Exploiting Users' Inconsistent Preferences in Online Social Networks to Discover Private Friendship Links

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1 Exploiting Users' Inconsistent Preferences in Online Social Networks to Discover Private Friendship Links Lei Jin*, Hassan Takabi #, Xuelian Long*, James Joshi* *School of Information Sciences University of Pittsburgh {lej17, xul10, # Department of Computer Science and Engineering University of North Texas ABSTRACT In a social network system, a friendship relation between two users is usually represented by an undirected link and it is visible in both users' friend lists. Such a dual visibility of a friendship link may raise privacy threats. This is because both the users of a friendship link can separately control its visibility to other users and their preferences of sharing such a friendship link may not be consistent. Even if one of them conceals the friendship link from a third user, that third user may find the link through the other user's friend list. In addition, as most social network users allow their friends to see their friend lists, an adversary can exploit these inconsistent policies caused by users' conflicting preferences to identify and infer many of a targeted user's friends and even reconstruct the topology of an entire social network. In this paper, we propose, characterize and evaluate such an attack referred as the Friendship Identification and Inference (FII) attack. In an FII attack scenario, an adversary first accumulates the initial attack relevant information based on the friend lists visible to him in a social network. Then, he utilizes this information to identify and infer a target's friends using a random walk based approach. We formally define the attack and present the attack steps, the attack algorithm and various attack schemes. Our experimental results using three real social network datasets show that FII attacks are effective in inferring private friendship links of a target and predicting the topology of the social network. Currently, most popular social network systems, such as Facebook, LinkedIn and Foursquare, are susceptible to FII attacks. Categories and Subject Descriptors K.4.1 [Computer and Society]: Public Policy Issues privacy General Terms Experimentation, Security Keywords Privacy; Social Networks; Friendship Attack Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. WPES '14, November 3, 2014, Scottsdale, Arizona, USA. Copyright 2014 ACM /14/11 $ INTRODUCTION In social network systems (SNSs), such as Facebook and LinkedIn, friendship links are usually undirected. A friendship link involves two users and the relationship is visible in both users' friend lists. Both users can separately control the visibility of their friend lists to other users, and hence, a privacy policy conflict may arise in terms of users' inconsistent preferences. Such an inconsistency of the policies could cause the violation of a user's privacy settings. In an SNS, such as Facebook, a friendship between users A and C is not visible to another user B only when both A and C do not allow B to view their friend lists. When only A (C) disallows B to see the link, B can find that A (C) is in C's (A's) friend list if C (A) allows B to view his friend list. Since the friendship graph is undirected, B can now conclude that A and C are friends. In this case, A's (C's) privacy policy for B is violated. Visible Link I II B F E G B Identified Link in an FII Attack Inferred Link in an FII Attack E Figure 1. Examples of the FII attack Most users in Facebook, LinkedIn and Foursquare allow their friends to see their friend lists since it is the default policy and users tend not to change the default privacy policies [30, 31]. An adversary can first target a friend or 2-distant neighbor who does not allow the adversary to see his friends. He can then exploit any inconsistency in policies related to the visibility of the friend lists to identify and infer many friends of the targeted user. For example, as shown in Figure 1.I, let user B be the adversary and user T be his target whose privacy settings do not allow B to see his friend list. Users A, D and E are B's friends and they allow B to view their friend lists. A, C and D are actually T's friends but initially B is not supposed to know that. However, by checking A's and D's friend lists, B can directly identify that A and D are T's friends. In addition, using the proposed random walk based approach (introduced in Section 2.4) on this social graph, B can infer that C has a high A D A C T C T D

2 probability of being a friend of T. Now, all of T's friends are identified and inferred by B. In an SNS, the friend lists of some users are public. An adversary can also target one user from these friend lists and then identify and infer friends of the targeted user. Note that the targeted user is not limited to be a friend or a 2-distant neighbor of the adversary in this attack scenario. For example, in Figure 1.II, let user B be the adversary and user T be his target whose privacy settings do not allow B to see his friend list. Users A, C, D and E are friends of T but B is initially not supposed to know that. The friend lists of A and C are public and T is included in both A's and C's friend lists. Then, B can directly see that A and C are friends of T from A's and C's public friend lists. Using the proposed random walk based approach, B can also infer that D and E are also highly possible to be T's friends. A user's friend list in an SNS is usually a valuable and sensitive piece of information for the user [1-4]. For instance, Facebook's move to suspend Google's Friend Connect program's access to Facebook's social graph in 2008 was motivated by the perceived importance of ensuring the privacy of friendship links [1]. In LinkedIn, when a user connects with several well-known professionals, it may help him in his search for jobs. At the same time, the exposure of a user's friend list may also play against his opportunities, for instance, if it includes an individual with a bad reputation. In addition, if an adversary finds a user's friend list, he may leverage it to launch further, more dangerous privacy attacks. For example, the adversary may infer the user's private attributes in his profile based on the public information of his friends [2, 5, 6]; the adversary can also successfully launch a profile cloning or identity clone attack on a targeted user whose friend list is exposed [7, 8]. In