PCT: Partial Co-Alignment of Social Networks

Transcription

1 PCT: Partia Co-Aignment of Socia Networks Jiawei Zhang University of Iinois at Chicago Chicago, IL, USA Phiip S Yu University of Iinois at Chicago, IL, USA Institute for Data Science Tsinghua University, Beijing, China psyu@csuicedu ABSTRACT Peope nowadays usuay participate in mutipe onine socia networks simutaneousy to enjoy more socia network services Besides the common users, socia networks providing simiar services can aso share many other kinds of information entities, eg, ocations, videos and products However, these shared information entities in different networks are mosty isoated without any known corresponding connections In this paper, we aim at inferring such potentia corresponding connections inking mutipe kinds of shared entities across networks simutaneousy Formay, the probem is referred to as the network Partia Co-aignmenT (PCT probem PCT is an important probem and can be the prerequisite for many concrete cross-network appications, ike socia network fusion, mutua information exchange and transfer Meanwhie, the PCT probem is aso very chaenging to address due to various reasons, ike (1 the heterogeneity of socia networks, (2 ack of training instances to buid modes, and (3 one-to-one constraint on the correspondence connections To resove these chaenges, a nove unsupervised network aignment framework, UNI- COAT (UNsupervIsed COncurrent AignmenT, is introduced in this paper Based on the heterogeneous information, UNICOAT transforms the PCT probem into a joint optimization probem To sove the objective function, the one-to-one constraint on the corresponding reationships is reaxed, and the redundant non-existing corresponding connections introduced by such a reaxation wi be pruned with a nove network co-matching agorithm proposed in this paper Extensive experiments conducted on rea-word coaigned socia network datasets demonstrate the effectiveness of UNICOAT in addressing the PCT probem Keywords Partia Network Co-Aignment, Mutipe Heterogeneous Socia Networks, Unsupervised Learning, Data Mining 1 INTRODUCTION Looking from a goba perspective, the andscape of onine socia networks is highy fragmented A arge number of onine socia networks have appeared and achieved prosperous deveopments in Copyright is hed by the Internationa Word Wide Web Conference Committee (IW3C2 IW3C2 reserves the right to provide a hyperink to the author s site if the Materia is used in eectronic media WWW 2016, Apri 11 15, 2016, Montréa, Québec, Canada ACM /16/04 recent years Some of these networks can even provide very comparabe network services and are of simiar network structures For instance, (1 Foursquare and Yep (two famous ocation-based socia networks can both offer ocation reated services for users; (2 Amazon and Ebay are both created for onine e-commerce; (3 Kickstarter 1 and Indiegogo 2 are both constructed to accumuate funding for projects from the pubic; and (4 Youtube and Vimeo 3 both provide arge amounts of video resources for users to either watch or share with friends In such an age of onine socia media, users usuay participate in mutipe socia networks simutaneousy to enjoy more socia networks services, who can act as bridges connecting different networks together Besides these common users, socia networks offering simiar services can aso share other common information entities, eg, ocations shared between Foursquare and Yep, and products sod in both Amazon and Ebay Formay, the shared information entities in different networks can act as anchors aigning these networks, which can be formay named as anchor instances (eg, the shared users can be caed anchor users, whie the shared ocations and products can be caed anchor ocations and anchor products respectivey What s more, the corresponding reationships between the anchor instances (indicating they are the same information entities across networks can be caed anchor inks For instance, the corresponding reationships between the shared users can be named as user anchor ink, whie those between shared ocations can be caed ocation anchor ink However, in the reaword, anchor instances in different networks are mosty isoated without any known anchor inks connecting them Probem Studied: In this paper, we want to infer different categories of anchor inks connecting various anchor instances across socia networks simutaneousy, which is formay defined as the network Partia Co-aignmenT (PCT probem PCT is a genera research probem and can be appied to different types of socia networks, ike Foursquare and Yep, Amazon and Ebay Meanwhie, as shown in Figure 1, in this paper, we wi mainy focus on the partia co-aignment of ocation based socia networks via shared users and ocations with the various connection and attribute information avaiabe in the networks PCT is an important research probem and can be the prerequisite for many concrete rea-word appications, ike network fusion [33, 29, 11, 31, 19], cross-network recommendation [27, 28, 34, 19], mutua community detection [32, 30], and inter-network information diffusion [26] Besides its importance, PCT is aso a nove probem and totay different from existing works on entity matching and network aignment, ike (1 supervised anchor ink inference [11], which focuses on inferring the user anchor inks ony with a supervised earning method; (2 user matching across networks [25], which

