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105
Consistency of spectral clustering
, 2004
"... Consistency is a key property of statistical algorithms, when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of a popular family of spe ..."
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Cited by 286 (15 self)
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Consistency is a key property of statistical algorithms, when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of a popular family of spectral clustering algorithms, which cluster the data with the help of eigenvectors of graph Laplacian matrices. We show that one of the two of major classes of spectral clustering (normalized clustering) converges under some very general conditions, while the other (unnormalized), is only consistent under strong additional assumptions, which, as we demonstrate, are not always satisfied in real data. We conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering in practical applications. We believe that methods used in our analysis will provide a basis for future exploration of Laplacianbased methods in a statistical setting.
Graph sparsification by effective resistances
 SIAM J. Comput
"... We present a nearlylinear time algorithm that produces highquality sparsifiers of weighted graphs. Given as input a weighted graph G = (V, E, w) and a parameter ǫ> 0, we produce a weighted subgraph H = (V, ˜ E, ˜w) of G such that  ˜ E  = O(n log n/ǫ 2) and for all vectors x ∈ R V (1 − ǫ) ∑ (x ..."
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Cited by 63 (4 self)
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We present a nearlylinear time algorithm that produces highquality sparsifiers of weighted graphs. Given as input a weighted graph G = (V, E, w) and a parameter ǫ> 0, we produce a weighted subgraph H = (V, ˜ E, ˜w) of G such that  ˜ E  = O(n log n/ǫ 2) and for all vectors x ∈ R V (1 − ǫ) ∑ (x(u) − x(v)) 2 wuv ≤ ∑ (x(u) − x(v)) 2 ˜wuv ≤ (1 + ǫ) ∑ (x(u) − x(v)) 2 wuv. (1) uv∈E uv ∈ ˜ E This improves upon the sparsifiers constructed by Spielman and Teng, which had O(n log c n) edges for some large constant c, and upon those of Benczúr and Karger, which only satisfied (1) for x ∈ {0, 1} V. We conjecture the existence of sparsifiers with O(n) edges, noting that these would generalize the notion of expander graphs, which are constantdegree sparsifiers for the complete graph. A key ingredient in our algorithm is a subroutine of independent interest: a nearlylinear time algorithm that builds a data structure from which we can query the approximate effective resistance between any two vertices in a graph in O(log n) time. uv∈E
On Social Networks and Collaborative Recommendation
"... Social network systems, like last.fm, play a significant role in Web 2.0, containing large amounts of multimediaenriched data that are enhanced both by explicit userprovided annotations and implicit aggregated feedback describing the personal preferences of each user. It is also a common tendency ..."
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Cited by 38 (0 self)
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Social network systems, like last.fm, play a significant role in Web 2.0, containing large amounts of multimediaenriched data that are enhanced both by explicit userprovided annotations and implicit aggregated feedback describing the personal preferences of each user. It is also a common tendency for these systems to encourage the creation of virtual networks among their users by allowing them to establish bonds of friendship and thus provide a novel and direct medium for the exchange of data. We investigate the role of these additional relationships in developing a track recommendation system. Taking into account both the social annotation and friendships inherent in the social graph established among users, items and tags, we created a collaborative recommendation system that effectively adapts to the personal information needs of each user. We adopt the generic framework of Random Walk with Restarts in order to provide with a more natural and efficient way to represent social networks. In this work we collected a representative enough portion of the music social network last.fm, capturing explicitly expressed bonds of friendship of the user as well as social tags. We performed a series of comparison experiments between the Random Walk with Restarts model and a userbased collaborative filtering method using the Pearson Correlation similarity. The results show that the graph model system benefits from the additional information embedded in social knowledge. In addition, the graph model outperforms the standard collaborative filtering method.
The slashdot zoo: Mining a social network with negative edges
 In WWW
, 2009
"... christian.bauckhage ..."
Audience selection for online brand advertising: privacyfriendly social network targeting
 In KDD ’09: Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining
, 2009
"... This paper describes and evaluates privacyfriendly methods for extracting quasisocial networks from browser behavior on usergenerated content sites, for the purpose of finding good audiences for brand advertising (as opposed to click maximizing, for example). Targeting socialnetwork neighbors re ..."
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Cited by 23 (2 self)
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This paper describes and evaluates privacyfriendly methods for extracting quasisocial networks from browser behavior on usergenerated content sites, for the purpose of finding good audiences for brand advertising (as opposed to click maximizing, for example). Targeting socialnetwork neighbors resonates well with advertisers, and online browsing behavior data counterintuitively can allow the identification of good audiences anonymously. Besides being one of the first papers to our knowledge on data mining for online brand advertising, this paper makes several important contributions. We introduce a framework for evaluating brand audiences, in analogy to predictivemodeling holdout evaluation. We introduce methods for extracting quasisocial networks from data on visitations to social networking pages, without collecting any information on the identities of the browsers or the content of the socialnetwork pages. We introduce measures of brand proximity in the network, and show that audiences with high brand proximity indeed show substantially higher brand affinity. Finally, we provide evidence that the quasisocial network embeds a true social network, which along with results from social theory offers one explanation for the increases in audience brand affinity.
Learning Spectral Graph Transformations for Link Prediction
"... We present a unified framework for learning link prediction and edge weight prediction functions in large networks, based on the transformation of a graph’s algebraic spectrum. Our approach generalizes several graph kernels and dimensionality reduction methods and provides a method to estimate their ..."
