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MiMAG: mining coherent subgraphs in multi-layer graphs with edge labels

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Abstract

Detecting dense subgraphs such as cliques or quasi-cliques is an important graph mining problem. While this task is established for simple graphs, today’s applications demand the analysis of more complex graphs: In this work, we consider a frequently observed type of graph where edges represent different types of relations. These multiple edge types can also be viewed as different “layers” of a graph, which is denoted as a “multi-layer graph”. Additionally, each edge might be annotated by a label characterizing the given relation in more detail. By simultaneously exploiting all this information, the detection of more interesting subgraphs can be supported. We introduce the multi-layer coherent subgraph model, which defines clusters of vertices that are densely connected by edges with similar labels in a subset of the graph layers. We avoid redundancy in the result by selecting only the most interesting, non-redundant subgraphs for the output. Based on this model, we introduce the best-first search algorithm MiMAG. In thorough experiments, we demonstrate the strengths of MiMAG in comparison with related approaches on synthetic as well as real-world data sets.

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Notes

  1. In the following, we use the terms “layers” and “dimensions” interchangeably.

  2. The contents of this paper are also included in the first author’s Ph.D. thesis [5]

  3. To avoid confusion, we use the term “vertex” for a vertex in the original graph and the term “node” for the nodes of the set enumeration tree, which represent sets of vertices.

  4. Note that for each layer we get a different set enumeration tree (cf. Fig. 4, top) as different subtrees might be pruned in each layer.

  5. If one is interested in generating all patterns, an arbitrary traversal strategy can be used. We, however, want to determine only a subset of the patterns (the non-redundant, high-quality ones).

  6. Due to our redundancy definition, it is possible that clusters of equal quality are redundant w.r.t. each other. As in this case, there is no indication that one of the clusters is “better” than the other(s), we simply keep the cluster in the result that was detected first.

  7. Please note that the pruning strategies in line 15 and 19 do not discard any valid clusters, but only vertices and dimensions that do not contribute to clusters (see Sect. 4.3).

  8. Assume there would exist an admissible algorithm expanding fewer subtrees. Then, after termination of this algorithm, there exists at least one subtree that has not been investigated by this algorithm but whose estimated quality is higher than the quality of the clusters in the result. Based on the information the algorithm has, however, it cannot rule out the possibility that this subtree contains a valid cluster of this quality. Thus, the algorithm cannot be admissible.

  9. Note: This property does not hold for the cluster model itself (neither for the set of vertices nor for the relevant dimensions). It is possible that O does not form a quasi-clique in dimension i, but \(O^{\prime } \supset O\) does.

  10. http://imdb.com.

  11. http://www.cs.cornell.edu/projects/kddcup/datasets.html.

  12. http://dblp.uni-trier.de.

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Authors and Affiliations

  1. Data Management and Data Exploration Group, RWTH Aachen University, Aachen, Germany

    Brigitte Boden, Holger Hoffmann & Thomas Seidl

  2. Department of Informatics, Technical University of Munich, Munich, Germany

    Stephan Günnemann

Authors
  1. Brigitte Boden
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  2. Stephan Günnemann
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  3. Holger Hoffmann
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  4. Thomas Seidl
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Correspondence to Brigitte Boden.

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Boden, B., Günnemann, S., Hoffmann, H. et al. MiMAG: mining coherent subgraphs in multi-layer graphs with edge labels. Knowl Inf Syst 50, 417–446 (2017). https://doi.org/10.1007/s10115-016-0949-5

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