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Based on our theory & methods for community detection and community comparison in graphs (Lyzinski et al. 2015),
we formulate a model selection procedure for deciding whether
a hierarchical stochastic blockmodel graph supports the hypothesis of repeated motifs.
Such a graph inference procedure provides a framework for addressing a fundamental outstanding question
regarding the atoms of neural computation (Marcus et al. 2014).


Consider RMHSBMs \(\mathcal{F}_{R,M_1,K_1}\) and \(\mathcal{F}_{R,M_2,K_2}\) with \(M_2 < M_1 = R\) & \(K_2 > K_1\) chosen such that the two model parameter spaces are of the same dimensionality. Then \(T = \| \hat{P}_1 - A \| ~/~ \| \hat{P}_2 -A \| > 1\) provides evidence in favor of the hypothesis that the graph possesses repeated motif structure.

References

Lyzinski, V., Tang, M., Athreya, A., Park, Y. & Priebe, C.E. (2015). Community Detection and Classification in Hierarchical Stochastic Blockmodels. ArXiv e-prints.

Marcus, G.F., Marblestone, A.H. & Dean, T.L. (2014). Frequently Asked Questions for: The Atoms of Neural Computation. ArXiv e-prints.