Maximizing Acceptance Probability for Active Friending in Online Social Networks

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1 Maximizing for Active Friending in Online Social Network De-Nian Yang, Hui-Ju Hung, Wang-Chien Lee, Wei Chen Academia Sinica, Taipei, Taiwan The Pennylvania State Univerity, State College, Pennylvania, USA Microoft Reearch Aia, Beijing, China {dnyang, ABSTRACT Friending recommendation ha uccefully contributed to the exploive growth of online ocial network. Mot friending recommendation ervice today aim to upport paive friending, where a uer paively elect friending target from the recommended candidate. In thi paper, we advocate a recommendation upport for active friending, where a uer actively pecifie a friending target. To the bet of our knowledge, a recommendation deigned to provide guidance for a uer to ytematically approach hi friending target ha not been explored for exiting online ocial networking ervice. To maximize the probability that the friending target would accept an invitation from the uer, we formulate a new optimization problem, namely, Maximization (APM), and develop a polynomial time algorithm, called Selective Invitation with Tree and In-Node Aggregation (), to find the optimal olution. We implement an active friending ervice with on Facebook to validate our idea. Our uer tudy and experimental reult reveal that outperform manual election and the baeline approach in olution quality efficiently. Categorie and Subject Decriptor K.4.2 [Computer and Society]: Social Iue ; F.2.2 [Analyi of Algorithm and Problem Complexity]: Non-numerical Algorithm and Problem Keyword Friending, ocial network, ocial influence 1. INTRODUCTION Due to the development and popularity of ocial networking ervice, uch a Facebook, Google+, and LinkedIn, the new notion of ocial network friending ha appeared in recent year. To boot the growth of their uer bae, exiting ocial networking ervice uually provide friending recommendation to their uer, encouraging them to end invitation to make more friend. Conventionally, friending recommendation are made following a paive friending trategy, i.e., a uer paively elect candidate from the provided recommendation lit to end the invitation. Moreover, the recommended candidate are uually friend-of-friend of the Permiion to make digital or hard copie of all or part of thi work for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie bear thi notice and the full citation on the firt page. Copyright for component of thi work owned by other than ACM mut be honored. Abtracting with credit i permitted. To copy otherwie, or republih, to pot on erver or to reditribute to lit, require prior pecific permiion and/or a fee. Requet permiion from KDD 13, Augut 11 14, 213, Chicago, Illinoi, USA. Copyright 213 ACM /13/8...$15.. uer, epecially thoe who hare many common friend with the uer. Thi trategy i quite intuitive becaue friendof-friend may have been acquaintance or friend offline. Furthermore, mot uer may feel more comfortable to end a friending invitation to friend-of-friend rather than a total tranger whom they have hared no ocial connection with at all. It i believed that given the ucce rate of uch a paive friending trategy to be high, it ha contributed to the exploive growth of online ocial networking ervice. In contrat to paive friending, the idea of active friending, where a peron may take proactive action to make friend with another peron, doe exit in our everyday life. For example, in a high chool, a tudent fan may like to make friend with the captain in the chool occer team or with the lead inger in a rock-and-roll band of the chool. A aleperon may be intereted in getting acquainted with a high-value potential cutomer in the hope of making a buine pitch. A young KDD reearcher may deire to make friend with the leader of the community to participate in organization and ervice of a conference. However, to the bet of our knowledge, the idea of providing friending recommendation to ait and guide a uer to effectively approach another peron for active friending ha not been explored in exiting online ocial networking ervice. We argue that ocial networking ervice provider, intereted in exploring new revenue and further growth of their uer bae, may be intereted in upporting active friending. One may argue that in exiting ocial networking ervice, an active friending initiator can end an invitation directly to the friending target anyway. 1 However, it may not work if the initiator i regarded a a tranger by the target, epecially when they are ocially ditant, i.e., they have no common friend. Therefore, to increae the chance that the target would accept the friending invitation, it may be a good idea for the initiator to firt know ome friend of the target, which in turn may require the initiator to know ome friend of friend of the target. In other word, if the initiator would like to plan for ome action, he may need the topological information of the ocial network between the target and himelf, which unfortunately i unavailable due to privacy iue. In thi ituation, it would thu be preferable if the ocial networking ervice provider, given a target pecified by the initiator, could provide a tep-by-tep guidance in the form of recommendation to ait the initiator to make friend toward the target. In thi paper, we are making a grand uggetion for the ocial networking ervice provider to upport active friending. Our ketch i a follow. By iteratively recommending a lit of candidate who are friend of at leat one exiting friend of the initiator, a ocial networking ervice provider may upport active friending, without violating the current practice of privacy preervation in recommendation. Con- 1 For the ret of the paper, we refer to the friending initiator and friending target a initiator and target for hort. 713

2 ider an initiator who pecifie a friending target. The ocial networking ervice, baed on it proprietary algorithm, recommend a et of friending candidate who may likely increae the chance for the target to accept the eventual invitation from the initiator. Similar to the recommendation for paive friending, the recommendation lit conit of only the friend of exiting friend of the initiator. By appearance, the initiator follow the recommendation to end invitation to candidate on the lit. The invitation i diplayed to a candidate along with the lit of common friend between the initiator and the candidate o a to encourage acceptance of the invitation. 