this paper, we propose a Friendship Identification and Inference (FII) attack that aims to identify and infer a target's friends in an SNS. In an FII attack, an adversary first constructs his initial attack knowledge based on the friend lists visible to him in a social graph. Based on this, his goal is to identify and infer a target's friends using our proposed random walk with restart approach; this approach has been shown to be probably the most effective approach for predicting links in social networks [27]. Here, we assume that an adversary uses it, as part of an FII attack, to infer as many private friendship relationships of a target as possible based on the initial knowledge he has. We demonstrate the effectiveness of FII attacks on three different real social network datasets using various measurements. The contributions of this paper can be summarized as follows: We first define an FII attack that aims to identify and infer a target's friends in SNSs and then present the detailed attack steps, the attack algorithm and various attack schemes which can make FII attacks more successful. Our experimental results show that FII attacks are effective in inferring private friendship links of a target and predicting the topology of the friendship network. We also evaluate the relationships between social network features and the attack results and discuss the potential defense solutions against such FII attacks. The remainder of the paper is organized as follows. We propose the FII attack, its associated algorithm and various attack schemes in Section 2 and present simulations of the FII attacks using three real social network datasets in Section 3. We present the related work in Section 4 and conclude the paper with a discussion of future work in Section FRIENDSHIP IDENTIFICATION & INFERENCE ATTACK In this section, we formalize the Friendship Identification and Inference (FII) attack and introduce the attack steps, the corresponding attack algorithm and various attack schemes. 2.1 Basic Definitions in SNSs An SNS is generally modeled as a graph G(V, E), where V represents a set of users and E={(x, y) x, y V} represents a set of friendship links among users. In a social graph, such as Facebook and LinkedIn, the friend link is usually undirected. An edge e = (x, y) is added to E when a friend request from x to y or from y to x is accepted. Adjacency Matrix. A social graph G can be represented by an adjacency matrix An n where n = V. For each aij A, aij = 1 if there is an edge e = (i, j) E; otherwise aij = 0. For each user i V, the user set F(i) ={ j j V, aij = 1} is the friend list of i. Note that in a social graph, when aij = 1, then aji = 1, i.e., j F(i) and i F(j). The user set D2(i) = { j k, j V, aij = 0, aik = 1 and akj = 1} represents the 2-distant neighbors of i. The 2-distant neighborhood relationship in a social graph is also undirected. Transition Matrix. Transition matrix is a matrix which describes the probability of transitioning from one node to another node in a social graph. It can be obtained based on the adjacency matrix. Let Pn n be a transition matrix, where pij represents the probability of stepping on node j from node i and pij = aij / j aij. The transition matrix is an important variable in the random walk based approach described in Section Adversary's Initial Attack Knowledge in an FII Attack In an SNS, such as Facebook, a user can see his friend list; most users share their friend lists with their friends [30, 31]; few users may hide their friend lists from specific friends; and some users make their friend lists public. Hence, given an adversary b, we categorize these types of friend lists that he can see: Type A: the friend lists of his friends who authorize b to view them. Type B: the friend lists of users whose friend lists are public. Note that these users are not friends of b. Hence, b's friend list, the friend lists in Type A and those in Type B form a sub graph which is an adversary's initial attack knowledge K(b) for an FII attack. K(b) = (Vkb, Ekb) where Vkb is a set of all users in K(b) and Ekb is a set of all edges in K(b). For example, in Figure 2, there are two connected components of a social graph. Let user B be the adversary. In the left part, we can see that users A and C are B's friends. Assume that A allows B to see his friend list while C hides his friend list from B. B does not know that the edge (E, F) exists as B is not a friend of user E and E does not allow B to see his friend list. B does not know that the edge (C, D) exists since C conceals his friend list from B. Note that B can identify the edge (B, C) = (C, B) from his friend list. B can also know edges (A, B), (A, E) and (A, D) from A's friend list. In the right part, assume that B cannot find a path to the sub graph composed of G, H, I and J in the social graph. Assume that the friend lists of users G and I are public and those of H and J are only visible to

3 their friends. From these public friend lists, B can know edges (G, H), (G, J), (H, I) and (I, J). Hence, VkB ={A, B, C, D, E, G, H, I, J}, EkB = {(A, B), (A, E), (A, D), (B, C), (G, H), (G, J), (H, I), (I, J)} in this case. The edge initially visible to B E F A B D C The edge initially invisible to B Figure 2. Adversary B's initial attack knowledge Assume there are m distinct nodes in Vkb for an adversary b. As mentioned in the previous subsection, we can use the adjacency matrix Am m (b) and the transition matrix Pm m (b) to represent K(b). 2.3 Friendship Identification and Inference (FII) Attack Definition We present the definition of an FII attack as follows: DEFINITION 1: Given a social graph G(V, E), an adversary b V and a target t V, Assumption on Target t's Privacy Setting: t defines a policy that does not authorize b to see F(t); Assumptions on Adversary's Initial Knowledge: A1: b's initial attack knowledge K(b) = (Vkb, Ekb) is constructed based on friend lists visible to him; A2: t is included in Vkb. Privacy Attack: We say that t is a victim (compromised 1 node) of an FII attack launched by b if b can identify and correctly infer at least e friends of t's friends in Fb(t) based on the proposed random walk based link predictions on K(b). In the definition, Fb(t) represents a set of t's friends identified and inferred by b through the FII attack and each user in Fb(t) is regarded as t's friend by b. Note that some users in Fb(t) may not be actual friends of t. Parameter e is a threshold set to distinguish the point at which privacy