We present a method to estimate block membership of nodes in a random graph generated by a stochastic blockmodel. We use an embedding procedure motivated by the random dot product graph model, a particular example of the latent position model. The embedding associates each node with a vector; these vectors are clustered via minimization of a square error criterion. We prove that this method is consistent for assigning nodes to blocks, as only a negligible number of nodes will be mis-assigned. We prove consistency of the method for directed and undirected graphs. The consistent block assignment makes possible consistent parameter estimation for a stochastic blockmodel. We extend the result in the setting where the number of blocks grows slowly with the number of nodes. Our method is also computationally feasible even for very large graphs. We compare our method to Laplacian spectral clustering through analysis of simulated data and a graph derived from Wikipedia documents.
We prove a central limit theorem for the components of the largest eigenvectors of the adjacency matrix of a finite-dimensional random dot product graph whose true latent positions are unknown. In particular, we follow the methodology outlined in Sussman et al. (2014) to construct consistent estimates for the latent positions, and we show that the appropriately scaled differences between the estimated and true latent positions converge to a mixture of Gaussian random variables. As a corollary, we obtain a central limit theorem for the first eigenvector of the adjacency matrix of an Erdös-Renyi random graph. arXiv:1305.7388v2 [math.ST]
Two-sample hypothesis testing for random graphs arises naturally in neuroscience, social networks, and machine learning. In this paper, we consider a semiparametric problem of two-sample hypothesis testing for a class of latent position random graphs. We formulate a notion of consistency in this context and propose a valid test for the hypothesis that two finite-dimensional random dot product graphs on a common vertex set have the same generating latent positions or have generating latent positions that are scaled or diagonal transformations of one another. Our test statistic is a function of a spectral decomposition of the adjacency matrix for each graph and our test procedure is consistent across a broad range of alternatives. We apply our test procedure to real biological data: in a test-retest data set of neural connectome graphs, we are able to distinguish between scans from di↵erent subjects; and in the C.elegans connectome, we are able to distinguish between chemical and electrical networks. The latter example is a concrete demonstration that our test can have power even for small sample sizes. We conclude by discussing the relationship between our test procedure and generalized likelihood ratio tests.
Objective To detect abnormal corneal thinning in keratoconus using pachymetry maps measured by high-speed anterior segment optical coherence tomography (OCT). Design Cross-sectional observational study. Participants Thirty-seven keratoconic eyes from 21 subjects and 36 eyes from 18 normal subjects. Methods The OCT system operated at a 1.3 μm wavelength with a scan rate of 2000 axial scans per second. A pachymetry scan pattern (8 radials, 128 axial scans each; 10 mm diameter) centered at the corneal vertex was used to map the corneal thickness. The pachymetry map was divided into zones by octants and annular rings. Five pachymetric parameters were calculated from the region inside the 5 mm diameter: minimum, minimum–median, inferior–superior (I-S), inferotemporal–superonasal (IT-SN), and the vertical location of the thinnest cornea. The 1-percentile value of the normal group was used to define the diagnostic cutoff. Placido-ring–based corneal topography was obtained for comparison. Main Outcome Measures The OCT pachymetric parameters and a quantitative topographic keratoconus index (keratometry, I-S, astigmatism, and skew percentage [KISA%]) were used for keratoconus diagnosis. Diagnostic performance was assessed by the area under the receiver operating characteristic (AROC) curve. Results Keratoconic corneas were thinner. The pachymetric minimum averaged 452.6±60.9 μm in keratoconic eyes versus 546±23.7 μm in normal eyes. The 1-percentile cutoff was 491.6 μm. The thinnest location was inferiorly displaced in keratoconus (−805±749 μm vs −118±260 μm ; cutoff, −716 μm). The thinning was focal (minimum–median: −95.2±41.1 μm vs −45±7.7 μm ; cutoff, −62.6 μm). Keratoconic maps were more asymmetric (I-S, −44.8±28.7 μm vs −9.9±9.3 μm ; cutoff, −31.3 μm ; and IT-SN, −63±35.7 μm vs −22±11.4 μm ; cutoff, −48.2 μm). Keratoconic eyes had a higher KISA% index (2641±5024 vs 21±19). All differences were statistically significant (t test, P<0.0001). Applying the diagnostic criteria of any 1 OCT pachymetric parameter below the keratoconus cutoff yielded an AROC of 0.99, which was marginally better (P= .09) than the KISA% topographic index (AROC, 0.91). Conclusions Optical coherence tomography pachymetry maps accurately detected the characteristic abnormal corneal thinning in keratoconic eyes. This method was at least as sensitive and specific as the topographic KISA. Financial Disclosure(s) Proprietary or commercial disclosure may be found after the references.
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