2 A uch, the aforementioned tep i repeated until the friending target appear in the recommendation lit and an invitation i ent by the initiator. It i clear, however, that the recommendation made for paive friending may not work well becaue active friending i target-oriented. The recommended candidate hould be carefully choen for the initiator, guiding him to approach the friending target tep by tep. To upport active friending, the key iue i on the deign of the algorithm that elect the recommendation candidate. A imple cheme i to provide recommendation by unveiling the hortet path between the initiator and the target in the ocial network, i.e., recommending one candidate at each tep along the path. A uch, the initiator can gradually approach the target by becoming acquainted with the individual on the path. However, thi hortet-path recommendation approach may fail a oon a a middle-peron doe not accept the friending invitation (ince only one candidate i included in the recommendation lit for each tep). To addre thi iue, it i deirable to recommend multiple candidate at each tep ince the initiator i more likely to hare more common friend with the target and thereby more likely to get accepted by the target. Thi i epecially true, if broadcating of the friending invitation can be made to all neighbor of the initiator friend. In other word, the probability to reach the friending target and get accepted can be effectively maximized a an enormou number of path are flooded with invitation to approach the target. Neverthele, the mechanim of friending invitation i abued here becaue the above undirectional broadcat aimlely involve many unneceary neighbor. Moreover, the initiator may not want to handle a large number of tediou invitation. In thi paper, we tudy a new optimization problem, called Maximization (APM), for active friending in online ocial network. The ervice provider, eager to explore new monetary tool to increae revenue, may conider charging uer for the active friending ervice. 3 Given an initiator, a friending target t, and the maximal number of invitation allowed to be iued by the initiator, APM find a et R of node, uch that can equentially end invitation to the node in R in order to approach t. The objective i to maximize the acceptance probability at t of the friending invitation when end it to t. The parameter control the trade-off between the expected acceptance probability of t and the anticipated effort made by for active friending t. 4 Again, R i not returned to a a whole due to privacy concern. Intead, only a ubet R of node that are adjacent to the exiting friend of are recommended 2 Thi i alo a common practice for paive friending in exiting ocial networking ervice uch a Facebook, Google+, and LinkedIn. 3 Recent new ha reported that Facebook now allow it uer to pay to promote their and their friend pot [1]. 4 Since i not aware of the network topology and the ditance to t, it i not reaonable to let directly pecify. Intead, it i more promiing for the ervice provider to lit a et of and the correponding acceptance probabilitie and monetary cot, o that the uer can chooe a proper according to her available budget. to, while other ubet of R will be recommended to a appropriate in later tep 5. The pread maximization problem [6, 7, 14, 17], which alo adopt a probabilitic influence model, i different from APM in thi paper. Given an initiator and hi friend, APM intend to dicover an effective ubgraph (i.e., R) between the eed and t. On the other hand, the pread maximization problem, given the topology of the whole ocial network, aim to find a given number of eed to maximize the ize of the whole pread t. To tackle the APM problem, we propoe three algorithm: i) Range-baed Greedy () algorithm, ii) Selective Invitation with Tree Aggregation (SITA) algorithm, and iii) Selective Invitation with Tree and In-Node Aggregation () algorithm. elect candidate by taking into account their acceptance probability and the remaining budget of invitation, leading to the bet recommendation for each tep. However, the algorithm doe not achieve the optimal acceptance probability of the invitation to a target due to the lack of coordinated friending effort. On the other hand, aiming to ytematically elect the node for recommendation, SITA i deigned with dynamic programming to find node which may reult in a coordinated friending effort to increae the acceptance probability of the target. SITA i able to obtain the optimal olution, yet ha an exponential time complexity. To addre the efficiency iue, further refine the idea in SITA by carefully aggregating ome information gathered during proceing to alleviate redundant computation in future tep and thu obtain the optimal olution for APM in polynomial time. The contribution of thi paper are ummarized a follow. We advocate the idea of active friending in online ocial network and propoe to upport active friending through a erie of recommendation lit which erve a a tep-to-tep guidance for the initiator. We formulate a new optimization problem, called Acceptance Probability Maximization (APM), for configuring the recommendation lit in the active friending proce. APM erve to maximize the acceptance probability of the invitation from the initiator to the friending target, by recommending elective intermediate friend to approach the target. We propoe a number of new algorithm for APM. Among them, Selective Invitation with Tree and In- Node Aggregation () derive the optimal olution for APM with O(n V r 2 R ) time, where n V i the number of node in a ocial network, and i the number of invitation budgeted for APM. We implement in Facebook in upport of active friending and conduct a uer tudy including 169 volunteer with varied background. The uer tudy and experimental reult indicate that efficiently outperform manual election and the baeline approach in olution quality. The ret of thi paper i organized a follow. Section 2 introduce a model for invitation acceptance and formulate APM. Section 3 review the related work. Section 4 preent the SITA and algorithm propoed for APM. Section 5 report our uer tudy and experimental reult. Finally, Section 6 conclude the paper. 5 In thi paper, APM i formulated a an offline optimization problem intending to maximize the expected acceptance probability. In an online cenario where the initiator doe not end invitation to ome node in R or ome node in R do not accept the invitation, a new APM with renewed invitation budget could be re-iued to obtain adapted recommendation. While thi cenario raie important iue, it i beyond the cope of thi paper. 714