violation becomes a significant concern (a successful attack). Note that in an FII attack, we assume that additional actions, such as mutual friend based queries, are not needed to extend K(b) defined above. We note that b may be even more successful in carrying out the FII attack using some additional features and the related attack schemes are discussed in Section 2.5. Also note that an FII attack can also be used to identify and infer the absence of friendship links between t and other users. The exposure of absence of friendship links (potential friends) between t and other users can cause various problems for t. For example, in an identity clone attack [8], an adversary can connect with the potential friends of a targeted user before the targeted user actually does that. Such an action will make it difficult for the targeted user to become friends with the potential friends. However, in this paper, G H J I we only focus on identifying and inferring the existing friends of a targeted user Targets In an FII attack, a target t can be considered as one of these three types based on the relationship with an adversary b: 1) a friend; 2) a 2-distant neighbor; 3) other users who are more than 2-distant away from b in a social graph. Note that a target t should be included in Vkb according to one of the assumptions from an adversary b's initial knowledge. t can be any of b's friends since t is included in b's friend list and it is included in Vkb. When t is a 2-distant neighbor of b, 1) b should be able to see at least one friend list of his friend who is friend with the 2-distant neighbor; or 2) there is at least one public friend list that contains t. In these two cases, t can also be included in Vkb. When t is more than 2-distant away from b in a social graph, b has to search the public friend lists within the SNS to see if b is included in one of them. To discover users' public friend lists in a social graph, an adversary 1) can utilize the public search feature built in an SNS to discover the users whose friend lists are public; and 2) can discover the groups where the members are publicly available and then to check if some members' friend lists are public [4] Property Below we propose and prove a key property related to the FII attack based on the following definitions. DEFINITION 2: Given a social graph G(V, E), an adversary b V, a target t V, K(b) = (Vkb, Ekb) and t's friend list F(t), we define a user set Ib(t) = Vkb F(t). DEFINITION 3: Given a social graph G(V, E), an adversary b V, a target t V and Fb(t) which is generated by b from an FII attack, we define a user set FCb(t) which represents correctly identified and inferred friends of t. FC b(t) = Fb(t) F(t). Theorem: When b can compromise t in an FII attack, FCb(t) Ib(t). Proof: Since Fb(t) Vkb according to the definition of the FII attack, FC b(t) = Fb(t) F(t) Vkb F(t) = I b(t). From this theorem, we can know the specific set of t's friends who can be exposed to b in an FII attacks. We can also see that the maximum number of friends exposed to b is equal to Ib(t). E T s Friend Network F D B A T H G K(b) Figure 3. An example of the exposed friends to an adversary C User in I b (t) I J 1 In this paper, when we say b compromises t in an FII attack, it means that b can successfully identify and/or infer t's friends during the attack.

4 For example, in Figure 3, let user B be the adversary and user T be his target. We can see that only users A, C and D can be exposed to B in an FII attack. User I and J will not be exposed since they are not included in VkB. 2.4 Attack Algorithm & Schemes Random Walk & Attack Modeling Intuitively, the probability of a user A being a friend of another user C is related to the following conditions [14]: Condition 1: A's degree (number of friends); Condition 2: the network distance between A and C. When A is close to C in a social graph, it is more likely for them to be friends. Condition 3: number of the different paths between A and C. For example, in Figure 1.I, there are two different shortest paths (the length is 2) between users T and C and hence it is highly likely for them to be friends in this case. User F is a 4-distant neighbor of T in the social graph and the probability of the friendship between them is low. A simple random walk is a mathematical formalization of a path that consists of a succession of random steps. Assume there is a surfer walking in a network and let xl(i) be the probability that the surfer is at node i at time l. xl+1(i) is probability that the surfer is at node i at time l+1. Xl+1(i) = j (Probability of being at node j) Probability of walking from j to i = j xl (j) pji where pji is an element of the transition matrix for the network (see Section 2.1). Let Xl = (xl(i)) be a vector that represents the probabilities of the surfer being at any node in the network at time l. Hence, we have Xl+1 = XlP = Xl-1PP = =X0 P l where P represents the transition matrix for the network and X0 is a predefined vector. When Xl+1 XlP < ɛ, we say that the probability of the surfer being at any node i is stable for a long period of time. We model an FII attack as a random walk as follows: An adversary's initial attack knowledge K(b) is modeled as the network of the random walk. The probability of the surfer being at any node i in the random walk can be regarded as the probability of the friendship between the surfer and node i in an FII attack. In an FII attack, a target t is the surfer and Xl+1 is the probability indicating whether there is a friendship link between t and other users when Xl+1 XlP < ɛ. When the probability of the friendship between t and a user i is higher than that for other users, i is more likely to be friends with t. Random walk based approaches have been proved to be effective for predicting friend and users' trajectory in SNSs [12-15]. However, the simple random walk mentioned above is not appropriate for an FII attack. This is because the probability of the friendship between t and any user i is only related to the number of i's friends in the simple random work (Condition 1) [14]. Further, the related conditions 2 and 3 are also overlooked. By considering all the conditions, we adopt the random walk with restart approach [13, 28]. In such an approach, a node follows the simple random walk with the probability of β but it jumps back to the originator (it is a target t in an FII attack) with probability 1 β. That is Xl+1(i) = β j xl (j) pji + (1 β) X0. From this equation, we can