A Spectral Framework for Detecting Inconsistency across Multi-Source Object Relationships

Transcription

1 A Spectral Fraework for Detecting Inconsistency across Multi-Source Object Relationships Jing Gao, Wei Fan, Deepak Turaga, Srinivasan Parthasarathy, and Jiawei Han University of Illinois at Urbana-Chapaign, IL USA IBM T. J. Watson Research Center, Hawthorn, NY USA Abstract In this paper, we propose to conduct anoaly detection across ultiple sources to identify objects that have inconsistent behavior across these sources. We assue that a set of objects can be described fro various perspectives (ultiple inforation sources). The underlying clustering structure of noral objects is usually shared by ultiple sources. However, anoalous objects belong to different clusters when considering different aspects. For exaple, there exist ovies that are expected to be liked by kids by genre, but are liked by grown-ups based on user viewing history. To identify such objects, we propose to copute the distance between different eigen decoposition results of the sae object with respect to different sources as its anoalous score. We also give interpretations fro the perspectives of constrained spectral clustering and rando walks over graph. Experiental results on several UCI as well as DBLP and MovieLens datasets deonstrate the effectiveness of the proposed approach. Keywords-anoaly detection; ultiple inforation sources; spectral ethods I. INTRODUCTION Nowadays, there are usually several sources of inforation that describe different properties or characteristics of individual objects. For exaple, we can learn about a ovie fro its basic inforation including genre, cast, plots, etc., or the tags users give to the ovie, or the viewing histories of the users who watched the ovie. On each of the inforation source, a relationship graph can be derived to characterize the pairwise siilarities between objects where the edge weight indicates the degree of siilarity. As an exaple, Figure shows the siilarity relationships aong a set of ovies derived fro two inforation sources: ovie genres and users. The genre inforation ay indicate that two ovies that are both aniations are ore siilar than two other ovies where one is an aniation and the other is a roance ovie. Siilarly, ovies watched by the sae set of users are likely to be ore siilar than ovies that are watched by copletely different sets of users. Clearly, objects for a variety of clusters or counities based on individual siilarity relationship. For exaple, two clusters can be found fro both of the siilarity graphs in Figure. One cluster represents the ovies that are aniations, which are loved by kids; while the other cluster represents roance ovies, which are liked by grown-ups. Most of the ovies belong to the sae cluster even though different inforation sources are used. However, there are soe objects that fall into different clusters with respect to different sources. In this exaple, the aniated ovie Wall-E by genre is expected to be liked by kids, but it is liked by grown-ups based on user viewing history. Finding such inconsistent ovies can help fil distributors better understand the expected audiences of different ovies and ake sarter arketing plans. In this paper, we propose to detect objects that have inconsistent behavior aong ultiple inforation sources, which we refer to as horizontal anoaly detection. Soe other exaple scenarios of horizontal anoaly detection include detecting people who fall into different social counities with respect to different online social networks and detecting inconsistency across ultiple odule interaction graphs derived fro different versions of a software project. Furtherore, identifying horizontal anoalies can find applications in any fields including sarter planet, internet of things, intelligent transportation systes, arketing, banking, etc. To the best of our knowledge, this is the first work on identifying horizontal anoalies by exploring the inconsistencies aong ultiple sources. Traditional anoaly detection [4] approaches focus on identifying objects that are dissiilar to ost of the other objects fro a single source [9], [3], [2], [8]. On the other hand, ost of the existing work on ining ultiple inforation sources concerns erging and synthesizing odels, rules, patterns obtained fro ultiple sources by reconciling their differences, such as ulti-view learning [2], eerging or contrast patterns [5], ulti-view clustering [], [8] and consensus clustering [5], [6]. As for ulti-source anoaly detection, the studies focus on how to identify anoalies within a specific context where the pre-defined contextual attributes include spatial attributes [3], neighborhoods in graphs [6], social counities [7], and contextual attributes [7], [4]. Although these studies take two types of attributes (behavioral and contextual [4]) into consideration, they cannot be easily generalized to horizontal anoaly detection spanning ultiple sources. The reason is that they siply detect anoalies fro the behavioral attributes while the contextual attributes only provide the context in which the anoalies are detected. In soe sense, these contextual anoalies are still extracted fro one source, whereas the proposed ethod can identify