3 2. INVITATION ACCEPTANCE The notion of acceptance probability refer to an invitation. Thu, here we firt dicu two important factor that may affect the acceptance probability of a friending invitation in the environment of online ocial networking ervice and decribe how in thi work we determine whether an individual would accept a received invitation. Next, we explain why the iue of deriving the acceptance probability over a ocial network i very challenging and how we addre thi iue by adopting an approximate probability baed on a maximum influence in-arborecence (MIIA) tree. We formulate the acceptance probability maximization (APM) problem baed on the MIIA tree. The invitation acceptance model follow the exiting ocial influence and homophily model, which have been jutified in the literature. Later in Section 5, the invitation acceptance model will be validated by a uer tudy with 169 volunteer. 2.1 Factor for Invitation Acceptance In the proce of active friending, while friending candidate are recommended for the initiator to end invitation, whether the invitee will accept the invitation remain uncertain. Baed on prior reearch in ociology and online ocial network [12, 2, 21], we argue that when a peron receive an invitation over an online ocial network, the deciion of the invitee depend primarily on two important factor: i) the ocial influence factor [13, 14], and ii) the homophily factor [9, 12, 21]. Here, the ocial influence factor repreent the influence from the urrounding (i.e., common friend) of individual in the ocial network on the deciion. On the other hand, the homophily factor capture the fact that each individual in a ocial network ha a ditinctive et of peronal characteritic, and the imilaritie and compatibilitie among the characteritic of two individual can trongly influence whether they will become friend [12]. Between them, ocial influence come from etablihed ocial link, while the homophily between two individual may exit without a prerequiite of etablihed ocial relationhip. Thu, we conider thee two factor eparately but aim to treat them in a uniformed fahion in our derivation of the acceptance probability for an invitation. A the ocial influence factor involve the tructure of the ocial network (i.e., the common friend of the individual), we firt conider the acceptance probability of an invitation in term of ocial influence. 6 Let the ocial network be repreented a a ocial graph G(V, E) where V conit of all the uer in the ocial networking ytem and E be the etablihed ocial link among the uer. An edge weight w u,v [, 1] on the directed edge (u, v) E probabilitically denote the ocial influence of u upon v. The probability can be derived according to an exiting method [13, 14] according to the interaction in online ocial network, while the etting of negative ocial influence ha alo been introduced in [4]. Thu, if u i aociated with an invitation from a uer to v (i.e., u i a common friend of and v), w u,v i the probability for v to be ocially influenced by u to accept the invitation. 7 Hence, the acceptance probability for an invitation can be derived by taking into account the ocial influence of all the exiting common friend aociated with an invitation. It i aumed that each common friend u ha an independent ocial influence on the invitee v to accept the friending invitation [9, 12, 2] and thu the overall acceptance probability can be obtained by aggregating the individual ocial influence. Later, the uer tudy in Section 6 We intend to extend it with homophily factor later. 7 The ocial influence probability ha been extenively ued to quantify the probability of ucce in the proce of conformity, aimilation, and peruaion in Social Pychology [9, 12, 2]. While how to obtain the edge weight i an active reearch topic [13, 24], it i out of cope of thi paper. 5 demontrate that the influence probability and homophily probability derived according to the literature are conitent to the real probabilitie meaured from the uer. While obtaining the acceptance probability for a given invitation (a decribed above) i imple, deriving the acceptance probability for a friending target t who doe not have any common friend with the initiator become very challenging becaue more than one invitation need to be iued (o a to make ome common friend firt), and there are complicated correlation among uer acceptance event for uer between and t. Moreover, our ultimate tak i to find a et R of intermediate uer between and t with ize at mot for to end invitation to, o a to maximize the acceptance probability of t. We call thi problem the acceptance probability maximization (APM) problem. Due to the combinatorial nature of thi invitation et R, it i till difficult to find uch a et to maximize the acceptance probability of t even in cae where computing the acceptance probability i eay. The following theorem make the above two hardne precie. Theorem 1. Given the et of neighbor S of the initiator, computing the acceptance probability of t i #P-hard. Moreover, finding a et R with ize that maximize the acceptance probability of target t i NP-hard, even for cae when computing acceptance probability i eay. Proof. We prove the theorem in [3]. 2.2 Approximate The pread maximization problem in the Independent Cacade (IC) model [17] alo face the challenge in Theorem 1. To efficiently addre thi iue, an approximate IC model, called MIA, ha been propoed [4, 5, 6]. The ocial influence from a peron u to another peron v i effectively approximated by their maximum influence path (MIP), where the ocial influence w u,v on the path (u, v) i the maximum weight among all the poible path from u to v. MIA create a maximum influence in-arborecence, i.e., a directed tree, MIIA(t, θ) including the union of every MIP to t with the probability of ocial influence at leat θ from a et S of leaf node. The MIA model ha been widely adopted to decribe the phenomenon of ocial influence in the literature [4, 5, 6] with the following definition on activation probability, which i baically the ame a the acceptance probability if broadcat friending invitation to all node in MIIA(t, θ). Definition 1. The activation probability of a node v in MIIA(t, θ) i ap (v, S, MIIA(t, θ))) = 1, if v S, if N in (v) = 1 u N in (v) (1 ap (u, S, MIIA(t, θ)) w u,v ), otherwie where N in (v) i the et of in-neighbor of v. Note that ap (u, S, MIIA(t, θ)) w u,v i the joint probability that u i activated and uccefully influence v, and u can never influence v if it i not activated. Therefore, the activation probability of a node v can be derived according to the activation probability of all it in-neighbor, i.e., the child node in the tree. Since S i the et the leaf node, the activation probabilitie of all node in MIIA(t, θ) can be efficiently derived in a bottom-up manner from S toward t. In light of the imilarity between the IC model and the deciion model for invitation acceptance in active friending with no budget limitation of invitation, we alo exploit MIA to tackle the APM problem. MIIA(t, θ) i contructed by the MIP from all friend of to t, i.e., S i the et of friend of. In other word, θ i et a to enure that the ocial influence from every friend i fully incorporated. Neverthele, different from the activation probability in the literature, which allow the influence to propagate via every node 715