see that the probability of t and a user i being friends tends to be higher when i is closer to t. In addition, when there are more paths between t and i, the probability of t and i being friends also tends to be higher. This is because the value of β j xl (j) pji is higher (j represents the number of 2-distant paths between t and i). Additionally, since i is closer to t (2-distant away), the part of (1 β) X0 will also make the Xl+1(i) be a higher value Attack Steps & Algorithm In an FII attack, there are three steps: collection of adversary's initial attack knowledge, friend identification and friend inference. The adversary b's initial attack knowledge K(b) is constructed based on users' privacy settings for their friend lists. When a user is b's friend, b can find him from his friend list and check whether his friend list is visible. When a user is b's 2-distant neighbor, b may find the user from one of friend lists of his friends and then check whether the user's friend list is visible. For a user who is neither b's friend nor 2-distant neighbor, b may find the user using the approaches mentioned in Section and then check whether the user's friend list is available. Note that when b's initial attack knowledge contains several separated sub-graphs, we only consider the sub-graph that includes the target. For example, in Figure 2, there are two separate subgraphs. Sub-graph 1 consists of users A, B, C, D, E and F while subgraph 2 consists of users G, H, I and J. There is no edge connecting these two sub-graphs. Assume B is the adversary and J is the target in an FII attack. In this scenario, b's initial attack knowledge includes only sub-graph 2. Such a setting can make sure that the random walk will be stable for a long period of time and it can also help in improving the attack result. Additionally, an adversary needs to set a parameter λ, which is used to determine the total number of users identified and inferred as t's friends (the size of Fb(t)) before launching an FII attack. The purpose of the friend identification step is to identify a target's friends based on the initial attack knowledge that an adversary has. In this step, an adversary b is able to check whether a target t exists in the edges in Ekb of K(b). When (i, t) or (t, i) Ekb, i is t's friend. In the friend inference step, an adversary b adopts the random walk with restart approach to infer friends of a target t based on K(b). In particular, an adversary b has to complete the following steps to infer t's friends: 1. Compute the adjacency matrix A and the transition matrix P based on K(b). 2. Get FIDb(t) which represents the friends identified in the friend identification step. 3. Define the initial vector V0 =(vi)m, where m is the number of distinct nodes in A. For each vi, when i is the index of t in A, vi = 1; otherwise, vi = Define a parameter β (0, 1). Define Vl as a distribution for t after l times walking. Vl = β P Vl-1 + (1 β) V0. Set second parameter ɛ which represents a very small value, e.g., Keep walking till Vl Vl-1 < ɛ. In this case, we say Vl is a stationary distribution and it reflects the probabilities of the friendship link between t and any other nodes in A. 5. Choose (λ FIDb(t) ) distinct nodes from A based on the result of Vl. These nodes have the higher probabilities of being friends with t than for those of the remaining nodes. However, the selected nodes should not be already included in FIDb(t) (note that t is excluded). These chosen nodes are inferred friends of t and we add them into FINb(t).

5 6. Finally, Fb(t) = FIDb(t) FINb(t). The attack algorithm used for an FII attack is shown below. Input: an adversary b, a target t, parameters λ, β and ɛ, users' privacy settings PS in a social graph Output: the identified and inferred friend set Fb(t) 1. Fb(t) = 2. K(b) = ConstructAdversaryInitialKnowledge(PS) 3. A(b) = converttoadjacencymatrix(k(b)) 4. P(b) = converttransitionmatrix(k(b)) 5. m = ncol(a(b)) // number of distinct nodes in A(b) 6. i = indexof(a(b), t) 7. // identify friends of t from A(b) 8. FIDb(t) = getnodefromindex(a(b), j) 9. // infer additional friends of t 10. // using the random walk with restart 11. if FIDb(t) 12. V0 = setinitialvector (i, m) 13. l=1 14. Vl= a P(b) Vl-1 + (1 a) V0 15. while Vl Vl-1 ɛ 16. l Vl= a P(b) Vl-1 + (1 a) V0 18. end while 19. // rank the probability from high to low 20. rankedprob = rank(vl) 21. k = λ FIDb(t) 22. FINb(t) = gettopprobabilityusers(rankedprob, k) 23. Fb(t) = FIDb(t) FINb(t) 24. end if Attack Schemes In this section, we introduce three attack schemes of FII attacks where an adversary has one or more attacker nodes and focuses on one or more targets, respectively. One attacker node and one target. We introduce three basic attack schemes using one attacker node here: Attack Scheme 1: It is the most basic attack scheme that is the same as the attack algorithm mentioned in Section In this scheme, an adversary chooses (λ FIDb(t) ) distinct users as friends of a target once (refer to the attack step 5 and line 22 in the attack algorithm). Attack Scheme 2: In this attack scheme, in the attack step 5, an adversary b chooses only one user who has the highest probability of being friends with a target t in one attack. Then, b assumes that the identified friendship link is correct and adds the link into Ekb of K(b). After that, b launches the attack again and chooses another user who has the highest probability of being friends with t and repeats this process until (λ FIDb(t) ) users are chosen. We can see that b's initial attack knowledge is expanded gradually in this attack scheme. Attack Scheme 3: In this attack scheme, besides a target t, there are also several users who hide their friend lists from an adversary b. b first tries to expand edges in K(b) by launching FII attacks on these users before launching the attack on t. For example, in Figure 2, let user B be the adversary and user C be the target. B can first try to attack user D and E who do not allow B to view their friend lists. Then, B adds the attack results (identified and inferred friendship links of D and E) to K(B). After that, B starts to launch the attack on C. In addition, when an adversary is attacking multiple other users to expand his initial attack knowledge, he can first target the user who may be the most vulnerable to the attack. For example, a user, who is closer to the adversary and has more paths to the adversary is more vulnerable to the attack. We expect that the attack schemes 2 