Automatic Grouping for Social Networks CS229 Project Report

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1 Automatic Grouping for Socia Networks CS229 Project Report Xiaoying Tian Ya Le Yangru Fang Abstract Socia networking sites aow users to manuay categorize their friends, but it is aborious to construct and keep updating those categories when a user s network grows. Leskovec et a. [2] defines an unsupervised mode to identify a user s socia circes. However, in rea ife users have persona preferences about how to group their friends. Indeed, it is possibe that two users have exacty the same socia networks but categorize their friends differenty. In such case, unsupervised methods wi fai to capture such persona preferences as they don t incorporate information about what kinds of socia circes the user finds vauabe. In this paper, we deveop a supervised mode for detecting socia circes that combines network structure as we as user information. Experiments show that our mode achieves significanty higher accuracy than K-means and Naive Bayes, and has comparabe overa performance to that in Leskovec et a. s work with ower computationa compexity. Our method aso turns out to have best performance on reativey sma networks. 1. Introduction As socia network sites get bigger and more cuttered, categorizing friends into different socia circes becomes a major mechanism for users to organize their socia networks and cope with overwheming voumes of information generated by their friends. Users in major socia network sites (e.g. Googe+, Facebook and Twitter) categorize their friends either manuay or simpy by grouping friends sharing a common attribute. The goa of our project is to set up a system which automaticay categorizes a user s friends. We incorporate concepts from socia network anaysis into machine earning techniques to sove the above probem. Research has been done on this topic via both conventiona machine earning approaches such as decision trees (Baatarjav et a. [1]), and aso socia network techniques (Leskovec & McAuey [2]). Leskovec et a. [2] proposed an unsupervised method to tacke this probem. We propose a new mode that uses this method as a component. Given a singe user, a network is formed by his/her friends. Foowing [2], we refer to this user as the ego and this network as its ego-network. In our project, we formuate this probem as a supervised earning probem and take into account both the profie information and the network structure. Our method aso differs from conventiona custering methods in the sense that the custers can overap with each other. We introduce a discriminative mode to identify socia circes based on the fact that circes tend to be densey Figure 1: Sampe circe diagram connected with members sharing some common traits. With maximum ikeihood estimation, our agorithm can earn the structure of the socia circes as we as common features within each circe. Additionay, we compare our agorithm with both the K-means agorithm and Naive Bayes as baseines. 2. Dataset Description The dataset we used is the Facebook dataset in [2], which contains 9 ego networks comprised of 4039 users and an undirected socia network with friendship connections. The profie information is coected in 26 categories, incuding anguages, hometowns, birthdays, ocations, etc. Socia circe abes were obtained by asking the 9 egos to 1

2 Figure 3: Test error v.s Number of preassigned centroids Figure 2: Feature space diagram manuay identify a the circes to which their friends beong. On average, there are 19 circes in each ego-network with an average size of 22 friends. 3. Feature Construction The profie of a singe user can be represented as a tree where each eve encodes increasingy specific information. (See Figure 2). We construct the feature space by aggregating a the user attributes in a ego network and represent a singe user s profie information as a binary vector, where 1 indicates the user has this attribute. For exampe (Figure 2), user x has profie [Gender: Mae, Education: Degree: Undergrad, Education: Schoo: Stanford, Education: Major: CS, Education: Major: Math, Language: French]. Then his profie vector is: [0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0]. Note that such profie vectors are defined per ego-network. For exampe, athough thousands of companies exist in the whoe Facebook network, ony a few appear among any particuar ego network. Let T x = (T1 x,..., Tn x ) denote user x s profie vector. We define the difference vector σ x,y = δ(t x, T y ), as an indicator of whether the two users x and y differ at feature. We define s x,y = 1 σ x,y as the simiarity vector. Suppose the ego user is x 0. We construct the foowing two features, one associated with nodes and the other associated with edges: φ(x) = s x,x0, the simiarity vector between user x and the ego ony and aso ψ(x, y) = s x,y, pairwise simiarity vector between x and y. 4. Methodoogy 4.1. K-means We use the K-means method as our unsupervised baseine mode to detect socia circes in the Facebook data. We impement the agorithm using the feature mapping φ(x) Figure 4: Number of resuting centroids v.s K ony and et the preassigned number of custers range from 1 to 16. Figure 3 shows that as the preassigned custers number increases, the test error increases as we. Notaby, K-means works the best in the degenerate case where there is ony one custer. This indicates that K-means is not the right mode for this probem and/or featurization. Another issue with K-means is that the number of resuting centroids may be ess than K because some of the custers merge. In this probem, we observe this as we increase the parameter K. (See Figure 4) This indicates that a ot of the custers formed are garbage custers, and increasing K hurts accurate prediction. This situation occurs because: first, the feature vector for each user is sparse with binary outcome (as opposed to continuous outcome which is more appropriate for the K-means method) and second, in reaity, the socia circes in a socia network do overap, thus a custering agorithm is not proper here Naive Bayes We impement Naive Bayes with feature map φ(x) as our supervised baseine mode. Recognizing that socia circes may overap, we encode the socia circes to which a user x beongs as c x = (c x 1,..., c x K ) where cx is a binary variabe indicating whether user x is in circe C, and K is the tota

