner.

A. Data Sets

The datasets in our experiments are from Epinions, Flixster,Flickr, and Digg. We remove those users who have no friendsand the users who have not performed any actions. For the sakeof efficiency, we used samples of the data sets. Our samplespreserve the distributions of the number of actions for usersas well as for items.

We present relevant statistics for four data sets in Table II,and describe the data sets in the following:

∙ In the Epinions5 data set, an action is users’ ratingson products, and item adoption is that friends haverated the same item. Notice that the items in Epinionsare from different categories including digital cameras,music, books, etc.

∙ The Flixster data set was crawled by us fromFlixster.com, which is publicly available6 . Actions aredefined as ratings on movies by users, and item adoptionis defined as ratings on the same movies by a user and afriend. Unlike Epinions, all the items in the Flixster dataset are movies, but there are different genres of movies:action, romance, fantasy, and so on.

∙ The Flickr data set was collected by [4] from Flickr.com.The action is defined as a user tagging photos as favorite.We regard two friends tagging the same photo as itemadoption.

∙ The Digg data set was collected by [22] from Digg.com.The action is defined as digging stories, i.e., users statethat they like a story. We regard two friends digging thesame story as item adoption.

Notice that the statistics of social and individual actionsshow that Flixster and Digg have much higher proportions ofsocial actions than Flickr and Epinions.

B. Experimental Setup

In our experiments, we split the whole data set into atraining data set and a test data set. For each user, we pick atimestamp 𝑡 ∈ 𝑇 so that the training data set contains all theuser states from the beginning time to time 𝑡, which contains90% of all user actions, while the test data set contains theremaining user states from time 𝑡 to the end 𝑡𝑒𝑛𝑑, whichcontains the 10% most recent user actions.

Our goal is to learn the factor weights based on likelihoodfunction on the training data set, and then we use the learnedfactor weights to predict the probability of user states in thetest data set. Given the learned parameters, we calculate theprediction probability using Equation 1. We label the useractive if the predicted probability of an action is greaterthan the activation threshold and inactive otherwise. In ourexperiments, we used 1000 different threshold values from 0to 1 with step 0.001.

As in [12], the evaluation metric used in our experi-ments is the ROC curve, which plots the true positive rate(TPR = TP/(TP+FN)) versus the false positive rate (FPR

5http://alchemy.cs.washington.edu/data/epinions/6http://www.cs.sfu.ca/s̃ja25/personal/datasets/

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TABLE IISTATISTICS OF THE EPINIONS, FLIXSTER, FLICKR, AND DIGG DATA SETS

Statistics Epinions Flixster Flickr DiggUsers 11K 20K 13K 1.2KItems 11K 13K 14K 32K

Social Relations 280K 130K 1.3M 28KFriends per User 25 7 101 23Actions per Item 25 62 61 35Actions per User 25 39 61 959

Actions 277K 770K 729K 1.1MSocial Actions 27K(10%) 208K(27%) 140K(19%) 297K(27%)

Individual Actions 250K(90%) 562K(73%) 589K(81%) 803K(73%)Beginning Time November 2000 November 2005 November 2006 August 2 2008

Ending Time February 2002 November 2009 March 2007 August 27 2008Time Interval 60 Days 90 Days 15 Days 1 Day

Number of Timestamps 14 10 10 25

= FP/(FP+TN)), where TP, FP, FN, and TN represent truepositives, false positives, false negatives, and true negativesrespectively. The closer the curve is to the point (0,1), thebetter the performance. Each point on the ROC curve corre-sponds to one possible value of the activation threshold, whichis the same for all users. To measure the contribution of afactor, we compute the AUC (area under ROC curve) of thecorresponding partial model and compare it to the AUC of𝐹𝑊 model.

We compare the following methods for factor weight learn-ing:∙ 𝐼𝐶. This is the discrete Bernoulli model proposed in [12],

which achieved the best performance in their experiments(refer to Section V for more details).

∙ 𝐹𝑊𝑤. This method is a version of our model which usesonly the social correlation effect.

∙ 𝐹𝑊𝑤+𝑡𝑒𝑚𝑝. This is a version of 𝐹𝑊𝑤, which considerstemporal social correlation weights.