and 3 will be able to produce better results on a specific target. We will demonstrate these attack schemes and compare the attack results in Section 3. Multiple attacker nodes and one target. It is possible that there are multiple adversaries or an adversary can control multiple accounts in an SNS. These accounts can be used to conduct an FII attack on the same target. In this case, we believe more friends of the target will be identified and correctly inferred as the adversary can gain more knowledge related to the target. To launch the attack using multiple accounts, the adversary can first combine the initial attack knowledge from each attacker node to form a network that potentially includes more users related to the target. Then, the adversary can choose any of the basic attack schemes mentioned in the previous paragraphs to launch an FII attack on the target using the combined network. We will demonstrate such an attack scheme and compare its performance to those of the basic attack schemes (proposed in previous paragraph) in Section 3.3. Topology of the entire social network (multiple attacker nodes and multiple targets). When an adversary is able to have enough attacker nodes to launch the FII attacks, he may be able to infer the topology of an entire social network. At first, based on the initial attacker node, the adversary can first choose the target, who is the most vulnerable for the FII attack, from a set of targets and then launch an FII attack on the selected target. For example, the target that has many more paths to the attacker node may be more vulnerable for the FII attack. After more private friend links are identified and inferred from the initial attack, similarly, the adversary can continue to find such vulnerable targets and further conduct more FII attacks on them. Finally, after enough FII attacks have been conducted, the adversary may be able reconstruct the topology of the entire social network using his identified and inferred friendship links. We will show the attack results of such an attack scheme in Section Discussion The edge initially visible to B Inferred Friend of T from an FII attack B C A D The edge initially invisible to B Identified Friend of T from an MFB attack Figure 4. The FII attack vs. the MFB attack FII and Mutual-friend Based (MFB) attacks. In a mutual-friend based attack [4], an adversary can identify a target s private friends by conducting various queries of mutual friends. We find that an FII attack and a mutual-friend based attack are complimentary and they can be used together for more successful privacy attacks. For example, in Figure 4, let user B be the adversary and user T be the target who does not allow B to view his friend lists. B has three E T

6 friends A, C and D. Only C allows B to see his friend lists. By launching an MFB attack, B immediately knows that A and D are friends of T through the mutual friend query between B and T. In an FII attack, B can infer that user E is more likely to be a friend of T. Using only MFB attack, B cannot infer that E is a T's friends with a high probability. Using just an FII attack, B can only acquire that A and D are friends of T with low probabilities. Defense Approaches. The problem causing FII attacks is similar to the resource sharing problem among multiple users or policy conflict problem in an SNS. The intuitive approach to defend against FII attacks is to design a mechanism to address the conflicts of privacy policies among users. In the literature, there are several existing works that can be used to address conflicts in policies. XACML provides the algorithms including Deny-overrides, Permit-overrides, First-applicable and Only-one-applicable to address policy conflicts [34]. For instance, when Policy A (decision of deny) and Policy B (decision of allow) have conflicting decisions, an SNS will ignore Policy B and accept Policy A using Deny-overrides. However, the mechanisms in XACML do not take users' requirements into account. For example, if user A hides his friend lists from user B and Deny-overrides is applied, all of A's friends have to hide their friend lists from B in order to conceal their friendships from A. However, most of these friends may be likely to share their friend lists with B and these users' preferences are sacrificed. When Permit-overrides is applied in the above example, B can have a high probability to identify and correctly infer A's friends using FII attacks and hence A's preference will be violated. Squicciarini et al. propose an auction mechanism to address the policy conflicts in shared resources based on a voting algorithm (Clarke-Tax mechanism) and game theory [32]. Hu et al. propose a conflict resolution approach that balances the need for privacy protection and the users' desire for information sharing by quantitative analysis of privacy risk and sharing loss [33]. Both approaches can be applied to address policy conflicts and defend against FII attacks. 3. EXPERIMENTS In this section, we introduce the setup of the experiments for the FII attacks and demonstrate various FII attacks. 3.1 Datasets In this paper, we use three real social network datasets to demonstrate FII attacks: D1: It is a Facebook friend network dataset used in [9]. We remove the isolated nodes in the original dataset since they are useless in our experiments. D2: It is another Facebook friend network dataset used in [10]. Since it is not a well-connected network, we filter the nodes and choose the giant component of the original dataset for our experiments. D3: It is a Foursquare friend network dataset used in [11]. This network is a well-connected network. The basic statistical information of these datasets is shown in Table 1. We can see that users in D1 have the highest degrees and they are most connected with each other according to the average clustering coefficient and average path length. Users in D3 have the least degrees but they are more connected with each other than those in D2, although users in D2 have higher degrees than those in D3. Table 1. Basic statistical information of the datasets D1 D2 D3 Nodes 3,963 63,392 10,326 Edges 88, ,886 52,974 Ave. Degree Ave. Clustering Coefficient Ave. Path Length Attacks using One Attacker Node Initializations To simulate FII attacks using one attacker node, in each dataset, we randomly choose 1,000 users as attacker nodes. Each attacker node has three targets: Target 1: we randomly choose one target from the attacker node's friend lists. Target 