3 number of circes. For each circe C, we use c x s as cassification abes, and perform the Naive Bayes agorithm for this particuar circe. In this way, we obtain K cassifiers (h 1,..., h K ), with h denoting the cassifier for circe C. The agorithm yieds an average test error of 47.05%. The high test error is the resut of some particuary big circes in the network; some circes cover up to 70% of the users. Naive Bayes is very ikey to identify these circes whie ignoring other smaer circes. In some extreme cases, the agorithm wi assign users apparenty at random to each circe according to their size in the training data, regardess of the user s feature vector Our mode In this section, we improve the featurization and propose a more sophisticated mode to better sove the probem Featurization 1. Feature Space Dimension Reduction Both the previous two agorithms suffer from highdimensiona feature spaces. Noticing that simiarity vectors are sparse and that each entry of the vectors corresponds to a eaf node in the profie tree (Figure 2), we address the issue by summing up the entries beonging to the same category. More specificay, s x,y p = chidren(p) sx,y, where p denotes category p. This achieves a reduction in feature space dimension from over 300 to 26 for the Facebook data. 2. Network Structure At this point, we have ony used φ(x), the profie information for each user as our feature vectors. However, we woud aso ike to take into account the simiarity between the users to improve our mode. More specificay, we wi aso incorporate the simiarity vector ψ(x, y) between two users x and y to expore the network structure. As members of the same socia circe tend to be densey connected, this wi provide important information about the socia circe formation Proposed Mode We propose a discriminative mode which considers both the profie information and the network structure in order to identify the socia circes. The input to our mode is an ego-network G = (V, E), aong with the feature vectors φ(x) and ψ(x, y) and circe abes. V and E denote the node set and the edge set of the ego-network. Suppose the users are {x (1),..., x (m) }, with corresponding circe abes {c (1),..., c (m) }. We denote the feature vectors of a users as Φ and Ψ. For each circe C, et θ denote the parameter vector associated with shared features within the circe and et α denote some trade-off parameter which wi be expained ater. Our agorithm wi yied θ, α by maximizing the foowing og-ikeihood: (θ, α) = og(p(c, G Φ, Ψ; θ, α)) = og(p(c Φ; θ)p(g C, Ψ; θ, α)). (1) The og-ikeihood consists of two parts: the first part is the ikeihood of the circe abe C based ony on the node features φ(x), and the second part is the ikeihood of the edge set E based on the edge features ψ(x, y) and the different circes C. Since the circes C and the edges e = (x, y) are generated independenty, we wi have: 1 = og p(c Φ; θ) m = og p(c (i) φ(x (i) ); θ) = m i=1 K i=1 =1 2 = og p(g C, Ψ; θ, α) = og = + e / E og p(c (i) φ(x (i) ); θ ) (2) p(e E C, Ψ; θ, α) e / E p(e / E C, Ψ; θ, α) og p(e E C, Ψ; θ, α) og p(e / E C, Ψ; θ, α) (3) We use the ogistic regression mode to form the ikeihood of the circe abes, i.e., p(c (i) = 1 φ(x (i) ); θ ) = g(θ T φ(x(i) )), where g is the sigmoid function. For the ikeihood of the edge set E in the graph, we observe that an edge between x and y is ikey to form if they beong to the same circe C in which case θ T ψ(x, y) tends to be high. [2]. Thus the probabiity of e = (x, y) E is: p(e E C, ψ(e); θ, α) exp{ θ T ψ(e) C :{x,y} C C :{x,y} C α θ T ψ(e)} where α determines the amount we penaize if x, y C. Aso et: (4) d (e) = δ({x, y} C ) α δ({x, y} C ) (5) K D(e) = θ T ψ(e)d (e) (6) =1 Then with the fact p(e E) + p(e / E) = 1, we got: p(e E) = ed(e) 1 p(e / E) = (7) 1 + e D(e) 1 + e D(e)