2 user profie user tempora activity user text usage geo-ocation 8 AM 4 PM 11 PM ocation visiting pattern 8 AM 12 PM 4 PM 8 PM 11 PM User Anchor Link ocation text descriptions???? Location Anchor Link user profie user tempora activity user text usage geo-ocation 8 AM 4 PM 11 PM ocation visiting pattern 8 AM 12 PM 4 PM 8 PM 11 PM Figure 1: Exampe of the PCT probem ocation text descriptions expores various user attribute information ony to match users between different socia networks; (3 bipartite graph aignment [12], which aims at matching two bipartite graphs merey with the ink information; and (4 homogeneous bioogica network matching [21], which studies the matching probem between two homogeneous PPI (protein-protein interaction networks based on the structure information ony As shown in Figure 1, different from a these reated works, in the PCT probem, (1 few known anchor inks connecting anchor instances between networks are avaiabe, and the ack of training instances wi make the supervised methods [11] fai to work, (2 anchor instances contain heterogeneous information, incuding both attribute and ink information (eg, users have connections with other users, and aso have attribute information, ike profie information, tempora activity and text usage patterns; whie ocations have connections to users, and aso have attribute information, eg, geo-ocation, user visiting patterns and text descriptions, and (3 mutipe different types of anchor inks (ie, user and ocation anchor inks are to be inferred simutaneousy More information about other reated works is avaiabe in Section 5 To hep iustrate the difference, we aso summarize the differences of this paper from existing works in Tabe 1 Despite its importance and novety, the PCT probem is very chaenging to sove due to: heterogeneity of socia networks: anchor instances in onine socia networks can be associated with heterogeneous information, ike various types of attributes and compex inks How to utiize such heterogeneous information to improve the network aignment resuts is very difficut unsupervised network co-aignment: socia network aignment with mutipe types of anchor inks has never been studied before and the PCT probem is sti an open probem to this context so far Furthermore, the unsupervised earning setting (due to the ack of known anchor inks, ie, training instances poses extra chaenges on addressing the PCT probem one-to-one property: the anchor inks to be inferred are assumed to have an inherent one-to-one constraint, ie, each user/ocation can have at most one account in one network (The case that users/ocations have mutipe accounts in one network can be addressed with [23], where dupicated accounts can be aggregated in advance to form one unique virtua account and the anchor inks connecting these virtua accounts wi sti be one-to-one How to preserve and utiize the constraint to improve network aignment resuts can be a great chaenge To address the above chaenges in PCT, a nove unsupervised network aignment framework UNICOAT (UNsupervIsed COncurrent AignmenT is proposed in this paper Based on both attribute and ink information, UNICOAT formuates the aignment probem as a joint optimization probem to infer both potentia user and ocation anchor inks By reaxing the one-to-one constraint on anchor inks, UNICOAT soves the optimization objective function with an aternative updating schema Meanwhie, the introduced nonexisting anchor inks (by such a reaxation can be further pruned with a minimum cost network fow based co-matching agorithm effectivey The rest of the paper is organized as foows We first introduce the terminoogy definitions and formuate the probem in Section 2 In Section 3, we propose the UNICOAT framework in detai Section 4 presents the experiment resuts on rea-word co-aigned socia networks Finay, in Sections 5-6, we describe the reated works and concude this paper 2 PROBLEM FORMULATION Before introducing the UNICOAT framework, we wi give the definitions of some important concepts and formuation of the PCT probem first in this section 21 Terminoogy Definition Definition 1 (Attribute Augmented Socia Network: Information entities (eg, users and ocations in socia networks studied in this paper can have both ink and attribute information and such kind of networks can be formuated as attribute augmented socia networks, G =(V, E, B, where node set V = U[Lcontains both user and ocation nodes, ink set E = E u,u [E u, contains the inks among users and those between users and ocations Attribute set B = B u [B, where B u and B are the sets of attributes about users and ocations respectivey Definition 2 (Co-Aigned Attribute Augmented Socia Networks: Socia networks that share both common users and ocations can be represented as co-aigned attribute augmented socia networks G = ((G (1,G (2, (A (1,2 u, A (1,2, where G (1 and G (2 are two attribute augmented socia networks respectivey and A (1,2 u, A (1,2 are the sets of undirected user anchor inks and ocation anchor inks between networks G (1 and G (2 respectivey Link (u (1,v (2 2A (1,2 u iff u (1 and v (2 are the accounts of the same user in G (1 and G (2 respectivey, and simiar for inks in A (1,2 Traditiona anchor inks introduced in existing works [11, 34] normay represent the inks connecting the same users accounts in different networks (ie, the user anchor inks mentioned above In this paper, we extend the definition of anchor inks to any kinds of common information entities (eg, users and ocations shared between networks and specify the definitions about user and ocation anchor inks more ceary 22 Probem Statement Based on the above terminoogy definitions, we can present the PCT probem formay as foows: Co-Aignment Probem: For any two given attribute augmented socia networks G (1 and G (2, with the ink and attribute information in both G (1 and G (2, the PCT probems aims at inferring

3 Tabe 1: Summary of reated probems PCT: Partia Anchor Link User Matching Bipartite Network PPI Network Property Co-Aignment Inference [11] across Networks [25] Aignment [3] Aignment [21] network heterogeneous heterogeneous heterogeneous bipartite homogeneous information used ink&attribute ink&attribute attribute ink ink setting unsupervised supervised supervised unsupervised unsupervised # anchor inks mutipe kinds singe kind singe kind singe kind singe kind the potentia anchor inks between users and ocations across G (1 and G (2 respectivey In other words, PCT expores the inference of both user anchor ink and ocation anchor ink sets A (1,2 u and between G (1 and G (2 concurrenty A (1,2 3 PROPOSED METHOD In this section, we wi introduce the UNICOAT framework to address the PCT probem in detai Based on the ink and attribute information, we wi formuate the PCT probem as a joint optimization probem in Section 31 to infer potentia user and ocation anchor inks across networks To sove the objective equation, we propose to reax the one-to-one constraint