2 5 7 2 grown-ups kids grown-ups 7 4 kids X-The Lion King; X2-Toy Story; X3-Kungfu Panda; X4-Wall-E; X5-Casablanca; X6-Titanic; X7-The Notebook Figure : A Horizontal Anoaly Detection Exaple objects with inconsistent behavior across ultiple sources. In this paper, we propose a systeatic approach to identify horizontal anoalies fro ultiple inforation sources. We assue that each individual inforation source captures soe siilarity relationships between objects that ay be represented in the for of a siilarity atrix (whose entries represent the pairwise quantitative siilarity between objects). We cobine the input atrices into one large siilarity atrix and adopt spectral techniques to identify the key eigenvectors of the graph Laplacian of the cobined atrix. Horizontal anoalies are identified by coputing cosine distance between the coponents of these eigenvectors. The ethod can be regarded as conducting spectral clustering on ultiple inforation sources siultaneously with a joint constraint that the underlying clustering structures are siilar, and objects that are clustered differently are categorized as horizontal anoalies. The horizontal anoalies can also be regarded as those having long coute tie in the rando walk defined over the graph. We validate the proposed algorith on both synthetic and real data sets, and the results deonstrate the advantages of the proposed approach in finding horizontal anoalies. II. METHODOLOGY Suppose we have a set of N objects X = {x,x 2,...,x N } and there are P inforation sources that describe different aspects of these objects. The objective is to assign an anoalous score s i to each object x i, which represents how likely the object is anoalous when its behavior differs aong the P different inforation sources. In this section, we present a HOrizontal Anoaly Detection (HOAD) algorith to solve the proposed proble. An object can be regarded as a horizontal anoaly if it is assigned to different clusters when using various inforation sources, and thus we calculate the anoalous degree of an object based on how uch its clustering solutions differ fro each other. To siplify the notations, we start with the cases having two distinct inforation sources (Section II-A), give spectral clustering and rando walk interpretations in 6 3 Section II-B, and explain how it is generalized to ultiple inforation sources in Section II-C. A. HOAD Algorith Suppose we have two N N siilarity atrices on the N objects: A and W, where a ij and w ij define the siilarity between x i and x j fro different aspects. The algorith consists of two ajor steps: First, we conduct soft clustering on A and W together with the constraint that an object should be assigned to the sae cluster; Second, we quantify the difference between the two clustering solutions to derive anoalous scores. The details are as follows. We construct a cobined graph by connecting the nodes which correspond to the sae object in the two siilarity graphs with an edge weighted., a large positive nuber, is a penalty paraeter. An exaple of such a graph is illustrated in Figure. The set of nodes in the cobined graph consists of two copies of the objects: {x,...,x N,x,...,x N } (2N nodes in total). Let M be an N N diagonal atrix with on the diagonal: M = I where I is an N N identify atrix. Let Z be the adjacency atrix of the cobined graph, which is a 2N 2N atrix: [ ] A M Z =. () M W First, we copute the graph Laplacian L as: L = D Z (2) using degree atrix D (a 2N 2N diagonal atrix): D = diag ( 2N { z ij } 2N i=). (3) j= Secondly, copute the k sallest eigenvectors of L (with sallest eigenvalues) and let H R 2N k be the atrix containing these eigenvectors as coluns. We divide H into two subatrices U and V each with size N k so that H =[U V] T. Therefore, the i-th and (i + N)-th rows of H are represented as: u i = h i, v i = h i+n, (4) which correspond to two soft clustering representations of x i with respect to A and W respectively. As can be seen, with the help of the edge between the copies of the sae object, we try to cluster the objects in the sae way across different sources. In Section II-B, we give a theoretical justification of this clai. Finally, copute the anoalous score for object x i using cosine distance between the two vectors: s i = u i v i u i v i. (5) The algorith flow is suarized in Algorith.

