Abstract
Determining if a graph displays a clustered structure prior to subjecting it to any cluster detection technique has recently gained attention in the literature. Attempts to group graph vertices into clusters when a graph does not have a clustered structure is not only a waste of time; it will also lead to misleading conclusions. To address this problem, we introduce a novel statistical test, the \(\delta \)-test, which is based on comparisons of local and global densities. Our goal is to assess whether a given graph meets the necessary conditions to be meaningfully summarized by clusters of vertices. We empirically explore our test’s behavior under a number of graph structures. We also compare it to other recently published tests. From a theoretical standpoint, our test is more general, versatile and transparent than recently published competing techniques. It is based on the examination of intuitive quantities, applies equally to weighted and unweighted graphs and allows comparisons across graphs. More importantly, it does not rely on any distributional assumptions, other than the universally accepted definition of a clustered graph. Empirically, our test is shown to be more responsive to graph structure than other competing tests.
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Notes
- 1.
Also, note that in this article we assume undirected graphs with no self-loops.
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Acknowledgments
Pierre Miasnikof was supported by a Mitacs-Accelerate PhD award IT05806. He also wishes to thank Lasse Leskelä of Aalto University, for the introduction to the work of Gao and Lafferty. Liudmila Prokhorenkova and Andrei Raigorodskii were supported by The Russian Science Foundation (grant number 16-11-10014).
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Miasnikof, P., Prokhorenkova, L., Shestopaloff, A.Y., Raigorodskii, A. (2020). A Statistical Test of Heterogeneous Subgraph Densities to Assess Clusterability. In: Matsatsinis, N., Marinakis, Y., Pardalos, P. (eds) Learning and Intelligent Optimization. LION 2019. Lecture Notes in Computer Science(), vol 11968. Springer, Cham. https://doi.org/10.1007/978-3-030-38629-0_2
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