∙ 𝐹𝑊 −𝑤. This is a partial 𝐹𝑊 model, where the socialcorrelation effect is removed.

∙ 𝐹𝑊 − 𝜃𝑢. This is the model that takes all factors intoaccount except the user effect.

∙ 𝐹𝑊 −𝜃𝑖. This is a 𝐹𝑊 partial model, where we removethe item effect from our full model.

∙ 𝐹𝑊 . This is our full 𝐹𝑊 model with all effects.Note that all the above different versions of the 𝐹𝑊 model

include the sparsity factor. All the methods were implementedin C++, and all experiments were performed on a serverrunning Windows 7 with an Intel Xeon E5630 2.53 GHz CPUand 8GB RAM. Based on our preliminary experiments, weset the stop criterion parameter 𝑇𝑜𝑙 to 1000, the maximumiterations to 300, the 𝑆𝑡𝑒𝑝 to 1e-07 for 𝜌 and to 1e-03 for theremaining variables.

C. Experimental Results

In this subsection, we present the results of our experiments,the prediction performance comparison, the factor contributionanalysis, and the temporal social correlation weight analysis.Since the 𝐼𝐶 model is only applicable to predict social actions,in our experiments, we ignore the individual actions in the testdata set.

1) Prediction performance comparison: Figure 1 reportsthe ROC curves of our 𝐹𝑊 model and the state-of-the-art𝐼𝐶 model in the Epinions, Flixster, Flickr, and Digg datasets. The 𝐹𝑊 model consistently outperforms the 𝐼𝐶 modelin Epinions, Flixster, and Flickr data sets by a substantialmargin, while it performs similar in the Digg data set. Ourexplanation is that in Digg the user action of digging a storyis usually influenced by her friends’ actions (see also TableIII). Considering only the social correlation factor, we observesimilar results of the 𝐹𝑊𝑤 model and the 𝐼𝐶 model in theFlixster and Digg data sets, while there is a slight gain inEpinions and Flickr. This indicates that the sparsity factorcan help improve the “social correlation only” model 𝐹𝑊𝑤,because some actions that fail to give rise to item adoption canbe attributed to the social correlation, while others may causedby the sparsity factor (i.e., users missed recommendations fromfriends). Besides, the gain increases after we add either useror item effects into the 𝐹𝑊𝑤 model. These results prove ourhypothesis that social actions can be attributed to the user,item, and social correlation factors, not only to the socialcorrelation factor as done in [12], [26], [27].

2) Factor contribution analysis: To analyze the contribu-tion of individual factors in our 𝐹𝑊 model, we compare the𝐹𝑊 model with the 𝐹𝑊 − 𝑤, 𝐹𝑊 − 𝜃𝑢, 𝐹𝑊 − 𝜃𝑖, and𝐹𝑊𝑤 models. Our 𝐹𝑊 model contains the social correlation,user, and item effects. Specifically, we remove one of thethree factors from our full model, and we train and evaluatethe performance of the partial models. Figure 2 presentsthe resulting AUC, and shows that all three factors make asignificant difference on all four data sets. We also observethat the item factor is strong in the Epinions and Flixster datasets, but is very weak in the Flickr data set. Surprisingly, the𝐹𝑊−𝜃𝑖 model, which has no item factor, slightly outperformsthe 𝐹𝑊 model in Flickr. Why does the item factor contributein the Epinions and the Flixster data sets, but does not helpin the Flickr data set? In the Epinions and Flixster data sets,the items (products in Epinions and movies in Flixster) arepublic and all users can perform actions on them, while inthe Flickr data set the items (photos) are normally private onwhich only friends check. Hence, Flickr users are not aware

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Fig. 1. ROC comparison of 𝐼𝐶 and 𝐹𝑊 models.

of the popularity of items, which likely explains the absenceof an item factor. In Digg, the AUC remains the same whenwe remove either the user or item effect, but it decreases whenwe remove both of them. The reason may be that user actionsof digging a story are based on their friends’ behavior anddisregard the user or item effects.