And the non-existing redundant anchor inks introduced by such a reaxation wi be pruned with the network co-matching agorithm to be introduced in Section Anchor Links Co-Inference As introduced in Section 2, et A (1,2 u be the set of inferred user anchor inks between networks G (1 and G (2, which maps users between networks G (1 and G (2 Considering that users in different socia networks are associated with both inks and attribute information, the quaity of the inferred anchor inks A (1,2 u can be measured by the costs introduced by such mappings cacuated with users ink and attribute information, ie, cost(a (1,2 u = cost in inks (A (1,2 u + cost in attributes(a (1,2 u, where denotes the weight of the cost obtained from the attribute information ( is set as 1 in the experiments for simpicity, ie, the ink and attribute information is treated to be of the same importance Considering that ocations are aso attached with ink and attributes, simiar cost function can be defined for the inferred ocation anchor inks in A (1,2 : cost(a (1,2 = cost in inks (A (1,2 + cost in attributes(a (1,2 The optima user and ocation anchor inks (A (1,2 u and (A (1,2 to be inferred in the PCT probem that can minimize the cost functions can be represented as (A (1,2 u, (A (1,2 = arg min A (1,2 u,a (1,2 cost(a (1,2 u +cost(a (1,2 To resove the objective function, in the foowing parts of this section, we wi introduce the (1 isoated user anchor ink inference in subsection 311, (2 isoated ocation anchor ink inference in subsection 312, and (3 the joint co-inference framework of user and ocation anchor inks in subsection User Anchor Links Inference Socia connections among users ceary iustrate the socia community structures of users in onine socia networks Meanwhie, attribute information (eg, profie information, text usage patterns, tempora activities can revea users unique persona characteristics Common users in different networks tend form simiar community structures [32] and have very cose persona characteristics [25] As a resut, ink and attribute information about the users both pays very important roes in inferring potentia user anchor inks across networks In this part, we wi introduce how to use such information to improve the user anchor ink inference resuts User Anchor Link Inference with Link Information Based on the socia inks among users in both G (1 and G (2 (ie, E u,u (1 and E u,u (2 respectivey, we can construct the binary socia adjacency matrices [17] S (1 2 R U(1 U (1 and S (2 2 R U(2 U (2 for networks G (1 and G (2 respectivey Entries in S (1 and S (2 (eg, S (1 (i, j and S (2 (, m wi be assigned with vaue 1 iff the corresponding socia inks (u (1 i,u (1 j and (u (2,u (2 m exist in G (1 and G (2, where u (1 i,u (1 j 2U (1 and u (2,v m (2 2 U (2 are users in networks G (1 and G (2 Via the inferred user anchor inks A (1,2 u, users as we as their socia connections can be mapped between networks G (1 and G (2 We can represent the inferred user anchor inks A (1,2 u with binary user transitiona matrix P 2 R U(1 U (2, where the (i th,j th entry P(i, =1iff ink (u (1 i,u (2 2A (1,2 u Considering that the constraint on user anchor inks is one-to-one, each coumn and each row of P can contain at most one entry being assigned with vaue 1, ie, P1 U(2 1 appe 1 U(1 1, P > 1 U(1 1 appe 1 U(2 1, where P1 U(2 1 and P > 1 U(1 1 can get the sum of rows and coumns of matrix P respectivey Equation P1 U(2 1 appe 1 U(1 1 denotes that every entry of the eft vector is no greater than the corresponding entry in the right vector Matrix P is an equivaent representation of user anchor ink set A (1,2 u Next, we wi infer the optima user transitiona matrix P, from which we can obtain the optima anchor ink set A (1,2 u The optima user anchor inks are those which can minimize the inconsistency of mapped socia inks across networks and the cost introduced by the inferred user anchor ink set A (1,2 u with the ink information can be represented as cost in ink(a (1,2 u =cost in ink(p = P > S (1 P S (2 2, F where k k F denotes the Frobenius norm of the corresponding matrix and P > is the transpose of matrix P User Anchor Link Inference with Attribute Information Besides socia inks, users in socia networks can be associated with a set of attributes, which can provide extra hints for identifying the correspondence reationships about users across networks In this part, we wi introduce the method to infer the user anchor inks with attribute information, which incudes username information, text usage patterns and tempora activity information Username that can differentiate users from each other in onine socia networks is ike their onine ID, which is an important factor in inferring potentia anchor inks Let (u (1 be a potentia i,u (2 anchor ink between G (1 and G (2, the usernames of u (1 can be represented as two sets of characters n(u (1 i and n(u (2

4 respectivey, based on which, various metrics proposed by Liu [25] can be appied to measure the simiarity between u (1 In this paper, we propose to cacuate the simiarity between the usernames with measure Jaccard s Coefficient [14], ie, sim(n(u (1 i,n(u (2 = n(u(1 i \ n(u (2 n(u (1 i [ n(u (2 Users usuay have their unique active tempora patterns in onine socia networks [11] For exampe, some users ike to sociaize with their onine friends in the eary morning, but some may prefer to do so in the evening after work Users onine active time can be extracted based on their post pubishing timestamps effectivey Let t(u (1 i and t(u (2 be the normaized tempora activity distribution vectors of users u (1, which are both of ength 24 Entries of t(u (1 i and t(u (2 contain the ratios of posts being pubished at the corresponding hour in a day For exampe, t(u (1 i (3 denotes the ratio of a posts written by u (1 1 at 3AM Based on vectors t(u (1 i and t(u (2, we can cacuate the inner product of the tempora distribution vectors [11] as the simiarity scores between u (1 in their tempora activity patterns, ie, sim(t(u (1 i, t(u (2 = t(u (1 i > t(u (2 Besides profie and onine activity tempora distribution information, peope normay have very different text usage habits onine [25], which can revea