4 By pugging eq. 7 into eq. 3, we get 2 = D(e) og (1 + e D(e) ) (8) Both 1 and 2 are concave, thus we are abe to optimize = through gradient ascent. The update rue goes as foows: (θ, α) θ = (θ, α) α m [c (i) g(θ T φ(x (i) ))]φ(x (i) ) i=1 + d (e)ψ(e) e D(e) 1 + e D(e) d (e)ψ(e) = θ T ψ(e)δ({x, y} C ) + e D(e) 1 + e D(e) θt ψ(e)δ({x, y} C ) (9) (10) We randomy seect 70% of the users in an ego-network as our training data and obtain θ s and α s by maximizing eq.1 using the gradient ascent update rues defined above. To predict the circe abes of some user x i in the test dataset, we compute the ikeihood of p(x i C, G Θ, Φ, Ψ) for each circe C, where G is the new network after adding x i. Then x i is predicted to beong to the top J circes that have the argest ikeihoods. Our resuts show that J = 3 usuay gives very good predictions, whie one can aso seect J via cross-vaidation Evauation Metrics We evauate our method by examining the differences between the circes our agorithm seects Ĉ = {Ĉ1,..., Ĉ ˆK} and the true circe abes C = {C 1,..., C K }. We adopt the Baanced Error Rate (BER) as a difference measure between the two circes [3], and take the average BER of a the circes as our error rate. BER(Ĉ, C) = 1 2 ( Ĉ\C Ĉ + C\Ĉ ). (11) C For unsupervised earning methods ike the K-means agorithm, we don t know the correspondence between the circes in Ĉ and C. As a matching heuristic, we aign the circes of these two types by minimizing f(i) = argmin j ( ˆµ i µ j 2 ), (12) where ˆµ i and µ i are the centroids of Ĉi and C i respectivey. Therefore f defines a correspondence between Ĉ and C, i.e., C f(i) is the corresponding circe for Ĉi. Figure 5: Accuracy comparison Figure 6: θ i for circe 6 in ego-network 1 5. Experiment & Resuts During the impementation, we annea the earning rate α to acceerate the earning speed. The comparison of the three methods we impemented is shown in Figure 5. As expected, we observe that the K-means method performs the worst, and our method outperforms the Naive Bayes method for 6 ego-networks out of 9. Figure 6 pots the parameter vector θ 6 for circe C 6 in ego-network 1. The 5 th, 6 th, 10 th and 14 th entries in the vector are significanty arger than the other entries. We further examine the corresponding categories in ego-network 1 and find that those entries correspond to Education: Schoo, Education: Type, Gender and Locae (i.e. Location), which are important features for socia network detection. We aso pot the prediction resuts of a circe on ego-network 3 as in Figure 7. In the pot, densey connected nodes form a custer. The resut shows that our mode successfuy detects amost a the members of the circe.

5 [5] M. Handcock, A. Faftery, and J. Tantrum. Mode-based custering for socia networks. Journa of the Roya Statistica Society. Serires A, [6] J. Yang and J. Leskovec. Defining and evauating network communities based on ground-truth. In ICDM, [7] T. Hastie, R. Tibshirani, and J. Friedman. The Eements of Statistica Learning. Springer Series in Statistics Springer New York Inc., New York, NY, USA, 2001 [8] J. A. Hartigan and M. A. Wong. A K-Means Custering Agorithm. Journa of the Roya Statistica Society. Series C (Appied Statistics), Vo. 28, No. 1, pp , Figure 7: Prediction graph on ego-network 3 6. Concusion and Future Work We introduce a way of combining the user profie information and the socia network structure to detect the socia circes to which a user beongs in an ego-network. As a supervised mode, our method captures ego users persona preferences in grouping their friends, and it aso outperforms the methods which ony consider the user profie information. Aso, it is reasonaby common that users in the same socia circe are aso friends with each other, which wi resut in interesting graph structures that we can take advantage of in circe detection. For prediction we now pick the top J circes of the highest probabiities as the circes a user beongs to. In order to improve the mode, we can use cross vaidation to decide the number of the circes each user beongs to. Aso we can boost the efficiency of the agorithm by eiminating the irreevant features in feature space reduction. References [1] E. Baatarjav, S. Phithakkinukoon and R. Dantu. On the Move to Meaningfu Internet Systems. OTM 2008 Workshops Lecture Notes in computer Science. Vo. 5333, pp , [2] J. McAuey and J. Leskovec. Discovering socia circes in ego networks. arxiv: , [3] Y. Chen and C. Lin. Combining SVMs with various feature seection strategies. Springer, [4] J. Friedman. Stochastic gradient boosting. Computationa Statistics & Data Anaysis, 2002.