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To rank the contribution of each factor, we first compute theAUC difference between the full model 𝐹𝑊 and the partialmodels 𝐹𝑊−𝑤, 𝐹𝑊−𝜃𝑢 and 𝐹𝑊−𝜃𝑖, and rank the factorsin descending order of their AUC differences, e.g., the social

correlation factor is more important than the user factor if theAUC difference between 𝐹𝑊 and 𝐹𝑊 − 𝑤 is larger thanthe AUC difference between 𝐹𝑊 and 𝐹𝑊 − 𝜃𝑢. Table IIIpresents the ranking of the social correlation, user, and itemeffects on all four data sets. The most important factor is theitem, user, user, and correlation factor in Epinions, Flixster,Flickr, and Digg respectively. This shows that all these threefactors make significant contributions in some, but differentdata sets. As also discussed in [1], in some data sets the socialcorrelation factor is not as important as we might expect, suchas Epinions and Flixster.

TABLE IIIRANKING OF SOCIAL CORRELATION, USER, AND ITEM FACTORS

Data set First Second Third

Epinions item correlation userFlixster user item correlationFlickr user correlation itemDigg correlation item user

3) Temporal social correlation weight analysis: We analyzethe models with temporal social correlation weights. On Epin-ions, Flixster, and Flickr, our experiments show only minordifferences between the temporal and static versions for the𝐹𝑊 model, likely because the correlation factor is not themajor effect impacting the user actions. Figure 3 reports theROC of the 𝐹𝑊𝑤 and 𝐹𝑊𝑤+𝑡𝑒𝑚𝑝 models in the Digg data

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set. It turns out that the 𝐹𝑊𝑤+𝑡𝑒𝑚𝑝 model has clearly betterperformance than the 𝐹𝑊𝑤 model, which indicates that thesocial correlation is indeed decreasing in the course of thetime. These results confirm similar results observed for the𝐼𝐶 model in [12].

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V. RELATED WORK

In this section, we review related work from the followingthree areas: (1) maximization of social influence; (2) testingof existence for homophily and social influence; (3) modelingof social correlation and the strength of social correlation.

Maximization of social influence in social networks hasbeen widely investigated. Domingos and Richardson [9], [24]are the first to describe the problem of choosing influential setsof users given a social network and considering propagationof the social influence from a data mining perspective. Kempeet al. [17] prove that the problem formulated as discreteoptimization is NP-hard, and present a greedy approximationalgorithm. Leskovec et al. [21] present an optimized greedyalgorithm. Chen et al. [5], [6] also propose two faster greedyalgorithms for influence maximization. All the above methodsonly find the top-k influential users and do not model thesocial influence at the interaction level. Besides these modelsfind the top-k influential users to predict the future actions,while we exploit the actions in the past to learn the socialinfluence weight among pairs of users.

Some work tests for the existence of homophily and socialinfluence. While we determine the local strength of correlationbetween two users, these methods test whether globally inthe entire data set there is significant homophily or socialinfluence. Anagnostopoulos et al. [1] propose a randomizationmethod to show that there is significant social correlation intagging behavior data set, which cannot be attributed to influ-ence. [20] also present a randomization procedure for assessingwhether a data set exhibits social influence and homophilyeffects. [29] distinguish social influence from homophily foritem adoption using match sampling.

A growing work of research [7], [8] has focused on model-ing social networks and user behavior. [8] studies the temporalevolution of the interplay between attribute similarity andsocial ties. They identify and model the interaction betweensocial influence and selection effects in Wikipedia data set,and find feedback effects between these two factors. Froma different perspective, Cosley et al. [7] considers snapshotobservations of a social network and the detailed temporaldynamics of a social network. A method which infers thedetailed dynamics from observed snapshots is proposed. Themain focus of these works is modeling user behavior by takingsocial influence into account, and they do not model thestrength of all effects. Furthermore, none of them considersall effects influencing the user actions.

The works most relevant to our proposed method arethose of [12], [23], [25], [26], [27], [28]. Tang et al. [26]introduce the problem of topic-based social influence analysisand propose a Topical affinity Propagation approach. Theirrecent work [23] further investigates the topic-based socialinfluence and propagation on heterogeneous networks andproposes a generative graphical model. Their work is based onheterogeneous networks with user nodes and content nodes.They assume that the user is going to perform an action,and their method predicts on which item the user is going toperform that action. This problem definition is different fromours: given an item, we predict whether a user is going toperform an action or not. Another work [25] from the sameauthors also proposes a graphical model to model the actionsand predict users future actions, however, they do not consideritems in their problem definition. Hence, they ignore the itemand sparsity effects.