persona unique characteristics and can be appied in inferring the user anchor inks across networks We represent the text content used by users u (1 as bag-ofwords vectors [11], w(u (1 i and w(u (2, weighted by TF-IDF [9] respectivey Commony used text simiarity measure: Cosine simiarity [5] can be appied to measure the simiarities in text usage patterns between u (1, ie, sim(w(u (1 i, w(u (2 = w(u(1 i > w(u (2 w(u (1 i w(u (2 With these different attribute information (ie, username, tempora activity and text content, we can cacuate the simiarities between users across networks G (1 and G (2 We represent such simiarity matrix as 2 R U(1 U (2, where entry (i, is the simiarity between u (1 (i, can be represented as a combination of sim(n(u (1 i,n(u (2, sim(t(u (1 i, t(u (2 and sim(w(u (1 i, w(u (2 and inear combination is used in in this paper due to its simpicity and wide usages The optima weights of simiarity scores cacuated with different attribute information can be earnt from the data theoreticay, but it wi make the mode too compicated To focus on the co-aignment probem itsef, in this paper, we assume they are a of the same importance and propose to assign them with the same weight for simpicity concerns In other words, (i, = 1 sim(n(u (1 3 i,n(u (2 + sim(t(u (1 i, t(u (2 + sim(w(u (1 i, w(u (2 Simiar users across socia networks are more ikey to be the same user and user anchor inks A (1,2 u that aign simiar users together shoud ead to ower cost In this paper, the cost function introduced by the inferred user anchor inks A (1,2 u in attribute information is represented as cost in attribute(a (1,2 u =cost in attribute(p = kp k 1, where k k 1 is the L 1 norm [18] of the corresponding matrix, entry (P (i, can be represented as P(i, (i, and P denotes the Hadamard product [4] of matrices P and User Anchor Link Inference with Link and Attribute Information Both ink and attribute information is important for user anchor ink inference By taking these two categories of information into consideration simutaneousy, we can represent the cost introduced by the inferred user anchor ink set A (1,2 u as cost(a (1,2 u =cost in ink(a (1,2 u + cost in attribute(a (1,2 u = P > S (1 P S (2 2 F kp k 1 The optima user transitiona matrix P which can ead to the minimum cost can be represented as P = arg min P cost(a(1,2 u = arg min P P > S (1 P S (2 2 F st P 2{0, 1} U(1 U (2, kp k 1 P1 U(2 1 appe 1 U(1 1, P > 1 U(1 1 appe 1 U( Location Anchor Links Inference Simiar to users, ocations in onine socia networks are aso associated with both ink and attribute information (ike the ocation inks between users and ocations, profie information and text descriptions about the ocations, as we as the (ongitude, atitude coordinate information The (ongitude, atitude pairs of the same ocation in different networks are usuay not identica and various nearby ocations can have very cose coordinates, which pose great chaenges in addressing the probem Location Anchor Link Inference with Link Information Let L (1 and L (2 be the sets of ocations in networks G (1 and G (2 respectivey Based on the ocation inks between users and ocations in networks G (1 and G (2 (ie, E (1 u, and E (2 u,, we can construct the binary ocation adjacency matrices L (1 2 R U(1 L (1 and L (2 2 R U(2 L (2 for networks G (1 and G (2 respectivey Entries in L (1 and L (1 eg, L (1 (i, j and L (2 (, m are fied with vaue 1 iff user u (1 i has visited ocation (1 j in G (1 and user u (2 has visited ocation m (2 in G (2 Besides the user transitiona matrix P which maps users between G (1 and G (2, we can aso construct the binary ocation transitiona matrix Q 2{0, 1} L(1 L (2 based on the inferred ocation anchor ink set A (1,2, which maps ocations between G (1 and G (2 The cost introduced by the inferred ocation anchor ink set A (1,2 can be defined as the number of mis-mapped ocation inks across networks, ie, cost in ink(a (1,2 = P > L (1 Q L (2 2 F Location Anchor Link Inference with Attribute Information In ocation-based socia networks, each ocation has their own profie page, which shows the name and a the review comments about the ocation Simiar to the simiarity scores for user anchor inks, for any two ocations i 2L (1 and m 2L (2, based on the names of ocations i and m, we can cacuate the simiarity scores between i and m to be sim(n( i,n( m = n(i \ n(m n( i [ n( m

5 Users review comments can summarize the unique features about ocations, which are aso very important hints for inferring potentia ocation anchor inks Simiary, we represent users review comments posted as ocations i and m as bag-of-words vectors weighted TF-IDF, w( i and w( m And the simiarity between i and m based on the review comments can be represented as sim(w( i, w( m = w( i > w( i Coser ocations are more ikey to the same site than the ones which are far away Based on the (atitude, ongitude information, we propose to define the simiarity score between ocations i and m as foows: sim((at( i,ong( i, (at( m,ong( m = p (at(i at( m (ong( i ong( m p 2 (180 ( (90 ( 90 2 Furthermore, we can aso construct the simiarity matrix between ocations in G (1 and G (2 as 2 R L(1 L (2, where entry (j, m = 1 sim(n( 3 i,n( m + sim(w( i, w( m + sim((at( i,ong( i, (at( m,ong( i The optima ocation transitiona matrix Q which can minimize the cost in attribute information can be represented as cost in attribute(a (1,2 = kq k 1 Location Anchor Link Inference with Link and Attribute Information By considering the ocation inks and attributes attached to ocations simutaneousy, the cost function of inferred ocation anchor inks A (1,2 can be represented as cost(a (1,2 =cost in ink(a (1,2 + cost in attribute(a (1,2 = P > L (1 Q L (2 2 F kq k 1 The optima user and ocation transitiona matrices P and Q that can minimize the mapping cost wi be P, Q = arg min P,Q cost(a(1,2 = arg min P> L (1 Q L (2 P,Q st Q 2{0, 1} L(1 L (2, 2 F kq k 1, Q1 L(2 1 appe 1 L(1 1, Q > 1 L(1 1 appe 1 L(2 1, 313 Co-Inference of Anchor Links User transitiona matrix P is invoved in the objective functions of inferring both user anchor inks and ocation anchor inks, and these two different anchor ink inference tasks are strongy correated (due to P and can be inferred simutaneousy By integrating the objective equations of anchor ink inference for both users and ocations, the optima transitiona matrices P and Q can be obtained simutaneousy by soving the foowing objective function: P, Q = arg min P,Q cost(a(1,2 u +cost(a (1,2 = arg min P> S (1 P S (2 P,Q kp k 1 kq k 1, 2 F + P > L (1 Q L (2 2 F st P 2{0, 1} U(1 U (2, Q 2{0, 1} L(1 L (2, P1 U(2 1 appe 1 U(1 1, P > 1 U(1 1 appe 1 U(2 1, Q1 L(2 1 appe 1 L(1 1, Q > 1 L(1 1 appe 1 L(2 1 The objective function is an constrained 0 1 integer programming probem, which is hard to address mathematicay Many reaxation agorithms have been proposed so far [1] To sove the probem, in this paper, we propose to reax the binary constraint of matrices P and Q to rea numbers in range [0, 1] and entries in P and Q wi denote the existence probabiities/confidence scores of the corresponding anchor inks Redundant anchor inks introduced by such a reaxation wi be pruned with the co-matching agorithm to be introduced in the next section Meanwhie, the Hadamard product terms P and Q can be very hard to dea with when soving the optimization probem Considering that matrices P,, Q and are a positive matrices, we wi repace the L 1 norm of Hadamard product terms with the foowing Lemmas Lemma 1: For any given matrix A, the square of its Frobenius norm equas to the trace of AA >, ie, kak 2 F = tr(aa> Lemma 2: For two given positive matrices A and B of the same dimensions, the L 1 norm of the Hadamard product about A and B equas to the trace of A > B or AB >, ie, ka Bk 1 = tr(a > B= tr(ab > PROOF According to the definitions of matrix trace, terms tr(a > B and tr(ab > equas to the Frobenius product [18] of matrices A and B, ie, Meanwhie, ka tr(a > B=tr(AB > = i,j Bk 1 = i,j (A B(i, j = i,j A(i, jb(i, j A(i, j B(i, j Considering that both A and B are positive matrices, so the foowing equation can aways hod: ka Bk 1 = i,j A(i, j B(i, j =tr(a > B=tr(AB > where ocation anchor inks aso have one-to-one constraint, and the ast two equations are added to maintain such a constraint To sove the objective function, in this paper, we wi foow the Aternating Projected Gradient Descent (APGD method introduced in [12] and the one-to-one constraint is reaxed, where constraints P1 appe 1, P > 1 appe 1 wi be repaced with kpk 1 appe t instead, where t is a sma constant Simiary, the one-to-one constraint on Q is aso reaxed and repaced with kqk 1 appe t Furthermore, by incorporating terms kpk 1 and kqk 1 into the minimization objective function Based on the reaxed constraints as we as

6 Lemmas 1-2, the new objective function can be represented to be arg min f(p, Q =tr (P > S (1 P S (2 (P > S (1 P S (2 > P,Q + tr (P > L (1 Q L (2 (P > L (1 Q L (2 > User Preference Bipartite Graphs Co-Matching Network Fow Graph tr(p > tr(q > + kpk 1 + µ kqk 1 st 0 U(1 U (2 appe P appe 1 U(1 U (2, 0 L(1 L (2 appe Q appe 1 L(1 L (2, where and µ denote the weights on kpk 1 and kqk 1 respectivey As we can see, the objective function is with respect to P and Q and we cannot give a cosed-form soution for the objective function In this paper, we propose to cacuate the optima P and Q with aternative updating procedure based on the gradient descent agorithm: (1 fix Q and minimize the objective function wrt P; and (2 fix P and minimize the objective function wrt Q If during these two updating procedures, entries in P or Q become invaid, we use a projection to guarantee the [0, 1] constraint: (1 if P(i, j > 1 or Q(i, j > 1, we project it to 1; and (2 if P(i, j < 0 or Q(i, j < 0, we project it to 0 [12] Matrices P and Q can be initiaized with the method introduced in the Experiment Setting Section, and the aternative updating equations of these two matrices are avaiabe as foow: P = P 1 (P 1, Q = P S (1 PP > (S (1 > P +(S (1 > PP > S (1 P + L (1 QQ > (L (1 > P S (1 P(S (2 > (S (1 > PS (2 L (1 Q(L (2 > >, Q = Q 1 (P, Q = Q (L (1 > PP > L (1 Q (L (1 > PL ( µ11>, where 1 and 2 are the search steps in updating P and Q respectivey Such a updating process wi continue unti both P and Q converge The optima earning rates 1 and 2 obtaining the minimum f(p, Q can be represented as ( 1 = arg 1 min f(p, Q, ( 2 = arg 2 min f(p, Q The functions can be addressed by taking derivative of f( with regards to 1 (or 2 and make it equa to 0, we can obtain a cubic equation invoving 1 (or 2 Mutipe roots may exist when addressing the equation and the representation of the roots is very compicated In this paper, for simpicity, we propose to assign 1 and 2 with a constant vaue (ie, 005 in the experiments 32 Network Fow based Co-Matching To sove the objective function, the one-to-one constraints on both user anchor inks and ocation anchor inks are reaxed, which can take vaues in range [0, 1] As a resut, users and ocations in each network can be connected by mutipe user/ocation anchor inks of various confidence scores across networks simutaneousy and the one-to-one constraint can no onger hod any more To maintain such a constraint on both user and ocation anchor inks, Location Preference Bipartite Graphs S Figure 2: User and Location Preference Bipartite Graphs and Co-Matching Network Fow Graph we propose to prune the redundant ones introduced due to the reaxation with network fow based network co-matching agorithm in this subsection Based on user sets U (1 and U (2, ocation sets L (1 and L (2, as we as the existence confidence scores of potentia user and ocation anchor inks between networks G (1 and G (1 (ie, entries of P and Q, we can construct the user and ocation preference bipartite graphs as shown in the eft pots of Figure 2 User Preference Bipartite Graph The user preference bipartite graph can be represented as BG U = (U (1 [U (2, U (1 U (2, W U, where U (1 [U (2 denotes the user nodes in G (1 and G (2, U (1 U (2 contains a the potentia user anchor inks between G (1 and G (2, and W U wi map inks in U (1 U (2 to their confidence scores (ie, entries in P inferred in the previous section Location Preference Bipartite Graph Simiary, we can aso represent the ocation preference bipartite graph to be BG L =(L (1 [L (2, L (1 L (2, W L, where the weight mapping of