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Cited by 19 (2 self)
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We present a unified framework for learning link prediction and edge weight prediction functions in large networks, based on the transformation of a graph’s algebraic spectrum. Our approach generalizes several graph kernels and dimensionality reduction methods and provides a method to estimate their parameters efficiently. We show how the parameters of these prediction functions can be learned by reducing the problem to a onedimensional regression problem whose runtime only depends on the method’s reduced rank and that can be inspected visually. We derive variants that apply to undirected, weighted, unweighted, unipartite and bipartite graphs. We evaluate our method experimentally using examples from social networks, collaborative filtering, trust networks, citation networks, authorship graphs and hyperlink networks. 1.
An experimental investigation of graph kernels on a collaborative recommendation task
 Proceedings of the 6th International Conference on Data Mining (ICDM 2006
, 2006
"... This paper presents a survey as well as a systematic empirical comparison of seven graph kernels and two related similarity matrices (simply referred to as graph kernels), namely the exponential diffusion kernel, the Laplacian exponential diffusion kernel, the von Neumann diffusion kernel, the regul ..."
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Cited by 19 (6 self)
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This paper presents a survey as well as a systematic empirical comparison of seven graph kernels and two related similarity matrices (simply referred to as graph kernels), namely the exponential diffusion kernel, the Laplacian exponential diffusion kernel, the von Neumann diffusion kernel, the regularized Laplacian kernel, the commutetime kernel, the randomwalkwithrestart similarity matrix, and finally, three graph kernels introduced in this paper: the regularized commutetime kernel, the Markov diffusion kernel, and the crossentropy diffusion matrix. The kernelonagraph approach is simple and intuitive. It is illustrated by applying the nine graph kernels to a collaborativerecommendation task and to a semisupervised classification task, both on several databases. The graph methods compute proximity measures between nodes that help study the structure of the graph. Our comparisons suggest that the regularized commutetime and the Markov diffusion kernels perform best, closely followed by the regularized Laplacian kernel. 1
A SketchBased Distance Oracle for WebScale Graphs
"... We study the fundamental problem of computing distances between nodes in large graphs such as the web graph and social networks. Our objective is to be able to answer distance queries between pairs of nodes in real time. Since the standard shortest path algorithms are expensive, our approach moves t ..."
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Cited by 16 (1 self)
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We study the fundamental problem of computing distances between nodes in large graphs such as the web graph and social networks. Our objective is to be able to answer distance queries between pairs of nodes in real time. Since the standard shortest path algorithms are expensive, our approach moves the timeconsuming shortestpath computation offline, and at query time only looks up precomputed values and performs simple and fast computations on these precomputed values. More specifically, during the offline phase we compute and store a small “sketch ” for each node in the graph, and at querytime we look up the sketches of the source and destination nodes and perform a simple computation using these two sketches to estimate the distance. Categories and Subject Descriptors G.2.2 [Graph Theory]: Graph algorithms, path and circuit problems
A family of dissimilarity measures between nodes generalizing both the shortestpath and the commutetime distances
 in Proceedings of the 14th SIGKDD International Conference on Knowledge Discovery and Data Mining
"... This work introduces a new family of linkbased dissimilarity measures between nodes of a weighted directed graph. This measure, called the randomized shortestpath (RSP) dissimilarity, depends on a parameter θ and has the interesting property of reducing, on one end, to the standard shortestpath d ..."
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Cited by 14 (7 self)
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This work introduces a new family of linkbased dissimilarity measures between nodes of a weighted directed graph. This measure, called the randomized shortestpath (RSP) dissimilarity, depends on a parameter θ and has the interesting property of reducing, on one end, to the standard shortestpath distance when θ is large and, on the other end, to the commutetime (or resistance) distance when θ is small (near zero). Intuitively, it corresponds to the expected cost incurred by a random walker in order to reach a destination node from a starting node while maintaining a constant entropy (related to θ) spread in the graph. The parameter θ is therefore biasing gradually the simple random walk on the graph towards the shortestpath policy. By adopting a statistical physics approach and computing a sum over all the possible paths (discrete path integral), it is shown that the RSP dissimilarity from every node to a particular node of interest can be computed efficiently by solving two linear systems of n equations, where n is the number of nodes. On the other hand, the dissimilarity between every couple of nodes is obtained by inverting an n × n matrix. The proposed measure can be used for various graph mining tasks such as computing betweenness centrality, finding dense communities, etc, as shown in the experimental section.
Graph nodes clustering based on the commutetime kernel
 In Proceedings of the 11th PacificAsia Conference on Knowledge Discovery and Data Mining (PAKDD 2007). Lecture notes in Computer Science, LNCS
, 2007
"... This work presents a kernel method for clustering the nodes of a weighted, undirected, graph. The algorithm is based on a twostep procedure. First, the sigmoid commutetime kernel (KCT), providing a similarity measure between any couple of nodes by taking the indirect links into account, is compute ..."
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Cited by 13 (6 self)
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This work presents a kernel method for clustering the nodes of a weighted, undirected, graph. The algorithm is based on a twostep procedure. First, the sigmoid commutetime kernel (KCT), providing a similarity measure between any couple of nodes by taking the indirect links into account, is computed from the adjacency matrix of the graph. Then, the nodes of the graph are clustered by performing a kernel kmeans or fuzzy kmeans on this CT kernel matrix. For this purpose, a new, simple, version of the kernel kmeans and the kernel fuzzy kmeans is introduced. The joint use of the CT kernel matrix and kernel clustering appears to be quite successful. Indeed, this methodology provides good results, outperforming the spherical kmeans, on a document clustering problem involving the newsgroups database. 1