Xiang et al. [27] propose a latent variable model whichestimates relationship strength between users in online socialnetworks from user behavior. Their model mainly considersthe effect of homophily. [28] present a predictive model thatmaps social media data to tie strength. The model builds ona real data set of over 2000 social ties, and it distinguishesbetween strong and weak ties. While in this paper, we focuson all effects.

[11] maximizes the social influence weight based on userbehavior. However, the social network is the output of theirmethod, while it is the input in our model. Moreover, in theirexperiments, the model only considers one item, while ourmodel involves multiple items.

Goyal et al. [12] learn the influence probabilities using theindependent cascade model (𝐼𝐶). At a given timestamp, eachuser is either active or inactive, and each individual’s tendencyto become active increases as more of its friends becomeactive. Scine there is an assumption that the probability ofdifferent friends influencing the user are independent, eachfriend has a probability to trigger the user to perform theaction. The influence probability 𝑝𝑣,𝑢 from user 𝑣 to user 𝑢is computed by Equation 9, where ∣𝐼𝑣,𝑢∣ is the number ofitems adopted from 𝑣 by 𝑢, and ∣𝐼𝑣∣ is the number of actions

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performed by 𝑣.

𝑝𝑣,𝑢 =∣𝐼𝑣,𝑢∣∣𝐼𝑣∣ (9)

Therefore, the joint probability 𝑝𝑢(𝑁𝑢) can be defined asfollows.

𝑝𝑢(𝑁𝑢) = 1−∏

𝑣∈𝑁𝑢

(1− 𝑝𝑣,𝑢) (10)

The authors of [12] claim that the influence of a particularaction of a user on her friends is decreasing over time.Accordingly, a temporal model (𝐼𝐶𝑡𝑒𝑚𝑝) is introduced and thetemporal influence probability 𝑝𝑡𝑣,𝑢 is computed in equation11, where 𝑝𝑣,𝑢 is the probability in equation 9 and 𝜏𝑣,𝑢 is theaverage time difference between 𝑡𝑢,𝑖 and 𝑡𝑣,𝑖 over all items 𝑖that 𝑢 adopted from 𝑣, where 𝑡𝑢,𝑖 is the time of user 𝑢 performan action on item 𝑖.

𝑝𝑡𝑣,𝑢 = 𝑝𝑣,𝑢 × 𝑒− 𝑡−𝑡𝑣,𝑖

𝜏𝑣,𝑢 (11)

VI. CONCLUSION AND FUTURE WORK

With the rapid growth of interest in online social networks,such as Flixster and Facebook, social networks are changingthe way we live. Social correlation plays an important rolein social networks and leads people to adopt the behavior oftheir friends. However, not all friends have the same degree ofsocial correlation on a user. Users follow the behavior of someof their friends more than other friends. Moreover, the user,item, and sparsity effects impact the user behavior as well.In this paper, we propose a probabilistic model, called FactorWeight Model (𝐹𝑊 ), to learn the social correlation, user, item,and sparsity weights from user actions. Our model maximizesthe joint probability of the observed user actions. We performexperiments on four real life data sets (Epinions, Flixster,Flickr, and Digg). We evaluate the quality of action prediction,and experimental results demonstrate that the 𝐹𝑊 modelachieves better performance than a state-of-the-art method.

This paper suggests several interesting directions for futureresearch. First, besides the problem of predicting whethera user will perform an action on specific items, predictinga rating for a given item is also a natural task in recom-mendation. Although learning and inference will be morecomplex, considering rating values may improve the accuracyof learning the factor weights. Intuitively, the social correlationweight between two users who have the similar ratings is largerthan that one of two users who have a large rating difference.Second, the social correlation we consider in this paper iscontext independent. However, it would be interesting toinvestigate the social correlation weight in different contexts.Users may influence friends in some contexts while in othercontexts they may not. For example, in Flixster a user mayinfluence friends in action movies if she is an action moviefan, but if she has no interest in other types of movies, sheprobably has no social influence on other types of movies.

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