4 and friend of non-friend of v 7 v 3 v 8 v 12 v 1 v 4 t v 2 v 5 v 6 ocial influence link homophily link v 9 v 1 v 11 Figure 1: Combining the ocial influence and homophily factor in MIIA(t, θ), the acceptance probability for active friending allow the ocial influence to take effect on the invitation acceptance only via a et R of node to be elected in our problem. Thu, we define the acceptance probability for an invitation to node v a follow. Definition 2. The acceptance probability for an invitation of a node v in MIIA(t, θ) i ap(v, S, R, MIIA(t, θ))) = 1, if v S, if v / R or N in (v) = 1 u N in (v),u R (1 ap(u, S, R, MIIA(t, θ)) w u,v), otherwie where N in (v) i the et of in-neighbor of v. Equipped with MIA, we are able to derive the acceptance probability of t efficiently with a imple iterative approach from the leaf node to the root (i.e., t). The above MIA arborecence incorporate only the ocial influence factor. A dicued earlier, the homophily factor between the initiator and the receiver of an invitation i alo crucial for friending. Homophily in [9, 12, 21] repreent the probability for two individual u and v to create a new ocial link due to hared common peronal characteritic. Homophily in Sociology manifet the general tendency of people to aociate with other and imilar other can be quantified with variou approache [2, 16, 29]. The homophily probability can be et according to [3]. To extend MIA, we attach a duplicated to each node with a directed edge, with a parameter pecifying the homophily factor from to v. The MIP from each candidate to t, together with the directed edge from to the candidate, i incorporated in the extended MIA. Therefore, the extended MIA i alo an arborecence, where each leaf node i a friend of or herelf, and thoe leaf node make up the et S. Figure 1 how an example of the extended MIA. For each internal node, uch a v 1, it acceptance probability factor are not only the ocial influence from v 3 and v 4 but alo the homophily factor between and v 1. In thi paper, the influence probability and homophily probability are derived according to the above literature without aociating them with different weight. Later, uer tudy will be preented in Section 5, and the reult how that the real acceptance probability complie with the acceptance probability of the above model. 2.3 Problem Formulation In thi work, we formulate an optimization problem, called Maximization (APM), to elect a given number of intermediate people to ytematically approach the friending target t baed on MIIA(t, θ). The APM problem i formally defined a follow. Maximization (APM). Given a ocial network G(V, E), an initiator and a friending target t, elect a et R of uer for to end friending invitation uch that the acceptance probability ap(t, S, R, MIIA(v, θ)) i maximized, where S i the friend of, including itelf. A analyzed later, the optimal olution to APM can be obtained in O(n V r 2 R ) time 8, where n V i the number of node in a ocial network, and i the total number of invitation allowed. The etting of ha been dicued in Section 1. It i worth noting that APM maximize the acceptance probability of t, intead of minimizing the number of iteration to approach t, which can be achieved by the hortet path routing in an online ocial network. Neverthele, it i poible to extend APM by limiting the number of edge in an MIP of MIIA(t, θ), to avoid incurring an unaccepted number of iteration in active friending. 3. RELATED WORK Recommendation for paive friending ha been explored in the recent few year. Chen et al. [3] demontrated that friending recommendation baed on the topology of an online ocial network are the eaiet way to reult in the acceptance of an invitation. In contrat, recommendation baed on content poted by uer are very powerful for dicovering potential new friend with imilar interet [3]. Meanwhile, reearch how that preference extracted from ocial networking application can be exploited for recommendation [15]. To avoid recommending ocially ditant candidate, uer are allowed to pecify different ocial contraint [23], e.g., the ditance between a uer and the recommended friending target, to limit the cope of friending recommendation. Moreover, community information ha been explored for recommendation [25]. It i important to note that the aforementioned reearch work and idea are propoed for paive friending, where the friending target are determined by the recommendation engine of ocial networking ervice provider in accordance with variou criteria (e.g., preference and ocial cloene). Thu, the uer can conveniently (but paively) end an invitation to target on the recommendation lit. Complementary to the conventional paive friending paradigm, in thi paper, we propoe the notion of active friending where a friending target can be pecified by the initiator. Accordingly, the recommendation ervice may ait and guide the initiator to actively approach a target. The impact of ocial influence ha been demontrated in variou application, uch a viral marketing [6, 17, 18] and interet inference [28]. Given an online ocial network, a major reearch problem i the eed election problem, where the eed correpond to the leaf node of MIA (i.e., initiator and her friend) in our problem. In contrat, APM elect the topology between the friend and t, intead of electing the eed. The homophily factor, capturing the tendency of uer to connect with imilar other, ha been conidered in everal application, uch a by identifying either truted uer [26] or uer relationhip [31] in ocial network. Notice that ome work develop algorithm to return a ubgraph or path, uch a community detection [19], hortet path [1], pattern matching [11], or graph iomorphim query [8]. In contrat to the hortet path query, our algorithm for the APM problem emphaize the returning of a graph, intead of a path. The topology of the returned graph contain valuable neighborhood information of ome common friend who can be leveraged to effectively increae the acceptance probability of a friending invitation. The initiator of a pattern matching or a graph iomorphim query need to pecify a ubgraph a the query input. However, the goal of thi tudy i to find an unknown graph between and t to maximize the acceptance probability of a invitation to a friending target. 8 MIA wa propoed to implify the IC model, which i computation intenive and not calable. Neverthele, we prove that APM in the IC model i NP-hard for general network and not ubmodular in [3]. 716

5 4. ALGORITHM DESIGN To tackle the APM problem, we deign efficient algorithm in upport of the invitation recommendation for active friending. From our earlier dicuion, it can be eaily oberved that the et of intermediate node in R, i.e., thoe to be recommended for invitation, play a crucial role in maximizing the acceptance probability for active friending. Here we firt introduce a range-baed greedy algorithm which provide ome good inight for our other algorithm. The algorithm, given an invitation budget, aim to find the et of invitation candidate for recommendation to an initiator who would like to make friend with a target t. Let R denote the anwer et, which i initialized a empty at the beginning. The algorithm iteratively elect a node v from the neighbor of current friend and add it to R baed on two heuritic: 1) the highet acceptance probability and 2) the number of remaining invitation. The purpoe of the former i to minimize the potential wate of a friending invitation, while the latter avoid electing a node too far away to reach t by contraining that v can only be at mot R 1 hop away from t. A a reult, the range-baed greedy algorithm i inclined to firt expand the friend territory of and then aggreively approach toward the neighborhood of