Abstract
Existing block modeling methods can detect communities as blocks. However it remains a challenge to easily explain to a human why nodes belong to the same block. Such a description is very useful for answering why people in the same community tend to interact cohesively. In this paper we explore a novel problem: Given a block model already found, describe the blocks using an auxiliary set of information. We formulate a combinatorial optimization problem which finds a unique disjunction of the auxiliary information shared by the nodes either in the same block or between a pair of different blocks. The former terms intra-block description, the latter inter-block description. Given an undirected graph and its \(k-\)block model, our method generates \(k + \frac{k(k-1)}{2}\) different descriptions. If the tags are descriptors of events occurring at the vertices, our descriptions can be interpreted as common events occurring within blocks and between blocks. We show that this problem is intractable even for simple cases, e.g., when the underlying graph is a tree with just two blocks. However, simple and efficient ILP formulations and algorithms exist for its relaxation and yield insights different from a state-of-the-art related work in unsupervised description. We empirically show the power of our work on multiple real-world large datasets.
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Notes
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Here \(\mathbf {D}\) denotes a diagonal matrix. We use \(\Vert \bullet \Vert _F\) to denote the Frobenius norm.
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- 3.
When this condition does not hold, our method can still be used but just with these edges removed as they are unexplainable.
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Proofs for our theorems, our codes and other supplementary materials are available on https://github.com/ZilongBai/ECMLPKDD2020TDBMG. .
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Acknowledgments
This research is supported by ONR Grant N000141812485 and NSF Grants IIS-1910306, IIS-1908530, OAC-1916805, ACI-1443054 (DIBBS), IIS-1633028 (BIG DATA) and CMMI-1745207 (EAGER).
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Bai, Z., Ravi, S.S., Davidson, I. (2021). Towards Description of Block Model on Graph. In: Hutter, F., Kersting, K., Lijffijt, J., Valera, I. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2020. Lecture Notes in Computer Science(), vol 12459. Springer, Cham. https://doi.org/10.1007/978-3-030-67664-3_3
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