2: we randomly choose one target from the attacker node's 2-distant neighbors. Target 3: we randomly choose one target that is more than 2- distant away from the attacker node in the social graph. Hence, an attacker node has to conduct three FII attacks on three targets and there are 3,000 FII attacks in each dataset. Note that all these targets do not allow the corresponding attacker nodes to view their friend lists. Also note that the selected targets have at least 10 friends since the corresponding attacker nodes need to identify and infer 10 ( Fb(t) = 10) of their friends. For each attacker node, besides the corresponding targets, we randomly choose additional 0 to 4 of his friends and set their friend lists invisible to the attacker node. The rest of the friends of the attacker node allow the attacker node to view their friend lists. Note that we set that Target 2 of an attacker node is included in the attacker node's initial attack knowledge. We also randomly choose 2 to 5 users who are not friends of the attacker node and whose friend lists are public and visible to the attacker node. We set that these selected users are friends of the Target 3 to make sure that the Target 3 is included in the attacker node's initial attack knowledge Initialization of Parameters In the experiments, we set Fb(t) = λ = 10, where λ is a parameter mentioned in the attack steps. A parameter e, which is mentioned in the definition of the FII attack and is used to evaluate the success of the FII attack, is dynamically set in our experiments. In Section 2.4.2, we mentioned other two parameters in the attack steps: β and ɛ. In all of our experiments, we set ɛ=10-6 for the equation Vl Vl-1 < ɛ. Note that β is the parameter to determine the probability that a node follows the random walk. Pan et al. show that β = 0.15 works on the web graph with the diameter 19 [29]. In our case, we assume the average diameter in the FII attacks is round 3. Theoretically, we can get a better value of β from (1 0.15) 19 =(1 β) 3 where it should be Hence, in all of our experiments, we set β = Measurements In our experiments, we use the following measures to evaluate the FII attacks: True Positive: It represents the number of identified and correctly inferred friends of a target. When it is larger than e (e is a pre-defined parameter in the definition of an FII attack), it indicates that the attack is successful.

7 Rate of Correctly Identified & Inferred Friends (RC): It indicates the percentage of exposed friends for a target. It can be captured as True Positive / Fb(t) (the number of identified and inferred friends). False Positive: It represents the number of incorrectly inferred friends of a target in Fb(t). It is equal to Fb(t) True Positive. Rate of Incorrectly Identified & Inferred Friends (RIC): It indicates the percentage of incorrectly exposed friends for a target. It can be captured as False Positive / Fb(t). Coverage: It represents the maximum number of exposed friends of a target for an attacker node. It is equal to Ib(t) mentioned in Section A higher value of the coverage value indicates that an adversary can find more friends of a target ideally in the attack. This measurement is used to analyze the attack results in Section Average True Positive for Attacker Nodes (Average RC) Average False Positive for Attacker Nodes (Average RIC) Table 2. Results of attack scheme 1 in the three datasets General Attack Results from Different Datasets In this subsection, we simulate the Attack Scheme 1 for the FII attacks in the three datasets to compare the attack results on these datasets. The attack results are shown in Table 2. From Table 2, we can see that FII attacks in D1 are the most successful one among the three datasets since attacker nodes from D1 can achieve the highest true positive and the lowest false positive. FII attacks in D2 are the least successful one and the attacker node gets the lowest true positive and highest false positive. Additionally, in D2, the average false positive is even higher than the average true positive, which indicates that the attack using one attacker node in D2 is not successful. Relationship Between an Attacker Node & a Target D1 D2 D3 Friend 7.82 (78.2%) 4.71 (47.1%) 5.48 (54.8%) 2-distant Neighbor 5.78 (57.8%) 2.85 (28.5%) 3.25 (32.5%) More than 2-distant Neighbor 4.03 (40.3%) 3.13 (31.3%) 3.19 (31.9%) Friend 2.18 (21.8%) 5.29 (52.9%) 4.52 (45.2%) 2-distant Neighbor 4.22 (42.2%) 7.15 (71.5%) 6.75 (67.5%) More than 2-distant Neighbor 5.97 (59.7%) 6.87 (68.7%) 6.81 (68.1%) Table 3. Results of attack schemes in the three datasets Average True Positive for Attacker Nodes Attack Scheme D1 D2 D3 Attack Scheme Attack Scheme Attack Scheme Table 4. Results from FII attacks on the topology of the entire network After Launching 1K FII Attacks After Launching 5K FII Attacks After Launching 10K FII Attacks After Launching 20K FII Attacks Average Correctly Inferred Friendship Links (Percentage of All Friendship Links) (7.76%) (33.88%) (60.84%) (71.53%) Average Incorrectly Friendship Links We can also see that the FII attacks are the most successful when the attacker nodes and targets are friends. Such a result is expected since the attacker nodes are very close to the targets and have more opportunities to have more mutual friends with the targets. When attacker nodes and targets are 2-distant neighbors, generally, the FII attacks are second most successful. However, in D2, the FII attacks where the attacker nodes and the targets are 2-distant neighbors, are less successful than those where the attacker nodes more than 2-distant away from the targets. To quantify the successful attacks in the simulations, we can set various values of e which is a pre-defined parameter to indicate a successful FII attack. When we set e = 3, we find out that most of FII attacks in D1 are successful. More than half of the FII attacks in D3 and around half of the FII attacks in D2 are successful. When we set e = 5, many FII attacks in D1 are still successful while around half of the FII attacks in D3 and around 20% of FII attacks in D2 are successful. By comparing the basic network statistics shown in Table 1, we may conclude that FII attacks are more effective in a social graph with a higher average clustering coefficient and a lower average path length. Other features for the social graph, such as number of nodes, number of edges, the average degree of nodes and the number of triangles, empirically, have insignificant impact