t. 4.1 Selective Invitation with Tree Aggregation While the range-baed greedy algorithm i intuitive, the node added to R at eparate iteration are not elected in a coordinated fahion. Thu, it i difficult for the rangebaed greedy algorithm to effectively maximize the acceptance probability. To addre thi iue, we propoe a dynamic programming algorithm, call Selective Invitation with Tree Aggregation (SITA), that find the optimal olution for APM by exploring the maximum influence in-arborecence tree rooted at t (i.e., MIIA(t, θ)) in a bottom-up fahion. SITA tart from the leaf node, i.e., node without inneighbor, to explore MIIA(t, θ) in a topological order until t i reached finally. In order to obtain the optimal olution, SITA need to invetigate variou allocation of the invitation to different node cloe to or t in MIIA(t, θ). However, it i not neceary for SITA to enumerate all poible invitation allocation. Thank to the tree tructure of MIIA(t, θ), for each node v, SITA ytematically ummarize the bet allocation for v, i.e. which generate the highet acceptance probability for v, correponding to the ubtree rooted at v. The ummarie will be exploited later by v parent node, which i the only out-neighbor of v, to identify the allocation generating the highet probability. The above procedure i repeated iteratively until t i proceed, and the allocation of invitation to the ubtree rooted at t i the olution returned by SITA. More pecifically, let f v,r denote the maximum acceptance probability for v to accept the invitation from while r invitation have been ent to the ubtree rooted at v in MIIA(t, θ). By firt orting all node in topological order to t, we proce f v,r of a node v after all f u,r of it in-neighbor u have been proceed. Apparently, f v, = for every node v that i not a friend of becaue no invitation will be ent to the ubtree rooted at v. On the other hand, for every leaf node v, which i a friend of (or itelf), f v,r = 1 for r =. For all other node v in MIIA(t, θ), SITA derive f v,r according to each in-neighbor f ui,r i a follow, f v,r = max {1 [1 f ui,r i w ui,v]}, (1) ri =r 1 u i N in (v) where N in (v) denote the et of in-neighbor of v with N in (v) = d v, u i i an in-neighbor of v, and r i i the number of invitation ent from to the ubtree rooted at u i. An invitation i ent to v, while the remaining r 1 invitation are ditributed to the in-neighbor of v. SITA effectively avoid examining all poible ditribution of the r 1 invitation to the node in the ubtree. Intead, Eq. (1) examine only.75 u 4 u 5 friend of non-friend of.96.8 u 6 u u 3.7 u 8 u u2.85 u 1 u u 12 u 15 u u 14 u 16 t.9.9 u 18 u 2 u u u 17 u u 22 u 23.9 u 25 u Figure 2: The running example(not including and her edge) f ui,r i of each in-neighbor u i of v on every poible number of invitation r i. In other word, only the in-neighbor of v, intead of all node in the ubtree, participate in the computation of f v,r to efficiently reduce the computation involved. For each node v, f v,r i derived in acending order of r until reaching r = min(, z v ), where z v i the number of node that are not friend of in the ubtree rooted at v. SITA top after f t,rr i obtained. In the following, we how that SITA find the optimal olution to APM. 9 Lemma 1. Algorithm SITA anwer the optimal olution to APM. Proof. We prove the lemma by contradiction. Aume that the olution from SITA, i.e., f t,rr, i not optimal. According to the recurrence, there mut exit at leat one inneighbor t 1 N in (t) together with the number of invitation r 1 uch that f t1,r 1 i not optimal. Similarly, ince f t1,r 1 i not optimal, there exit at leat one in-neighbor t 2 N in (t 1 ) of t 1 with the number of invitation r 2 uch that f t2,r 2 i non-optimal, r 2 < r 1. Here in the proof, let f ti,r i denote the non-optimal olution found in i-th iteration of the above backtracking proce, which will continue and eventually end with a probability f ti,r i uch that 1) t i i a friend of but f ti, 1 or f ti,r i = for r i >, or 2) t i i not a friend of but f ti,. The above two cae contradict the initial aignment of SITA. The lemma follow. Example. Figure 2 illutrate an example of MIIA(t, θ) with = 7, where the node denote the uer involved in deriving the maximal acceptance probability for t and the number labeled on edge denote the influence probability between two node. Without lo of generality, and her homophily edge are not hown in Figure 2. Note that the dark node at the leaf are and her exiting friend and thu have the acceptance probability a 1, while the white node are the recommendation candidate to be returned by SITA along with their acceptance probabilitie. SITA explore MIIA(t, θ) from the dark leaf node in a topological order, i.e., the f v,r of a node v i derived after all f u,r of it inneighbor u are proceed. Take u 4 a an example. f u4, = ince no invitation i ent, f u4,1 = 1 (1 f u5,.75) =.75. Similarly, for u 8, f u8, = and f u8,1 =.95. Conider u 6 which ha in-neighbor u 7 and u 8, f u6, =, f u6,1 =.8, and f u6,2 = 1 (1 f u7,.8)(1 f u8,1.7) =.933. Notice that for a node v, f v,r i derived for r [, min(z v, )], e.g., for u 6, we only derive r [, 2]. Neverthele, to find f v,r, SITA need to try different allocation by ditributing the number of invitation r i to each different neighbor u i and then combining the olution f ui,r i to acquire f v,r. For example, to derive f u17,5, it i neceary to ditribute 4 invitation to it in-neighbor, including u 18, u 2 and u 24. The poible allocation for (r 18, r 2, r 24 ) include (, 1, 3), (, 2, 2), (1,, 3), 9 Due to the pace contraint, we do not how the peudocode of SITA here but refer the reader to the next ection where a more general i preented. u 27.5 u 28 u

6 (1, 1, 2) and (1, 2, 1) 1, which will obtain acceptance probability.5738,.7539,.781, 674 and.7639 repectively. Eventually, we obtain f u17,5 = Notice that the number of poible allocation grow exponentially. After all the node are proceed, we obtain f t,7 = On the other hand, the greedy algorithm elect a uer v / R with the highet acceptance probability and at mot R 1 hop away from t. Accordingly, it elect u 8, u 6, u 15, u 12, and u 3 equentially. In the 6th tep, the node with the highet probability i u 1. However, u 1 i 3-hop away with 3 > R 1 = 2 and thu i not elected. Intead, it elect the node with the next highet acceptance probability, i.e., u 2. In the lat tep, only the root t can be elected, o obtain a olution with the acceptance probability a 13. A uch, SITA outperform. 