3 Algorith HOAD algorith Input: siilarity atrices A and W, nuber of eigenvectors k, penalty paraeter ; Output: anoalous score vector s; Algorith:. Copute atrix Z according to Eq. () 2. Copute graph Laplacian L as in Eq. (2) 3. Conduct eigen-decoposition of L and Let H be the k sallest eigenvectors with sallest eigenvalues 4. Copute anoalous score of each object s i based on Eq. (4) and Eq. (5) for i =,...,N return s B. Interpretations In this part, we explain the algorith fro the perspectives of spectral clustering and rando walk. Clustering on Cobined Graph. As can be seen, we first perfor spectral clustering on the cobined graph in Algorith. The basic idea of spectral clustering is to project the objects into a low-diensional space (defined by the k sallest eigenvectors of the graph Laplacian atrix) so that the objects in the new space can be easily separated. We call the projections as spectral ebeddings of the objects. It has been shown that the atrix fored by the k eigen vectors (H)ofLis the solution to the following optiization proble []: in H R N k Tr(H LH) s.t. H H = I (6) H is a 2N k atrix, which equals to [U V] T. The graph Laplacian L is defined as D Z (Eq. (2)), and Z is defined in Eq. (). Moreover, we suppose the degree atrices for A and W are D a and D w respectively: D a = diag ( N { a ij } N i=), D w = diag ( N { w ij } N i=). j= j= Then we can derive an equivalent forulation for the proble in Eq. (6): in U,V Tr(U (D a A)U)+Tr(V (D w W )V ) n k 2 u ij v ij s.t. U U + V V = I (7) i= j= The proof is oitted due to space liit. Clearly, each of the first two ters in Eq. (7) corresponds to the spectral clustering proble using A or W alone. The third ter acts as the constraint that the two clustering solutions should be siilar (cosine siilarity). Therefore, the first three lines in Algorith can be interpreted as conducting spectral clustering on the two input siilarity graphs siultaneously with a joint constraint. Our goal is to detect horizontal anoalies that have inconsistent behavior across sources, and thus the final step is to copute anoalous scores. Note that in Algorith, the i- th row vector in U (the first N rows of H) and V (the last N rows of H) contain the projections of object x i. Due to the principle of spectral clustering, if the spectral ebeddings u i and v i are close to each other, the corresponding object x i is ore likely to be assigned to the sae cluster with respect to two different sources. Therefore, the cosine siilarity between the two vectors u i and v i quantifies how siilar the clustering results of object x i on the two sources are, and thus represents its noral degree. In turn, the cosine distance as defined in Eq. (5) gives the anoalous degree of x i with respect to the two sources. The higher the score s i is, the ore likely x i is a horizontal anoaly. Rando Walk. Suppose we define a rando walk over the cobined graph, where the transition probability fro node x i to node x j is proportional to the edge weight in the graph. Let z ij be the edge weight between two nodes x i and x j in the graph, and vol(x) = 2N 2N i= j= z ij be the su of all the edge weights in the graph. Now we look at the coute distance between x i and x i, two copies of the sae object in the cobined graph. Coute distance is the expected tie it takes for the rando walk to travel fro x i to x i and back, and it can be coputed using the eigenvectors of the graph Laplacian L as defined in Eq. (2). Suppose L has eigenvalues λ,...,λ 2N, and U and V are two N N atrices containing all the eigenvectors for the two copies of the objects respectively. Let u i and v i denote the i-th row of U and V. We define γ as a length-2n vector with each entry γ l equal to (λ l ).5 if λ l, and otherwise. Now we divide γ into two length-n vectors: γ =[ γ u γ v ]. It can be derived that the coute distance c i between x i and x i is: c i = vol(x) u i γ u v i γ v 2. Recall that we copute the anoalous score of x i as u i v i u i v i. We can see that both the anoalous score and the coute distance can be represented as a distance function applied on the spectral ebeddings of the two copies of the object. The difference is that all the eigenvectors are used and they are scaled by (λ l ).5 in the coute distance coputation. Also, Euclidean distance is used instead of cosine distance. Although the connection is loose, coute distance can be a helpful intuition to understand the anoalous scores. If it takes longer tie to coute between the two copies of object x i in the graph, x i is ore likely to be a horizontal anoaly. C. Multiple Sources We can adapt Algorith to handle ore than two inforation sources as follows. Suppose we have siilarity atrices {W (),W (2),...,W (P ) } as the input. First, the cobined graph is constructed in a siilar fashion as discussed before: Duplicate the objects for P copies, in each copy retain the siilarity inforation fro each source, and connect each pair of the nodes corresponding to the sae object with an edge weighted. After that, we calculate its graph Laplacian and the k sallest eigenvectors following exactly the sae procedure as in Algorith. One concern

