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14
A local algorithm for finding wellconnected clusters
- CoRR
, 2013
"... Motivated by applications of large-scale graph clustering, we study random-walkbased local algorithms whose running times depend only on the size of the output cluster, rather than the entire graph. In particular, we develop a method with better theoretical guarantee compared to all previous work, b ..."
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Cited by 7 (2 self)
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Motivated by applications of large-scale graph clustering, we study random-walkbased local algorithms whose running times depend only on the size of the output cluster, rather than the entire graph. In particular, we develop a method with better theoretical guarantee compared to all previous work, both in terms of the clustering accuracy and the conductance of the output set. We also prove that our analysis is tight, and perform empirical evaluation to support our theory on both synthetic and real data. More specifically, our method outperforms prior work when the cluster is well-connected. In fact, the better it is well-connected inside, the more significant improvement we can obtain. Our results shed light on why in practice some random-walk-based algorithms perform better than its previous theory, and help guide future research about local clustering. 1.
Analyzing the Harmonic Structure in Graph-Based Learning
"... We find that various well-known graph-based models exhibit a common important harmonic structure in its target function – the value of a vertex is approximately the weighted average of the values of its adjacent neighbors. Understanding of such structure and analysis of the loss defined over such st ..."
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Cited by 3 (2 self)
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We find that various well-known graph-based models exhibit a common important harmonic structure in its target function – the value of a vertex is approximately the weighted average of the values of its adjacent neighbors. Understanding of such structure and analysis of the loss defined over such structure help reveal im-portant properties of the target function over a graph. In this paper, we show that the variation of the target function across a cut can be upper and lower bounded by the ratio of its harmonic loss and the cut cost. We use this to develop an analytical tool and analyze five popular graph-based models: absorbing random walks, par-tially absorbing random walks, hitting times, pseudo-inverse of the graph Lapla-cian, and eigenvectors of the Laplacian matrices. Our analysis sheds new insights into several open questions related to these models, and provides theoretical justi-fications and guidelines for their practical use. Simulations on synthetic and real datasets confirm the potential of the proposed theory and tool. 1
Coinciding walk kernels: Parallel absorbing random walks for learning with graphs and few labels
- In Asian Conference on Machine Learning
, 2013
"... Exploiting autocorrelation for node-label prediction in networked data has led to great suc-cess. However, when dealing with sparsely labeled networks, common in present-day tasks, the autocorrelation assumption is difficult to exploit. Taking a step beyond, we propose the coinciding walk kernel (cw ..."
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Cited by 2 (1 self)
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Exploiting autocorrelation for node-label prediction in networked data has led to great suc-cess. However, when dealing with sparsely labeled networks, common in present-day tasks, the autocorrelation assumption is difficult to exploit. Taking a step beyond, we propose the coinciding walk kernel (cwk), a novel kernel leveraging label-structure similarity – the idea that nodes with similarly arranged labels in their local neighbourhoods are likely to have the same label – for learning problems on partially labeled graphs. Inspired by the success of random walk based schemes for the construction of graph kernels, cwk is defined in terms of the probability that the labels encountered during parallel random walks coincide. In addition to its intuitive probabilistic interpretation, coinciding walk kernels outperform existing kernel- and walk-based methods on the task of node-label prediction in sparsely labeled graphs with high label-structure similarity. We also show that computing cwks is faster than many state-of-the-art kernels on graphs. We evaluate cwks on several real-world networks, including cocitation and coauthor graphs, as well as a graph of interlinked populated places extracted from the dbpedia knowledge base.
Scaling Graph-based Semi Supervised Learning to Large Number of Labels Using Count-Min Sketch
"... Graph-based Semi-supervised learning (SSL) algorithms have been successfully used in a large number of applications. These methods classify initially unlabeled nodes by propa-gating label information over the structure of graph starting from seed nodes. Graph-based SSL algorithms usually scale linea ..."
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Cited by 2 (0 self)
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Graph-based Semi-supervised learning (SSL) algorithms have been successfully used in a large number of applications. These methods classify initially unlabeled nodes by propa-gating label information over the structure of graph starting from seed nodes. Graph-based SSL algorithms usually scale linearly with the number of distinct labels (m), and require O(m) space on each node. Unfortunately, there exist many applications of practical sig-nificance with very large m over large graphs, demanding better space and time complexity. In this paper, we propose MAD-Sketch, a novel graph-based SSL algorithm which compactly stores label distribution on each node using Count-min Sketch, a random-ized data structure. We present theoretical analysis showing that under mild conditions, MAD-Sketch can reduce space complexity at each node from O(m) to O(logm), and achieve similar savings in time complexity as well. We support our analysis through exper-iments on multiple real world datasets. We observe that MAD-Sketch achieves simi-lar performance as existing state-of-the-art graph-based SSL algorithms, while requir-ing smaller memory footprint and at the same time achieving up to 10x speedup. We find that MAD-Sketch is able to scale to datasets with one million labels, which is be-yond the scope of existing graph-based SSL algorithms.
Coinciding Walk Kernels
"... Exploiting autocorrelation for node-label prediction in networked data has led to great success. However, when dealing with sparsely labeled networks, common in present-day tasks, the autocorrelation assumption is difficult to exploit. Taking a step beyond, we propose the coinciding walk kernel (cwk ..."