4.2 Selective Invitation with Tree and In-Node Aggregation Unfortunately, SITA i not a polynomial-time algorithm becaue in Eq. (1), O(r d v ) allocation are examined to ditribute r 1 invitation to the ubtree of the d v in-neighbor correponding to each node v. To remedy thi calability iue, we propoe the Selective Invitation with Tree and In- Node Aggregation () to anwer APM in polynomial time. effectively avoid the proceing of O(r dv ) allocation by iteratively finding the bet allocation for the firt k in-neighbor, which in turn i then exploited to identify the bet allocation for the firt k + 1 in-neighbor. The proce iterate from k = 1 till k = d v. Conequently, the poible allocation for ditributing r 1 invitation to all in-neighbor are returned by Eq. (1) in O(d v ) time, where d v i the in-degree of v in MIIA(t, θ). To efficiently derive f v,r in Eq. (1), we number the inneighbor of v a u 1, u 2... to u dv, where d v i the in-degree of v. Let m v,k,x denote the maximum acceptance probability by ending x invitation to the ubtree of the firt k neighbor of v, i.e., u 1 to u k. Initially, m v,1,x = f u1,x, x [, ]. derive m v,k,x according to the bet reult of the firt k 1 in-neighbor, m v,k,x = max {1 [1 m v,k 1,x x ][1 f uk x,x wu k,v]}, [,min(z uk,x)] (2) where f uk,x w u k,v i the acceptance probability for allocating x invitation to the k-th in-neighbor u k, and m v,k 1,x x i the bet olution for allocating x x invitation to the firt k 1 in-neighbor. By carefully examining different x, we can obtain the bet olution m v,k,x for a given k. tart from k = 1 to k = d v. For each k, begin with x = until x = min( i [1,k] z u i, 1), where i [1,k] z u i i the total number of node that are not friend of in the ubtree of the firt k in-neighbor. top after finding every m v,dv,x, x [, min(z v, 1)]. The peudocode i preented in Algorithm 1, and the following lemma indicate that the optimal olution of APM i m t,dt, 1. Lemma 2. For any v and r, f v,r = m v,dv,r 1. Proof. We prove the lemma by contradiction. Aume that m v,dv,r 1 i not optimal. According to the recurrence, there exit at leat one r dv uch that m v,dv 1,( 1 r d v ) i not optimal. Similarly, ince m v,dv 1,(r 1 r d v ) i not optimal, there exit at leat one r dv 1 uch that m v,dv 2,(r 1 r d v r dv 1) i not optimal. Therefore, let m v,dv i,(r 1 σi ), where σ i = j [,i 1] r d v j, denote the non-optimal olution obtained in the i-th iteration. The backtracking proce continue and eventually end with i = d v 1, where m v,1,r1 f u1,x. It contradict the initial aignment of m v,1,r1, and the lemma follow. 1 Some allocation are eliminated ince r i / [, min(, z ui )]. k x = 1 x = 2 x = 3 x = 4 x = 5 x = * * * * * * * * Table 1: All m u17,k,x The following theorem prove that the algorithm anwer the optimal olution to APM in O(n V r 2 R ) time, where n V i the number of node in a ocial network, and i the number of invitation in APM. Note that any algorithm for APM i Ω(n V ) time becaue reading MIIA(t, θ) a the input graph require Ω(n V ) time. Therefore, i very efficient, epecially in a large ocial network with n V ignificantly larger than d max and. Theorem 2. Algorithm anwer the optimal olution to APM in O(n V r 2 R ) time. Proof. According to Lemma 1 and Lemma 2, obtain the optimal olution of APM. Recall that n V i the number of node in the ocial network, and d v i the indegree of a node v in MIIA(t, θ). The algorithm contain O(n V ) iteration. Each iteration examine a node v to find m v,dv,x for every x [, min(z v 1, 1)], where i number of invitation ent by in APM. There are O(d v) cae to be conidered to explore all m v,dv,x for v in Eq. (2), and each cae require O( ) time. Therefore, finding m v,dv,x for a node v need O(d v r 2 R ) time, and for all node in MIIA(t, θ) it i O( v d vr 2 R ), where v d v = E. A MIIA(t, θ) i a tree (i.e. E = n V 1), the overall time complexity i O(n V r 2 R ). The theorem follow. Example. In the following, we illutrate how derive f u17,r, r [, z u17 ]. At the beginning, the in-neighbor of u 17 are ordered a u 18, u 2 and u Then, we find all m u17,1,x = f u18,x 1w u18,u 17, x [, min(z u18, 1)] firt, repreenting the maximum acceptance probability u 17 obtained by only ending x invitation to the ubtree rooted at the firt in-neighbor, i.e., u 18. Then we derive m u17,2,x for x [, min(z u18 + z u2, 1)] to acquire the maximum acceptance probability of u 17 by ending invitation to ubtree rooted at u 18 and u 2. Notice that different x, repreenting the invitation ditributed to the k-th ubtree, need to be examined in order to find the optimal olution. For intance, while deriving m u17,3,4, we compare 1 (1 m u17,2,1)(1 f u24,3 w u24,u 17 ) =.781, 1 (1 m u17,2,2)(1 f u24,2) w u24,u 17 ) =.7539 and 1 (1 m u17,2,3)(1 f u24,1) w u24,u 17 ) =.7417 and obtain m u17,3,4 = After deriving all f u17,x+1 = m u17,d v,x for x [, 6] (min( 1, z u17 1) = 6), the computation of u 17 finihe. Table 1 lit the detailed reult, where * denote the intance with r exceeding the number of people who are not the friend of in the firt k ubtree PERFORMANCE EVALUATION We implement active friending in Facebook and conduct a uer tudy and a comprehenive et of experiment to validate our idea of active friending and to evaluate the performance of the propoed algorithm. In the following, we firt detail the methodology of our evaluation and then preent the reult of our uer tudy and experiment. 5.1 Methodology We adopt a uer tudy and experiment, two complementary approache, for the performance evaluation. We aim to ue the uer tudy to invetigate how the recommendationbaed active friending approach fare with the approach baed on the uer own trategie (i.e., which they would 11 To avoid confuion, we keep their ID in thi example without renaming them a u 1, u 2, and u Note that m u7,k,x = when x =. 718

7 Algorithm 1 Selective Invitation with Tree and In-Node Aggregation () Require: The query iuer ; the targeted uer t; the influence tree MIIA(t, θ) rooted at t; the number of requet that can end. Enure: A et R of elected uer that end requet to, uch that the acceptance probability i maximized. 1: Obtain a topological order σ which order a node without in-neighbor firt. 2: for v σ do 3: //obtain all f v,r, r [, min(, n v )] 4: Order in-neighbor of v a u 1, u 2,... u dv 5: m v,,r for r [, min( 1, n r 1)] 6: for k = 1 to d v do 7: for x = to min( i [1,k] z u i, 1) do 8: m v,k,x 9: for x = to min(z uk, x) do 1: if m v,k,x < 1 [1 m v,k 1,x x ][1 f uk,x wu k,v] then 11: m v,k,x 1 [1 m v,k 1,x x ][1 f uk,x wu k,v] 12: π v,k,x x 13: f v, 14: f v,x+1 m v,k,x, x [, min( 1, i [1,k] zu i )] 15: Backtrack π v,k,x to obtain R 16: return R with maximized f t,rr follow under the exiting environment of ocial networking ervice). To perform the uer tudy, we implement an app. on Facebook. Through the app., the uer i able to decide whom to invite baed on their own trategie to approach the target. Meanwhile, according to the recommendation generated from the Range-Baed Greedy () algorithm and the Selective Invitation with Tree and In-Node Aggregation () algorithm, repectively, the uer alo end alternative et of invitation to carry out the active friending activitie for comparion. 