potentia ocation anchor inks (ie, W L can be obtained from ocation transitiona matrix Q in a simiar way as introduced before Co-Matching Network Fow Graph In this paper, we empoy traditiona network fow agorithm to match users and ocations across networks G (1 and G (2 simutaneousy, which are grouped together in an integrated network fow mode, named co-matching network fow As shown in the right pot of Figure 2, based on the user preference bipartite graphs and ocation preference bipartite graphs, we propose to construct the co-matching network fow graph by adding (1 a source node S, (2 a sink node T, (3 inks connecting node S and inks in U (1 [L (1 (ie, {S} (U (1 [L (1, and (4 inks connecting nodes in U (2 [L (2 and node T (ie, (U (2 [L (2 {T } Bound Constraint In the network fow mode, each ink in the co-matching network fow graph is associated with a upper bound and ower bound to contro the amount of fow going through it For exampe, the upper and ower bounds of potentia user anchor ink (u, v 2U (1 U (2 in the co-matching network fow graph can be represented as B(u, v appe F (u, v appe B(u, v, where F (u, v denotes the fow amount going through ink (u, v, B(u, v and B(u, v represent the ower bound and upper bound associated with ink (u, v respectivey T

7 Considering that the constraint on both user and ocation anchor inks is one-to-one and networks studied in this paper are partiay aigned, users in onine socia networks incude both anchor and non-anchor users; so is the case for ocations In other words, each user and ocation in onine socia networks can be connected by at most one anchor inks across networks, which can be achieved by adding the foowing upper and ower bound constraint on inks {S} (U (1 [L (1 and (U (2 [L (2 {T }: 0 appe F (u, v appe 1, 8(u, v 2{S} (U (1 [L (1 [(U (2 [L (2 {T } Among a the potentia user anchor inks in U (1 U (2 and ocation anchor inks in L (1 L (2, ony part of these inks wi be seected finay due to the one-to-one constraint To represent whether a ink (u, v is seected or not, we set the fow amount going through inks U (1 U (2 [L (1 L (2 as integers with upper and ower bounds to be 0 an 1 (1 denotes the ink is seected, and 0 otherwise respectivey, ie, F (u, v 2{0, 1}, 8(u, v 2U (1 U (2 [L (1 L (2 Mass Baance Constraint In addition, in network fow mode, for each node in the graph (except the source and sink node, the amount of fow going through it shoud meet the mass baance constraint, ie, for each node in the network, the amount of network fow going into it shoud equas to that going out from it: F (w, u = F (u, v, w2n F,(w,u2L F v2n F,(u,v2L F where N F = {S} [U (1 [U (2 [L (1 [L (2 [{T} denotes a the nodes in the co-matching network fow graph and L F = {S} (U (1 [L (1 [U (1 U (2 [L (1 L (2 [(U (2 [L (2 {T } represents a the inks in graph Maximum Confidence Objective Function A the potentia inks connecting users and ocations across networks are associated with certain costs in network fow mode, where inks with ower costs are more ikey to be seected In this paper, we modify the mode a itte and aim at seecting the inks introducing the maximum confidence scores instead from U (1 U (2 and L (1 L (2 respectivey, which can be obtained with the foowing objective functions: max F (u, v W U(u, v max (u,v2(u (1 U (2 (m,n2(l (1 L (2 F (m, n W L(m, n The fina objective equation of simutaneous co-matching of users and ocations across networks can be represented to be max F (u, v W U (u, v+ (u,v2(u (1 U (2 (m,n2(l (1 L (2 F (m, n W L (m, n, st 0 appe F (u, v appe 1, 8(u, v 2{S} (U (1 [L (1 [ (U (2 [L (2 {T }, F (u, v 2{0, 1}, 8(u, v 2U (1 U (2 [L (1 L (2, F (w, u = F (u, v w2n F,(w,u2L F v2n F,(u,v2L F The above network fow objective function can be soved with open-source tookits (eg, ScipyOptimization 4 and GLPK 5 and Tabe 2: Properties of the Heterogeneous Networks # node # ink network property Twitter Foursquare user 5,223 5,392 tweet/tip 9,490,707 48,756 ocation 297,182 38,921 friend/foow 164,920 76,972 write 9,490,707 48,756 ocate 615,515 48,756 Figure 3: Convergence anaysis of iterative updating method the detaied derivative steps wi not be introduced here due to the imited space In the obtained soution, the fow amount variabe of potentia user and ocation anchor inks achieving vaue 1 are the seected ones which wi be assigned with abe +1, whie the remaining (ie, those achieving vaue 0 are not seected which are assigned with abe 1 4 EPERIMENTS To test the effectiveness of the proposed UNICOAT mode, in this section, extensive experiments wi be done on two rea-word partiay co-aigned onine socia networks: Foursquare and Twitter We wi describe the datasets used in this paper at first and then introduce the experiment settings in detai Finay, we wi show the experiment resuts and give brief anaysis about the resuts 41 Dataset Descriptions The socia networks dataset used in this paper are Foursquare and Twitter, which are co-aigned by both users and ocations shared between these two networks These two socia network datasets are crawed during November, 2012, whose statistica information is avaiabe in Tabe 2 More detaied descriptions and the crawing method is avaiabe in [28, 34] 42 Experiment Settings In this part, we wi introduce the experiment settings in detai, which incude (1 comparison methods, (2 evauation metrics, and (3 experiment setups 421 Comparison Methods To show the advantages of UNICOAT in addressing the PCT probem, we compare UNICOAT with many different baseine methods Considering that no known user and ocation anchor inks are avaiabe actuay in the PCT probem, as a resut, no existing supervised network aignment methods (eg, MNA [11] can be appied A the comparison methods are based on unsupervised earning settings, which can be divided into 4 categories: Co-Aignment Methods UNICOAT: Method UNICOAT introduced in this paper can aign two onine socia networks based on the shared users and ocations simutaneousy, which consists of two steps:

8 (1 unsupervised potentia user and ocation anchor inks inference; (2 co-matching of socia networks to prune redundant anchor inks to maintain the one-to-one constraint Bipartite Graph Aignment Methods BIGALIGN: Method BIGALIGN is a bipartite network aignment methods introduced in [12], which can aign two bipartite graphs (eg, user-product bipartite graph simutaneousy with ink information ony BIGALIGNET: Method BIGALIGNET is a bipartite network aignment methods introduced in this paper BIGALIGNET can aign user-ocation bipartite networks with both ocation inks between users and ocations as we as attribute information about users and ocations across networks Isoated Aignment Methods ISO: Method ISO is an unsupervised network aignment method introduced in [12] ISO merey infers the user anchor inks ony based on the friendship information among users ISOET: Method ISOET is an unsupervised network aignment method proposed in this paper, which is identica to ISO but utiizes both friendship inks among users and attribute information of users Traditiona Unsupervised Link Prediction Methods Reative Degree Distance based Network Aignment: RDD is the heuristics based unsupervised network aignment method introduced in [12] to fi in the initia vaues of the crossnetwork transitiona matrices, eg, P and Q in this paper For any two users/ocation u (i and u (j m in networks G (i and G (j, the reative degree distance between them can be represented as RDD(u (i,u (j m = 1+ deg(u(i (deg(u (i deg(u (j m +deg(u (j m /2 High reative degree distance denotes ower confidence score of anchor ink (u (i,u (j m 422 Evauation Metrics Methods UNICOAT (the first step, BIGALIGN, BIGALIGNET ISO, ISOET and RDD can output the confidence scores of potentia inferred inks but no abes are avaiabe, whose performance can be evauated by metrics ike AUC and Precision@100, etc As to method UNICOAT, inks seected finay in the matching are assumed to achieve confidence score 10 and abe +1, whie the remaining can achieve confidence score 00 and abe 1 As a resut, UNICOAT can aso output the abes of potentia anchor inks, whose performance can be evauated by various metrics, eg, AUC, Precision@100, Precision, Reca, F1 and Accuracy simutaneousy 423 Experiment Setup In the experiments, a the known user anchor inks and ocation anchor inks are used for evauation ony, which are not used in buiding modes at a Initiay, a fuy co-aigned Foursquare and Twitter invoving 200 users and 200 ocations are randomy samped from the data To obtain networks of different partia aignment degrees, extra non-anchor users and ocations are added to the network controed by partia aignment rate = #tota item #anchor item 2 {1, 2, 3, 4, 5}, where = 1 denote fu aignment and = 5 means #tota item #anchor item = 5, ie, extra 800 non-anchor users and 1 Tabe 3: Performance comparison of different methods for inferring user anchor inks (UNICOAT here denotes the first step of UNICOAT ony measure AUC Prec@100 methods UNICOAT BIGALIGNET BIGALIGN ISOET ISO RDD UNICOAT BIGALIGNET BIGALIGN ISOET ISO RDD Tabe 4: Performance comparison of different methods for inferring ocation anchor inks (UNICOAT here denotes the first step of UNICOAT ony measure AUC Prec@100 methods UNICOAT BIGALIGNET BIGALIGN RDD UNICOAT BIGALIGNET BIGALIGN RDD non-anchor ocations are added to the network We first cacuate the socia adjacency matrices S (1, S (2 and ocation adjacency matrices L (1, L (2 based on the socia inks among users and ocation inks between users and ocations With the attribute information, we can represent the user simiarity matrix as and ocation simiarity matrix as respectivey Parameter is set as 1 in the experiments for simpicity Before co-updating the user and ocation transitiona matrices P and Q, entries in P and Q are initiaized with the reative degree distance scores between users and ocations across networks Matrices P and Q wi be updated with equations given in Section 313 unti convergence The vaues of earning rates 1 and 2 are set as constant 005 in the experiments Based on the updated matrices P and Q, we can get the scores of potentia user anchor inks and ocation anchor inks across networks and further prune the non-existing ones with the network co-matching method introduced in Section 32 Links seected finay are abeed as +1 inks with confidence 10 (to be rea anchor inks and the remaining are abeed as 1 inks with confidence 0 (to be nonexisting anchor inks instead 43 Convergence Anaysis To sove the objective function, we propose to update matrices P and Q iterativey unti convergence To show that with the co-

9 (a AUC (b Figure 4: Performance of methods without matching in inferring user anchor inks (UNICOAT here denotes the first step of UNI- COAT ony (a AUC (b Figure 5: Performance of methods without matching in inferring ocation anchor inks (UNICOAT here denotes the first step of UNICOAT ony updating equations, matrices P and Q can do converge, we show the L 1 norm of matrices P and Q in each iteration is shown in Figure 3, where parameter is set as 1 (ie, the networks are fuy co-aigned and a the users and ocations are anchor instances As shown in the figures, as the mutua updating continues, the L 1 norm of both P and Q can converge very quicky to around 200 in ess than 5 iterations 44 Experiment Resuts The experiment resuts of addressing the PCT probem are avaiabe in Tabes 3-4 and Figures 4-7 In Figure 4 and 5, we fix =1and show the resuts achieved by comparison methods without matching step (ie, methods UNI- COAT (the first step, BIGALIGN, BIGALIGNET, ISO,ISOET and RDD evauated by AUC and Precision@100 Methods ISO and ISOET can ony be appied to aign networks via user generated information, which are not compared in the aignment resuts of ocations (ie, Figure 5 In both Figure 4 and 5, we can observe that (1 UNICOAT performs the best among a the comparison methods in inferring user and ocation anchor inks evauated by both AUC and Precision@100 For exampe, in Figure 4, UNI- COAT can achieve AUC score of 087, which is over 6% better than BIGALIGNET and ISOET, and 50% higher than the AUC score achieved by BIGALIGN, ISO and RDD Simiar performance of UNICOAT is avaiabe in other pots It demonstrates that utiizing the heterogeneous information in the network to infer user and ocation anchor inks simutaneousy can improve the resuts a ot (2 BIGALIGNET and ISOET