4 is that, when the nuber of inforation sources increases, the size of the atrix L grows quadratically. Note that the graph Laplacian of Z is a sparse atrix, and also, we only need the k sallest eigenvectors instead of the full eigenspace. In fact, efficient packages such as ARPACK [], have been developed to copute a few eigenvectors of large-scale sparse atrix. Then we calculate the anoalous degree of an object x i based on the following P vectors: { h i, h i+n, h i+2n,..., h i+(p )N }. In the experient, we use the average pairwise distance as the easure: s i = P (P ) P P a= b= a b [ III. EXPERIMENTS h i+an h i+bn h i+an h i+bn We evaluate the HOAD algorith on synthetic data and real datasets including DBLP and MovieLens to validate its ability of identifying eaningful horizontal anoalies. A. Synthetic Data The concept of horizontal anoaly is new, and thus there are no benchark datasets for it. Therefore, we propose a ethod to convert a classification proble into a horizontal anoaly detection proble, and then apply this procedure on several UCI achine learning data sets. Data Generation. Suppose we have a training set fro a classification proble where each object consists of feature values and a class label. We siulate inconsistency across ultiple sources by swapping feature values of objects fro different classes. Suppose there are N objects in the training set: {x,...,x N }, and the features X can be partitioned into two views. We assue that objects within the sae class share siilar feature values in each feature set. Therefore, for two objects x i and x j fro different classes, if their feature values are swapped in one view but reain unchanged in the other, they have inconsistent behavior aong these two views, and thus represent horizontal anoalies. We apply the above ethod on four data sets obtained fro UCI achine learning repository : Zoo, Iris, Letter and Wavefor. On each data set, we repeat the transforation procedure 5 ties and at each tie, we generate a data set with around % anoalies. We evaluate the HOAD algorith on the 5 data sets and report the average accuracy. Evaluation Measure and Methods. For anoaly detection probles, one of the ost widely used evaluation approaches is ROC analysis, which represents the trade-off between detection rate and false alar rate. The area under ROC curve (), which is in the range [,], is a good evaluation etric. The higher the is, the better the algorith perfors. We show the perforance of the proposed HOAD algorith with various paraeter settings. Note that the first step of the proposed algorith is a constrained ] soft clustering on ultiple inforation sources. Instead of conducting a joint clustering, the baseline ethod clusters ultiple sources separately and calculates the anoalous scores based on the difference aong different clustering solutions. Specifically, in two-source probles, we conduct eigen decoposition on the graph Laplacian atrices of the two siilarity atrices A and W separately. Suppose the top k eigen representation of object x i are u i and v i respectively, then we use Eq. (5) to copute the anoalous score of x i for the baseline approach. Note that the ajor difference between the HOAD algorith and the baseline ethod is on how to copute u i and v i. Perforance. The experiental results on the four data sets are shown in Figure 2 where we vary the values of the paraeters and k. indicates the penalty we enforce when the two clustering solutions do not agree, and k represents the nuber of top eigenvectors. Neither nor k is used in the baseline and its perforance reains ostly stable except slight fluctuation due to rando sapling in data generation. Fro the experiental results, we can see that HOAD algorith generally outperfors the baseline, especially when k is sall (e.g., k =3). However, when the value of is higher, the difference in between the algoriths using different k is uch saller. Therefore, we focus on how to select the appropriate in the following discussion. On UCI datasets, it is clear that when increases, the proposed algorith has a higher. In the siulated study, the two feature sets are two disjoint subsets of the original features, and usually using all of the features leads to a better classification odel. Hence the two views are correlated and using a large captures this correlation well. However, this does not ean that we should assign a big nuber to in all cases because this pattern ay not always hold in real horizontal anoaly detection tasks. In the following experients on DBLP data sets, we illustrate the relationship between and the anoalous scores, and state soe principles in setting. In Figure 4, we show the running tie of HOAD algorith with respect to to 6 objects represented in two, three or four inforation sources. We conduct the experients on synthetic data sets where we randoly generate siilarity atrices for 5 trials, and report the average running tie. The eigenvectors are coputed using Matlab eigs function, which is based on ARPACK package []. As can be seen, the HOAD algorith can scale well to large data sets when the nuber of objects and nuber of sources both increase. B. Real World Data We discuss the issues of setting paraeters on DBLP data and present illustrative results on MovieLens data. DBLP. We define two horizontal anoaly detection tasks based on the DBLP 2 data where the objects are a set of 2 ley/db/