13 Note that Selective Invitation with Tree Aggregation (SITA) i not conidered becaue it make exactly the ame recommendation a. We recruited 169 volunteer to participate in the uer tudy. Each volunteer wa given 25 target with varied invitation budget to work on. The ocial ditance between the volunteer and the target were pre-determined in order to collect reult for comparion under controlled parameter etting. We further conducted experiment by imulation to evaluate the olution quality and efficiency of SITA,, and, implemented in an HP DL58 erver with four Intel Xeon E GHz CPU and 128 GB RAM. Two large real dataet, FacebookData and FlickrData were ued in the experiment. FacebookData contain 6,29 uer and 1,545,686 friend link crawled from Facebook [27], and FlickrData contain 1,846,198 uer and 22,613,981 friend link crawled from Flickr [22]. The initiator and target t are elected uniformly at random. An important iue faced in both of our uer tudy and the experiment i the ocial influence and homophily factor captured in the ocial network, which are required for, SITA and to make recommendation. Mot of the previou work adopted a fixed probability (e.g., 1/degree in [17, 6, 4, 7]) or randomly choe a probability from a et a value (e.g.,.1,.1,.1 in [6, 4]) due to the abence of real ocial influence probabilitie and homophily probabilitie. To addre thi iue, in the uer tudy, we obtained the ocial influence probability on each edge by mining the interaction hitory of volunteer in Facebook in accordance with [13, 14]. We alo derived the homophily probabilitie 13 To alleviate the burden of the participant, we end invitation on their behalve to the recommended candidate..8 Actual Derived # of common friend (a) Invitation Acceptance Actual Derived d,t (b) MIIA Tree Figure 3: Verifying the derived acceptance probability from to other node by mining the profile information in their Facebook page baed on [3]. The ocial network in the uer tudy i denoted a UerStudyData. A for the ocial network in FacebookData and FlickrData that were ued for the experiment, we unfortunately do not have peronal profile and hitorical interaction of the node. Thu, we could not generate the ocial influence probability and homophily probability by mining real data. A a reult, we choe to aign the link weight of the ocial network baed on: i) the ditribution of ocial influence and homophily probabilitie obtained from our uer tudy (denoted a US), and ii) the Zipf ditribution for it ability to capture many phenomena tudied in the phyical and ocial cience [32]. 5.2 Uer Study Through the uer tudy, we logged the repone of participant to invitation and thu were able to calculate the acceptance probabilitie correponding to invitation under variou circumtance. Uing the collected data, we made a number of comparion. Firt, we wanted to verify that the acceptance probability of an active friending plan derived baed on the MIIA tree (uing the mined ocial influence and homophily probabilitie a the link weight) are conitent with that of the plan being executed in the uer tudy. To thi end, we firt verified the accuracy of our invitation acceptance model (for ingle invitation) by comparing the derived acceptance probability and the actual acceptance probability obtained from real activitie in the uer tudy. Figure 3(a), in which reult obtained from the uer tudy and our model are repectively labeled a Actual and Derived, plot the comparion in term of the number of common friend in an invitation. A can be een, the acceptance probabilitie of both Actual and Derived increae a the number of common friend in invitation increae. Mot importantly, the reult are conitently cloe, indicating that our invitation acceptance model with it ocial influence and homophily weight ued i able to reaonably capture the deciion making proce for invitation in real life. Notice that the above comparion focue on the apect of invitation acceptance only, without taking into account the ocial network topology, which we approximate with the MIIA tree. To verify that uing the MIIA tree i ufficiently effective for active friending planning, we further compared acceptance probability derived uing our propoed algorithm with the actual acceptance probability obtained through executing the plan in the uer tudy. Figure 3(b) how that, under variou ditance between the initiator and target, the acceptance probabilitie derived uing MIIA tree i reaonably cloe to the actual acceptance probabilitie. Next, we compared the effectivene of trategie baed on, and the participant own heuritic. Figure 4(a) plot the comparion by varying the number of friending invitation,. and generally outperform uer heuritic (labeled a Uer) under all etting. We can oberve that the performance of i generally very good and improve a increae, while the performance of Uer and are marked with a leap from = 5 to 1 and remain cloe thereafter. Thi indicate the value of the extra computation effort required for deriving 719

8 Uer (a) Different (d,t = 3) Uer 2 3 d 4 5,t (b) Different d,t ( = 2) Figure 4: Acceptance probability in the uer tudy SITA Figure 5: Acceptance probability (d,t = 2) (ec.) SITA UerStudyData E-5 FacebookData.592 > 7 day.2 FlickrData Figure 6: Average Running time ( = 2) recommendation due to the increaed invitation budget are worthwhile, outperforming the heuritic trategie derived baed on and human intuition. Figure 4(b) decribe the evaluation of the acceptance probability of t under variou etting of d,t. When d,t i 2, it i more likely that there will be a lot of common friend (due to the nature of ocial network), and thu have greater acceptance probabilitie. When d,t increae, it become more difficult for an initiator to make effective deciion due to the maller number of common friend and the lack of knowledge about the ocial network topology that i larger and more complex. A i evident in Figure 4, ha the bet performance. 5.3 Experimental Reult While the uer tudy verifie that i able to achieve the bet performance, the ize of the ocial network i mall due to the limited number of volunteer participating in the tudy. To further validate our idea in a large-cale ocial network and to evaluate the calability of, we conducted an experimental tudy by imulation Scalability A proven earlier, SITA can obtain the optimal olution of APM. However, it i not calable a it need to examine all combination of invitation allocation. Here we ued it a a baeline to compare the efficacy and efficiency with over ocial network of different ize. Firt, we compared the reult by randomly ampling 5 (initiator, target) pair uing UerStudyData. A depicted in Figure 5, both SITA and ignificantly outperform in term of acceptance probability. Next, we compared their running time, not only uing UerStudyData but alo the large-cale FacebookData and FlickrData. A Figure 6 how, the SITA algorithm take more than 7 day without returning the anwer, and i thu not feaible for practical ue. For the ret of the experiment, we only compared with Senitivity Tet In thi ection, we conducted a erie of enitivity tet to examine the impact of different parameter, including the invitation budget ( ), the number of friend of (N), the ditance between the initiator and target (d,t ), and the kewne of ocial influence and homophily probabilitie (α). In experiment on the impact of, N, d,t, we teted both the FacebookData (US) and FlickrData (US). 