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Cited by 1 (1 self)
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Exploiting autocorrelation for node-label prediction in networked data has led to great success. However, when dealing with sparsely labeled networks, common in present-day tasks, the autocorrelation assumption is difficult to exploit. Taking a step beyond, we propose the coinciding walk kernel (cwk), a novel kernel leveraging label-structure similarity – the idea that nodes with similarly arranged labels in their local neighbourhoods are likely to have the same label – for learning problems on partially labeled graphs. Inspired by the success of random walk based schemes for the construction of graph kernels, cwk is defined in terms of the probability that the labels encountered during parallel random walks coincide. In addition to its intuitive probabilistic interpretation, coinciding walk kernels outperform state-ofthe-art kernel- and walk-based methods on the task of nodelabel prediction in sparsely labeled graphs. We also show that computing cwks is faster than many state-of-the-art kernels on graphs. We evaluate cwks on several real-world networks, including cocitation and coauthor graphs, as well as a network of interlinked populated places extracted from the dbpedia knowledge base. 1.
Graph-based Semi-supervised Learning: Realizing Pointwise Smoothness Probabilistically Yuan Fang † ‡
"... As the central notion in semi-supervised learn-ing, smoothness is often realized on a graph rep-resentation of the data. In this paper, we study two complementary dimensions of smoothness: its pointwise nature and probabilistic modeling. While no existing graph-based work exploits them in conjunctio ..."
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As the central notion in semi-supervised learn-ing, smoothness is often realized on a graph rep-resentation of the data. In this paper, we study two complementary dimensions of smoothness: its pointwise nature and probabilistic modeling. While no existing graph-based work exploits them in conjunction, we encompass both in a novel framework of Probabilistic Graph-based Pointwise Smoothness (PGP), building upon two foundational models of data closeness and label coupling. This new form of smoothness axiom-atizes a set of probability constraints, which ul-timately enables class prediction. Theoretically, we provide an error and robustness analysis of PGP. Empirically, we conduct extensive experi-ments to show the advantages of PGP. 1.
Σ-optimality for active learning on Gaussian random fields
- In Advances in Neural Information Processing Systems 26
, 2013
"... A common classifier for unlabeled nodes on undirected graphs uses label propaga-tion from the labeled nodes, equivalent to the harmonic predictor on Gaussian ran-dom fields (GRFs). For active learning on GRFs, the commonly used V-optimality criterion queries nodes that reduce the L2 (regression) los ..."
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A common classifier for unlabeled nodes on undirected graphs uses label propaga-tion from the labeled nodes, equivalent to the harmonic predictor on Gaussian ran-dom fields (GRFs). For active learning on GRFs, the commonly used V-optimality criterion queries nodes that reduce the L2 (regression) loss. V-optimality satis-fies a submodularity property showing that greedy reduction produces a (1 − 1/e) globally optimal solution. However, L2 loss may not characterise the true nature of 0/1 loss in classification problems and thus may not be the best choice for active learning. We consider a new criterion we call Σ-optimality, which queries the node that minimizes the sum of the elements in the predictive covariance. Σ-optimality directly optimizes the risk of the surveying problem, which is to determine the proportion of nodes belonging to one class. In this paper we extend submodularity guarantees from V-optimality to Σ-optimality using properties specific to GRFs. We further show that GRFs satisfy the suppressor-free condition in addition to the conditional independence inherited from Markov random fields. We test Σ-optimality on real-world graphs with both synthetic and real data and show that it outperforms V-optimality and other related methods on classification. 1
Local Network Community Detection with Continuous Optimization of Conductance and Weighted Kernel K-Means Twan van Laarhoven
, 2016
"... Abstract Local network community detection is the task of finding a single community of nodes concentrated around few given seed nodes in a localized way. Conductance is a popular objective function used in many algorithms for local community detection. This paper studies a continuous relaxation of ..."
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Abstract Local network community detection is the task of finding a single community of nodes concentrated around few given seed nodes in a localized way. Conductance is a popular objective function used in many algorithms for local community detection. This paper studies a continuous relaxation of conductance. We show that continuous optimization of this objective still leads to discrete communities. We investigate the relation of conductance with weighted kernel k-means for a single community, which leads to the introduction of a new objective function, σ-conductance. Conductance is obtained by setting σ to 0. Two algorithms, EMc and PGDc, are proposed to locally optimize σ-conductance and automatically tune the parameter σ. They are based on expectation maximization and projected gradient descent, respectively. We prove locality and give performance guarantees for EMc and PGDc for a class of dense and well separated communities centered around the seeds. Experiments are conducted on networks with ground-truth communities, comparing to state-of-the-art graph diffusion algorithms for conductance optimization. On large graphs, results indicate that EMc and PGDc stay localized and produce communities most similar to the ground, while graph diffusion algorithms generate large communities of lower quality.
Walking in the Cloud: Parallel SimRank at Scale
"... ABSTRACT Despite its popularity, SimRank is computationally costly, in both time and space. In particular, its recursive nature poses a great challenge in using modern distributed computing power, and also prevents querying similarities individually. Existing solutions suffer greatly from these pra ..."
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ABSTRACT Despite its popularity, SimRank is computationally costly, in both time and space. In particular, its recursive nature poses a great challenge in using modern distributed computing power, and also prevents querying similarities individually. Existing solutions suffer greatly from these practical issues. In this paper, we break such dependency for maximum efficiency possible. Our method consists of offline and online phases. In offline phase, a length-n indexing vector is derived by solving a linear system in parallel. At online query time, the similarities are computed instantly from the index vector. Throughout, the Monte Carlo method is used to maximally reduce time and space. Our algorithm, called CloudWalker, is highly parallelizable, with only linear time and space. Remarkably, it responses to both single-pair and single-source queries in constant time. CloudWalker is orders of magnitude more efficient and scalable than existing solutions for large-scale problems. Implemented on Spark with 10 machines and tested on the web-scale clue-web graph with 1 billion nodes and 43 billion edges, it takes 110 hours for offline indexing, 64 seconds for a single-pair query, and 188 seconds for a single-source query. To the best of our knowledge, our work is the first to report results on clueweb, which is 10x larger than the largest graph ever reported for SimRank computation.