5 (a) Zoo (b) Iris (c) Letter (d) Wavefor HOAD (k=3) HOAD (k=9) HOAD (k=3) HOAD (k=9) HOAD (k=3).4.2 HOAD (k=9) HOAD (k=3) HOAD (k=9) Figure 2: Anoaly Detection Perforance Coparison on Siulated Detection Tasks based on UCI Data conferences and authors respectively. 422 conferences are represented in two views: the keywords in the conferences and the authors who published in the conferences. Specifically, each conference x i has two vectors. In the first vector, each entry is the nuber of ties each word appeared in the paper titles of x i. In the second vector, each entry denotes the nuber of ties an author published in x i. The pairwise siilarity between two conferences x i and x j is defined as the cosine siilarity between the corresponding vectors. Therefore, the conferences that share lots of keywords, or share lots of authors are siilar. Siilarly, we select a set of 36 authors fro data ining related areas and extract two types of inforation fro DBLP: the publications and the co-authorships. Each author x i also has two vectors where in the first vector each entry denotes the occurrence of each word in the authors publications, and each entry corresponds to the nuber of ties two authors collaborate in the second one. Cosine siilarity is used, and siilar authors will share co-authors, or keywords in their publications. We study the effect of on the anoalous scores. For each, we apply the HOAD algorith to the data sets, and copute the ean and standard deviation of the objects anoalous scores. The results on conferences and authors are shown in Figure 3 where the points on the line are the average anoalous scores and the error bar denotes the standard deviation. Obviously, the average anoalous score decreases as increases. Recall that the anoalous scores indicate the degree of differences between the spectral ebeddings derived fro the two siilarity atrices. When we give a heavy penalty on different ebeddings by the two sources, we basically bias the two projections towards the ones that agree the ost. Therefore, when is larger, the spectral ebeddings fro the two sources are ore likely to be the sae, and thus the difference between the is saller. Another observation is that the variance aong the anoalous scores goes up first and then goes down as increases. When is quite large or quite sall, the two projections of all the objects would be very siilar or very different, and thus the objects receive siilar anoalous scores. There exists a large variability aong the anoalous scores only when is in the iddle of the spectru. Although can be drawn fro (, ), the average anoalous scores are within a fixed range: [,]. Therefore, we can choose which leads to an average Average Anoalous Score (a) Conferences Average Anoalous Score (b) Authors Figure 3: Analysis of Paraeter on DBLP Data anoalous score around.5 because the variance of the anoalous scores usually reaches the highest point here and this helps us identify the horizontal anoalies. MovieLens. We use the Movielens dataset 3 with ovies as objects. There are three sources of inforation to capture their relationships: ) Genre: Movies are classified as being of one or ore of 8 genres, such as Coedy and Thriller, which can be treated as binary vectors. 2) User viewing history: Each ovie has a list of users that watched the ovie. This ay also be represented as a vector (per ovie) across all users. 3) User tagging history: Movies are tagged by different users. Looking across all users, we can deterine a vector per ovie. In all three cases, we copute the pairwise siilarity using cosine siilarity across the vectors. The data set contains illion ratings and, tags for 68 ovies by 7567 users. We choose a set of 7595 ovies, each of which has both ratings and tags. We then have three siilarity atrices, corresponding to these three different sources. To evaluate the perforance of the HOAD algorith on MovieLens dataset, we label soe ovies as horizontal anoalies based on the list of weirdest ovies 4. There are 572 ovies listed as weirdest ovies and aong the 27 are found in the MovieLens dataset. These 27 ovies are labeled as anoalous and the reaining 7468 ovies are noral. Based on these labels, we calculate the area under ROC curve () of both HOAD and the baseline ethod based on their coputed anoalies scores for the 7595 ovies. HOAD algorith achieves a better (.4879) copared with that of the baseline ethod (.4423). This deonstrates the ability