14 A the obervation on both dataet are quite imilar, we only report both reult for the firt experiment and kip the FlickrData 14 US and ZF denote the link weight aigned repectively baed on model from the Uer Study and Zipfian Ditribution (a) Acceptance probability. # of edge in longet path (b) Number of edge Figure 7: Varying (FacebookData (US)) (a) Acceptance probability # of edge in longet path (b) Number of edge Figure 8: Varying (FlickrData (US)) reult for the ret due to pace contraint. Finally, in the lat experiment, we ue FacebookData (ZF) to oberve how α may potentially impact our algorithm. Impact of. By varying and etting the default d,t of ampled (, t) pair a 4, we compared and in term of the acceptance probability and the number of iteration uing FacebookData (US) and FlickrData (US) (ee Figure 7 and Figure 8, repectively). A can be een in Figure 7(a) and 8(a), exhibit a much better performance than, regardle of the. Meanwhile, Figure 7(b) and 8(b) reveal that the longet path in the olution R obtained by i horter than that in becaue tend to pend invitation on ome local uer with higher acceptance probabilitie. 15 Impact of N. We are intereted in finding out whether the number of friend of ha an impact on the performance. Thu, we choe four different group of initiator (who have around 1, 2, 3, and 4 friend, repectively) and ampled 1 different target t to compare their acceptance probabilitie. With et a 25, Figure 9 how that a the number of friend increae, the initiator have more choice to reach their target. In thi way, can find the optimal olution with a high acceptance probability, while the near-ighted tend to elect the friend of friend with higher acceptance probabilitie and eventually reult in a mall acceptance probability to t. Impact of d,t. We alo conducted an experiment to undertand the impact of d,t on the performance. Not urpriingly, the finding i conitent with our uer tudy (pleae refer to Section 5.2 and Figure 4(b)). A uch, we do not plot the reult here due to the pace limitation. Impact of α. In the experiment above, ocial influence and homophily factor are modeled baed on our uer tudy, but the ditribution in different ocial network may vary. Thu, through the kewne parameter α, we ued the Zipfian ditribution to examine the impact of α on our algorithm. A can be oberved in Figure 1, the ditribution of ocial influence and homophily become more kewed (i.e., α increae) and the acceptance probabilitie of and drop, becaue it become more difficult for invitation to get accepted when there are maller number of highly influential link while the number of le influential link increae. It i alo worth noting that, a the line in the figure indicate, the percentage of performance difference between and (i.e., the acceptance probability of divided by that of ) increae, which indicate that i able to handle a kewed ditribution much better than. 15 i inclined to take more time to reach t becaue invitation are equentially ent toward t. The latency of friending a new intermediate node i different for each node. 72

9 N Figure 9: Varying N (FacebookData (US)) α 1.5 Performance difference Figure 1: Varying α (FacebookData (ZF)) 6. CONCLUSION AND FUTURE WORK Oberving the need of active friending in everyday life, thi paper formulate a new optimization problem, named Maximization (APM), for making friending recommendation on online ocial network. We propoe the algorithm called Selective Invitation with Tree and In-Node Aggregation (), to find the optimal olution for APM and implement in Facebook. The uer tudy and experimental reult demontrate that active friending can effectively maximize the acceptance probability of the friending target. In our future work, we will firt explore the impact of the delay between ending an invitation and acquiring the reult in active friending. Thi i important when the uer want to make friend with the target within certain time frame. In addition, for multiple friending target, it i not efficient to configure recommendation eparately for each target. An idea i to give priority to the intermediate node that can approach many target imultaneouly. We will tudy active friending for a group of target in our future work. 7. ACKNOWLEDGMENTS Thi work wa upported in part by the National Science Council, Taiwan, under NSC E-1-3-MY3 and NSC E REFERENCES [1] ocial/facebook-promoted-pot/index.html. [2] V. Chaoji, S. Ranu, R. Ratogi, and R. Bhatt. Recommendation to boot content pread in ocial network. In Proc. WWW, page , 212. [3] J. Chen, W. Geyer, C. Dugan, M. J. Muller, and I. Guy. Make new friend, but keep the old: Recommending people on ocial networking ite. In CHI, page 21 21, 29. [4] W. Chen, A. Collin, R. Cumming, T. Ke, Z. Liu, D. Rincón, X. Sun, Y. Wang, W. Wei, and Y. Yuan. Influence maximization in ocial network when negative opinion may emerge and propagate. In SDM, page , 211. [5] W. Chen, W. Lu, and N. Zhang. Time-critical influence maximization in ocial network with time-delayed diffuion proce. In AAAI, 212. [6] W. Chen, C. Wang, and Y. Wang. Scalable influence maximization for prevalent viral marketing in large-cale ocial network. In KDD, page , 21. [7] W. Chen, Y. Yuan, and L. Zhang. Scalable influence maximization in ocial network under the linear threhold model. In ICDM, page 88 97, 21. [8] J. Cheng, Y. Ke, W. Ng, and A. Lu. FG-index: Toward verification-free query proceing on graph databae. In SIGMOD, page , 27. [9] R. B. Cialdini and N. J. Goldtein. Social influence: Compliance and conformity. Annual Review of Pychology, 55: , 24. [1] E. Cohen, E. Halperin, H. Kaplan, and U. Zwick. Reachability and ditance querie via 2-hop label. In SODA, page , 22. [11] W. Fan, J. Li, S. Ma, H. Wang, and Y. Wu. Graph homomorphim reviited for graph matching. In VLDB, page , 21. [12] T. L. Fond and J. Neville. Randomization tet for ditinguihing ocial influence and homophily effect. In WWW, page 61 61, 21. [13] A. Goyal, F. Bonchi, and L. V. S. Lakhmanan. Learning influence probabilitie in ocial network. In WSDM, page , 21. [14] A. Goyal, F. Bonchi, and L. V. S. Lakhmanan. A data-baed approach to ocial influence maximization. In VLDB, page 73 84, 211. [15] I. Guy, N. Zwerdling, D. Carmel, I. Ronen, E. Uziel, S. Yogev, and S. Ofek-Koifman. Peronalized recommendation of ocial oftware item baed on ocial relation. In RecSy, page 53 6, 29. [16] S. Hangal, D. MacLean, M. S. Lam, and J. Heer. All friend are not equal: Uing weight in ocial graph to improve earch. In Proc. SNA-KDD Workhop, 21. [17] D. Kempe, J. M. Kleinberg, and É. Tardo. Maximizing the pread of influence through a ocial network. In KDD, page , 23. [18] J. Lekovec, A. Kraue, C. Guetrin, C. Falouto, J. M. VanBrieen, and N. S. Glance. Cot-effective outbreak detection in network. In KDD, page , 27. [19] K. Macropol and A. K. Singh. Scalable dicovery of bet cluter on large graph. In VLDB, page , 21. [2] P. V. Marden and N. E. Friedkin. Network tudie of ocial influence. Sociological Method & Reearch, 22(1): , [21] M. McPheron, L. 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