on attack results. In theory, when the clustering coefficient is higher and the path length is smaller, it indicates that the nodes in the graph tend to cluster together. If nodes tend to cluster in a social graph, instinctively, an FII attack has a higher probability of success. This result suggests that an adversary should launch an FII attack in a more highly clustered network, in order to ensure a more successful attack Attack Results from Different Attack Schemes In this subsection, we first compare the attack results using three basic attack schemes (mentioned in Section 2.4.3) where an adversary only uses one attacker node. We simulate these three attack schemes in D1, D2 and D3. The results are shown in Table 3. Note that each attacker node has three different targets and we compute the average true positive from each attacker node on these three targets in Table 3. We can see that Attack Scheme 3 can significantly improve the attack results in each dataset. An attacker node using Attack Scheme 3 can identify and correctly infer at least one more friends of a target compared to the Attack Scheme 1. In addition, Attack

8 Scheme 2 can improve the attack results a bit compared to the result of Attack Scheme Attacks using Multiple Attacker Nodes To simulate the FII attacks using multiple attacker nodes, we first randomly choose 1000 targets in D3 and all of them have at least 10 friends. We then select four sets of attacker nodes for each selected target: M1: Attacker nodes in the set are one friend of the target, one 2-distant neighbor of the target and one user who is more than 2-distant away from the target. M2: Attacker nodes in the set are three friends of the target, one 2-distant neighbor of the target and one user who is more than 2-distant away from the target. M3: Attacker nodes in the set are three friends of the target, five 2-distant neighbors of the target and five users who are more than 2-distant away from the target. M4: Attacker nodes in the set are five friends of the target, five 2-distant neighbors of the target and five users who are more than 2-distant away from the target. For each selected attacker node, we randomly choose additional 0 to 4 of his friends and set their friend lists invisible to the attacker node. The rest of the friends of the attacker node allow the attacker node to view their friend lists. We also randomly choose 2 to 5 users who are not friends of the attacker node and whose friend lists are public and visible to the attacker node. The initialization of parameters in the FII attacks in these simulations is the same as those described in Section For each target, we adopt the Attack Scheme 1 and use the combine available network formed by the initial knowledge of all the attacker nodes to launch the attack. The attack results are shown in Figure 5. From the figure, we can see that the attack result highly depends on the number of attacker nodes that are friends of the corresponding targets. When the attacker nodes are 2-distant neighbors of the corresponding target or more than 2-distant away from the target, these attacker nodes contribute less for the attack result. Compared to the result of the FII attacks using one attacker node (shown in Table 2 and 3), we can see that the FII attacks using multiple attacker nodes can significantly identify and infer more private friends of a target Average True Positive for Targets Average False Positive for Targets 0 M1 M2 M3 M4 Figure 5. Attack results using multiple attacker nodes 3.4 Using FII Attacks to Infer the Topology of the Entire Network We present the results of using the FII attacks to infer the topology of the entire network in this subsection. First, we focus on D3. We randomly select an initial attacker node whose degree is larger than 10 in D3. For this attacker node, we set the initial attack knowledge of the attacker node by randomly choosing additional 0 to 4 of his friends whose friend lists are invisible to the attacker node. Then, a small set of the available friendship links forms the adversary's initial knowledge about the network topology. We then choose the most vulnerable target that has the most number of mutual friends (more 2-distant paths) with the attacker node and launch an FII attack using the Attach Scheme 1 (the adopted parameters in an FII attack are same as those mentioned in Section 3.2.2). We add the inferred friendship links into the adversary s knowledge. After that, from the adversary s knowledge about the network, we choose a pair of users who have the most number of mutual friends. We then set one of them as the attacker node and the other as the target in a new FII attack. Note that, before the new FII attack, we also initialize the attack knowledge for this attacker node using the same approach used for the initial attacker node. Lastly, we keep choosing the attacker nodes and targets to launch the FII attacks to infer the entire topology of the friendship network. We repeated each attack 10 times in D3 and the results are shown in Table 4. From the table, we can see that such an attack is effective and a large percentage of the network is exposed when an adequate number of of FII attacks are launched. For example, this attack can correctly infer more than 70% of the network topology when FII attacks were conducted. However, the number of incorrect friendship links included cannot be eliminated although it increases slowly with the increasing number of FII attacks launched. We argue that this is because the matrix indicating the friendship network in D3 is very sparse and there are many users who have fewer friends or even no friends. We evaluate this assumption by conducting the attack in D1, which has fewer users but has more friendship links. Our simulation of the attack in D1 confirms our assumption: when 1000 FII attacks were launched, such an attack scheme can correctly infer more than 85% of the network topology in D1 and achieves fewer than 2000 incorrectly inferred friendship links. 3.5 Discussions Attack Results & Attacker Nodes' Initial Attack Knowledge In this subsection, we try to find out the relationship between attack results and attacker nodes' initial attack knowledge. We simulate the three attack schemes proposed in Section using one attacker node. The results are shown in Figure 6 and Figure 7. Note that an attacker node's initial attack knowledge in our simulations is determined by 1) the number of additional friends who hide friend lists from the attacker nodes and 2) the number of users who are not friends of the attacker nodes and whose friend lists are public as mentioned in Section Figure 6 shows the average true positive of attacker nodes with regard to different number of friends who hide friend lists from the attacker nodes. We can see that the true positive does not decrease significantly with the increase of the number of additional friends hiding their friend lists from attacker nodes in D1. However, in D2 and D3, the true positive decreases significantly with the increase of the number of additional friends whose friend lists are invisible to the attacker nodes. It indicates that attacker nodes in D2 and D3 are more sensitive to the number of visible friends than those in D1. It is reasonable since users in D1 have more friends than those in