6 Running Tie (s) two sources three sources four sources Nuber of Objects Figure 4: Running Tie Table I: Anoalous Scores of 2 Popular Movies fro MovieLens Movie Score Movie Score One Flew Over the Cuckoos Nest.879 Seven Saurai.644 Pulp Fiction.773 Fight Club.6364 Casablanca.725 City of God.6278 The Shawshank Redeption.6949 The Lord of the Rings: The Return of the King.352 The Godfather: Part II.6822 The Lord of the Rings: The Fellowship of the Ring.3478 The Godfather.677 The Good, the Bad and the Ugly.394 Goodfellas.6768 Raiders of the Lost Ark.38 Schindlers List.6755 Rear Window Angry Men.673 Star Wars.2982 The Dark Knight.6535 Star Wars: Episode V-The Epire Strikes Back.2562 of the proposed HOAD algorith in detecting inconsistent ovies across various inforation sources. Moreover, we present the anoalous scores for the 2 ost popular ovies 5 as shown in Table I. As ay be seen, the ovies One Flew Over the Cuckoos Nest and Pulp Fiction are identified as horizontal anoalies, as they tend to show strong disagreeent between the genre classification, and the sets of users that watched and tagged the. Intuitively, this is expected as these two ovies do not really fit into one classification or user category. Borrowing reviews fro Wikipedia 6, Pulp Fiction is known for its rich, eclectic dialogue, ironic ix of huor and violence, and nonlinear storyline, which ake it different and anoalous aong ovies. For One Flew Over the Cuckoos Nest, the review says it is a coedy that cant quite support its tragic conclusion. These tell us the reasons why these two ovies are detected as being inconsistent. On the other hand, Star Wars receives the lowest anoalous score as it attracts a particular set of audiences. IV. CONCLUSIONS We propose to detect horizontal anoalies, or objects that have inconsistent behavior aong ultiple sources. Intuitively, they belong to different clusters when considering any aspects fro ultiple inforation sources. The proposed algorith has two intrinsic steps. In the first step, we construct a cobined siilarity graph based on the siilarity atrices and copute the k sallest eigenvectors of the graph Laplacian as spectral ebeddings of the objects. After that, we calculate the anoalous score of each object as the cosine distance between different spectral ebeddings. The physical eaning of the proposed algorith is explained fro both constrained spectral clustering and rando walk point of view. Experiental results show that the proposed algorith can find horizontal anoalies fro real-world datasets, where other anoaly detection ethods fail to identify these anoalies. REFERENCES [] S. Bickel and T. Scheffer. Multi-view clustering. In Proc. of ICDM4, pages 9 26, As listed on on Noveber [2] A. Blu and T. Mitchell. Cobining labeled and unlabeled data with co-training. In Proc. of COLT98, pages 92, 998. [3] M. M. Breunig, H.-P. Kriegel, R. T. Ng, and J. Sander. Lof: identifying density-based local outliers. In Proc. of SIGMOD, pages 93 4, 2. [4] V. Chandola, A. Banerjee, and V. Kuar. Anoaly detection: A survey. ACM Coput. Surv., 4(3), 29. [5] G. Dong and J. Li. Efficient Mining of Eerging Patterns: Discovering Trends and Differences. In Proc. of KDD99, pages 43 52, 999. [6] X. Z. Fern and C. E. Brodley. Solving cluster enseble probles by bipartite graph partitioning. In Proc. of ICML 4, pages , 24. [7] J. Gao, F. Liang, W. Fan, C. Wang, Y. Sun, and J. Han. On counity outliers and their efficient detection in inforation networks. In Proc. of KDD, pages , 2. [8] N. Khoa and S. Chawla. Robust outlier detection using coute tie and eigenspace ebedding. In Proc. of PAKDD, pages , 2. [9] E. M. Knorr, R. T. Ng, and V. Tucakov. Distance-based outliers: algoriths and applications. The VLDB Journal, 8(3-4): , 2. [] R. Lehoucq, D. Sorensen, and C. Yang. ARPACK users guide: Solution of large-scale eigenvalue probles with iplicitly restarted arnoldi ethods. SIAM Publications, 998. [] U. Luxburg. A tutorial on spectral clustering. Statistics and Coputing, 7(4):395 46, 27. [2] M. Markou and S. Singh. Novelty detection: a review-part : statistical approaches. Signal Processing, 83(2): , 23. [3] S. Shekhar, C.-T. Lu, and P. Zhang. Detecting graph-based spatial outliers: algoriths and applications (a suary of results). In Proc. of KDD, pages , 2. [4] X. Song, M. Wu, C. Jeraine, and S. Ranka. Conditional anoaly detection. IEEE Transactions on Knowledge and Data Engineering, 9(5):63 645, 27. [5] A. Strehl and J. Ghosh. Cluster ensebles a knowledge reuse fraework for cobining ultiple partitions. In Journal of Machine Learning Research, 3: , 23. [6] J. Sun, H. Qu, D. Chakrabarti, and C. Faloutsos. Neighborhood foration and anoaly detection in bipartite graphs. In Prof. of ICDM5, pages , 25. [7] X. Wang and I. Davidson. Discovering contexts and contextual outliers using rando walks in graphs. In Proc. of ICDM9, pages 34 39, 29. [8] D. Zhou and C. Burges. Spectral clustering and transductive learning with ultiple views. In Proc. of ICML7, pages 59 66, 27.