can achieve better performance than BIGALIGN and ISO Recaing that methods BIGALIGNET and ISOET use both the ink and attribute information, whie BI- GALIGN and ISO use the ink information It justifies that the attribute information of both users and ocations is hepfu for inferring anchor inks across networks (3 By comparing UNICOAT with RDD (ie, the initiaization method of matrices P and Q in UNICOAT, we observe that UNICOAT can outperform RDD with significant advantages It proves the effectiveness of the proposed network co-aignment mode, which can obtain better resuts than the initia vaue 45 Sensitivity Anaysis In Figures 4-5, parameter is fixed as 1 In Tabes 3-4, we further change it with vaues in {1, 2, 3, 4, 5} by adding more nonanchor users and ocations into the network Generay, with more non-anchor users and ocations, the PCT wi become more difficut and the performance of a the methods wi degrade, but UNICOAT can achieve the best performance consistenty For exampe, when = 5, the AUC score achieved by UNICOAT in inferring socia inks is 0799, which is 67%, 45%, 31%, 548% and 572% higher than that gained by BIGALIGNET,BIGALIGN, ISOET, ISO and RDD respectivey Simiar observations can be obtained from the user anchor inks inference resuts evauated by Precision@100, and ocation anchor ink inference by both AUC and Precision@100 in Tabes 3-4 In the previous part, we have shown the performance of methods without matching step, whie anchor inks inferred by which cannot meet the one-to-one constraint Next, we wi test the ef-

10 (a Precision (b Reca (c F1 (d Accuracy Figure 6: Performance of methods with matching in inferring user anchor inks (UNICOAT here incudes both two steps of UNI- COAT (a Precision (b Reca (c F1 (d Accuracy Figure 7: Performance of methods with matching in inferring ocation anchor inks (UNICOAT here incudes both two steps of UNICOAT fectiveness of the matching step in pruning the non-existing anchor inks and the resuts achieved by UNICOAT (the second step are shown in Figures 6-7 Parameter are assigned with vaues in {1, 2, 3, 4, 5} The anchor inks inferred by UNICOAT can a meet the one-to-one constraint and are of high quaity For exampe, when =1, the Precision, Reca, F1 and Accuracy achieved by UNICOAT are 073, 054, 062 and 075 respectivey in inferring user anchor inks As increases, Reca and F1 scores achieved by UNICOAT wi decrease as it wi be more hard to identify the rea anchor inks among arger number of potentia ones Meanwhie, the Precision and Accuracy of UNICOAT wi increase The potentia reason can be due to the cass imbaance probem By adding more non-anchor users to the network, more non-existing anchor inks (ie, the negative cass inks wi be introduced and UNICOAT can achieve higher Precision and Accuracy by predicting more negative instances correcty 5 RELATED WORKS Network aignment probem is an important research probem, which have been studied in various areas, eg, protein-proteininteraction network aignment in bioinformatics [10, 13, 20], chemica compound matching in chemistry [22], data schemas matching data warehouse [16], ontoogy aignment web semantics [7], graph matching in combinatoria mathematics [15], and figure matching and merging in computer vision [6, 2] In recent years, witnessing the rapid growth of onine socia networks, researchers start to shift their attention to aign mutipe onine socia networks Homogeneous network aignment was studied in [24], enightened by which the probem of aigning two bipartite networks is studied by Koutra [12], where a fast aignment agorithm which can be appied to arge-scae networks is introduced Users can have various types of attribute information in socia networks generated by their socia activities, based on which Zafarani et a study the cross-network user matching probem in [25] In addition to attribute information, Kong et a [11] propose to fuy aign socia networks with the heterogeneous ink and attribute information simutaneousy based on a supervised earning setting Besides fuy aigning different socia networks, Zhang et a propose a framework for partia socia network aignment in [29], where the constraint on anchor inks is one-to-one appe Anchor inks are very hard to obtain and to make use of the sma amount known anchor inks, Zhang et a formuate the network aignment as a PU earning probem instead [31] In addition, users nowadays are usuay invoved in more than two socia networks, a genera mutipe (more than two network aignment framework is introduced in [33], which utiize the transitivity aw property of anchor inks to identify the optima resuts Across the aigned networks, various appication probems have been studied Cross-site heterogeneous ink prediction probems are studied by Zhang et a [28, 27, 34, 31] by transferring inks across partiay aigned networks Besides ink prediction probems, Jin and Zhang et a proposes to partition mutipe arge-scae socia networks simutaneousy in [30, 32, 8] The probem of information diffusion across partiay aigned networks is studied by Zhan et a in [26], where the traditiona LT diffusion mode is extended to the mutipe heterogeneous information setting Shi et a give a comprehensive survey about the existing works on heterogeneous information networks in [19], which incudes a section taking about network information fusion works and reated appication probems in detai 6 CONCLUSION Mutipe kinds of information entities can be shared across networks, eg, users and ocations In this paper, simutaneousy inference of the anchor inks connecting common users and common ocations across heterogeneous networks is studied A nove unsupervised co-aignment framework UNICOAT is introduced in this paper, which consists of two phrases: (1 co-inference of potentia user and ocation anchor inks based on an unsupervised earning setting, and (2 co-matching of networks to prune nonexisting anchor inks and maintain the one-to-one constraint on anchor inks Extensive experiments conducted on rea-word socia network datasets demonstrate the outstanding performance of UNI- COAT 7 ACKNOWLEDGEMENT This work is supported in part by NSF through grants III , CNS , and OISE , and Googe Research Award