9 D2 and D3. For the attacker nodes in D1, the friends whose friend lists are invisible does not constitute a significant part in his friend list. However, such friends constitute the larger part of an attacker node's friend list in D2 and D3. Figure 6. The attack result vs. the number of additional friends whose friend lists are invisible to an attacker node Figure 7 shows the average true positive of attacker nodes with regards to the different number of public friend lists in the FII attacks. We can see that the true positive increases more significantly when there are more public friend lists in D1. However, in D2 and D3, with the increase of the number of public friend lists, the true positive does increase but the increase is not significant. It indicates that the number of public friend lists is more useful for attacker nodes in D1 than those in D2 and D3. The reason is that users in D1 tend to cluster with each other and it is more likely for the public friend lists to include the information related to the targets. On the other hand, users in D2 and D3 are not inclined to cluster with each other and hence it is less likely for the public friend lists to contain the information related to the targets. Average True Posituve Average True Positive D1 D2 D Number of Additional Friends Who Hide their Friend Lists from an Attacker Node D1 D2 D Number of Users Who are not Friends of an Attacker Node and Whose Friend Lists are Public Figure 7. The attack result vs. the number of users whose friend lists are public Analysis of Unsuccessful Attacks In this subsection, we aim to find out potential reasons that cause FII attacks to be unsuccessful. We set e=3 and analyze the attacks using Attack Scheme 1 in D3 using one attacker node. In the attack results, there are around 32% of unsuccessful attacks (the number of correct friends in Fb(t) is less than 3). The statistics about these attacks and the results are shown in Table 5. Table 5. Statistical information about the unsuccessful attacks Ave. Friends of Attacker nodes 8.27 Ave. Friends of Targets Ave. Mutual Friends between an Attacker Node and a Target 1.12 Ave. Coverage 6.28 By comparing the corresponding information in Table 5 and Table 1, we can see that the attacker nodes usually have fewer friends in unsuccessful attacks. The average number of friends for attacker nodes in unsuccessful attacks is 8.27 while the average number of friends for users in D3 is In addition, we find that the average number of mutual friends between an attacker node and a target is also limited. This may be another reason that leads to an unsuccessful attack. Furthermore, the most important reason is that the average coverage (6.28) is much less than 10 (the number of identified and inferred friends in an FII attack). Note that the coverage represents the maximum number ( Ib(t) ) of friends of a target exposed to an attacker node. According to the Theorem in Section 2.3.3, an attacker node cannot find more than Ib(t) friends of a target in an FII attack. In an unsuccessful attack, an attacker node can only correctly identify up to 6.28 friends of targets in average. However, in a successful attack in our experiments, Ib(t) is usually larger than 12. Hence, we believe that the small value of Ib(t) is another important reason that causes an attack to be unsuccessful. 4. RELATED WORK Many research efforts have focused on the link prediction/inference issues in SNSs based on the users' attributes. Hasan and Zaki summarize the supervised link prediction algorithms, including Bayesian probabilistic models, linear algebraic models and probabilistic relational models in [17]. Yin et al. introduce Social- Attribute Network (SAN), an attribute-augmented social network, to integrate network structure and node attributes to perform the link recommendations using the random walk with restart algorithm [12, 18]. Gong et al. extend the SAN framework based on their observation that inferring attributes could help predict links [19]. However, the above approaches usually require users' attribute information for link prediction. In an FII attack, an adversary can launch the attack without knowing users' attribute information. Recently, some efforts have focused on predicting missing links in a global SNS. Korolova et al. mention that users' behavior of sharing their friends to the public could allow an attacker to discover the topology of an SNS [1]. Effendy et al. extend the work by Korolova et al. and show that such an attack can be magnified substantially with the inference on user degrees [20]. Bonneau et al. report that the social network structure can be disclosed when a random sample of k edges from each node in the SNS is exposed [21]. Leroy et al. use group information to build a bootstrap probabilistic graph in order to perform friendship link inferences [22]. Kim and Leskovec adopt the Expectation Maximization (EM) algorithm to infer the missing nodes and edges by observing part of the network [23]. Erdös el al. propose a heuristic approach to discover the missing links based on common neighbors of subnetworks [24]. Chen et al. construct the Markov Logic Network to infer the friendship links on a large scale social graph [26]. However, these approaches focus on inferring few missing friendship links based on many observed friendship links. In an FII attacks, an adversary launches attacks with the enforcement of users' privacy settings and he is allowed to see friend lists that he is authorized to view. The adversary focuses on inferring many friendship links of a target from limited friendship information of the target.

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