Metrics for Graph Comparison: A Practitioner's Guide

Comparison of graph structure is a ubiquitous task in data analysis and machine learning, with diverse applications in fields such as neuroscience, cyber security, social network analysis, and bioinformatics, among others. Discovery and comparison of structures such as modular communities, rich clubs, hubs, and trees in data in these fields yields insight into the generative mechanisms and functional properties of the graph. Often, two graphs are compared via a pairwise distance measure, with a small distance indicating structural similarity and vice versa. Common choices include spectral distances (also known as $\lambda$ distances) and distances based on node affinities. However, there has of yet been no comparative study of the efficacy of these distance measures in discerning between common graph topologies and different structural scales. In this work, we compare commonly used graph metrics and distance measures, and demonstrate their ability to discern between common topological features found in both random graph models and empirical datasets. We put forward a multi-scale picture of graph structure, in which the effect of global and local structure upon the distance measures is considered. We make recommendations on the applicability of different distance measures to empirical graph data problem based on this multi-scale view. Finally, we introduce the Python library NetComp which implements the graph distances used in this work

Isodiametric inequality in Carnot groups

The classical isodiametric inequality in the Euclidean space says that balls maximize the volume among all sets with a given diameter. We consider in this paper the case of Carnot groups. We prove that for any Carnot group equipped with a Haar measure one can find a homogeneous distance for which this fails to hold. We also consider Carnot-Caratheodory distances and prove that this also fails for these distances as soon as there are length minimizing curves that stop to be minimizing in finite time. Next we study some connections with the comparison between Hausdorff and spherical Hausdorff measures, rectifiability and the generalized 1/2-Besicovitch conjecture giving in particular some cases where this conjecture fails.Comment: 14 page

On Nonrigid Shape Similarity and Correspondence

An important operation in geometry processing is finding the correspondences between pairs of shapes. The Gromov-Hausdorff distance, a measure of dissimilarity between metric spaces, has been found to be highly useful for nonrigid shape comparison. Here, we explore the applicability of related shape similarity measures to the problem of shape correspondence, adopting spectral type distances. We propose to evaluate the spectral kernel distance, the spectral embedding distance and the novel spectral quasi-conformal distance, comparing the manifolds from different viewpoints. By matching the shapes in the spectral domain, important attributes of surface structure are being aligned. For the purpose of testing our ideas, we introduce a fully automatic framework for finding intrinsic correspondence between two shapes. The proposed method achieves state-of-the-art results on the Princeton isometric shape matching protocol applied, as usual, to the TOSCA and SCAPE benchmarks

Model selection in High-Dimensions: A Quadratic-risk based approach

In this article we propose a general class of risk measures which can be used for data based evaluation of parametric models. The loss function is defined as generalized quadratic distance between the true density and the proposed model. These distances are characterized by a simple quadratic form structure that is adaptable through the choice of a nonnegative definite kernel and a bandwidth parameter. Using asymptotic results for the quadratic distances we build a quick-to-compute approximation for the risk function. Its derivation is analogous to the Akaike Information Criterion (AIC), but unlike AIC, the quadratic risk is a global comparison tool. The method does not require resampling, a great advantage when point estimators are expensive to compute. The method is illustrated using the problem of selecting the number of components in a mixture model, where it is shown that, by using an appropriate kernel, the method is computationally straightforward in arbitrarily high data dimensions. In this same context it is shown that the method has some clear advantages over AIC and BIC.Comment: Updated with reviewer suggestion

Interproximal Distance Analysis of Stereolithographic Casts Made by CAD-CAM Technology: An in Vitro Study

Statement of problem The accuracy of interproximal distances of the definitive casts made by computer-aided design and computer-aided manufacturing (CAD-CAM) technology is not yet known. Purpose The purpose of this in vitro study was to compare the interproximal distances of stereolithographic casts made by CAD-CAM technology with those of stone casts made by the conventional method. Material and methods Dentoform teeth were prepared for a single ceramic crown on the maxillary left central incisor, a 3-unit fixed dental prosthesis (FDP) on the second premolar for a metal-ceramic crown, and a maxillary right first molar for a metal crown. Twenty digital intraoral impressions were made on the dentoform with an intraoral digital impression scanner. The digital impression files were used to fabricate 20 sets of stereolithographic casts, 10 definitive casts for the single ceramic crown, and 10 definitive casts for the FDP. Furthermore, 20 stone casts were made by the conventional method using polyvinyl siloxane impression material with a custom tray. Each definitive cast for stereolithographic cast and stone cast consisted of removable die-sectioned casts (DC) and nonsectioned solid casts (SC). Measurements of interproximal distance of each cast were made using CAD software to provide mean ±standard deviation (SD) values. Data were first analyzed by repeated measures analysis of variance (ANOVA), using different methods of cast fabrication (stone and stereolithography) as one within subject factor and different cast types (DC and SC) as another within subject factor. Post hoc analyses were performed to investigate the differences between stone and stereolithographic casts depending upon the results from the repeated measures ANOVA (α=.05). Results Analysis of interproximal distances showed the mean ±SD value of the single ceramic crown group was 31.2 ±24.5 μm for stone casts and 261.0 ±116.1 μm for stereolithographic casts, whereas the mean ±SD value for the FDP group was 46.0 ±35.0 μm for stone casts and 292.8 ±216.6 μm for stereolithographic casts. For both the single ceramic crown and the FDP groups, there were significant differences in interproximal distances between stereolithographic casts and stone casts (P\u3c.001). In addition, the comparisons of DC with SC of stone and stereolithographic casts for the single ceramic crown and FDP groups demonstrated there was statistically significant differences among interproximal distances between DC stereolithographic casts and SC stereolithographic casts only for the FDP group (P\u3c.001). Conclusions For both the single ceramic crown and the FDP groups, the stereolithographic cast group showed significantly larger interproximal distances than the stone cast group. In terms of the comparison between DC and SC, DC stereolithographic casts for the FDP group only showed significantly larger interproximal values than those of the SC stereolithographic casts for the FDP group

Interproximal Distance Analysis of Stereolithographic Casts Made by CAD-CAM Technology: An in Vitro Study

Statement of problem The accuracy of interproximal distances of the definitive casts made by computer-aided design and computer-aided manufacturing (CAD-CAM) technology is not yet known. Purpose The purpose of this in vitro study was to compare the interproximal distances of stereolithographic casts made by CAD-CAM technology with those of stone casts made by the conventional method. Material and methods Dentoform teeth were prepared for a single ceramic crown on the maxillary left central incisor, a 3-unit fixed dental prosthesis (FDP) on the second premolar for a metal-ceramic crown, and a maxillary right first molar for a metal crown. Twenty digital intraoral impressions were made on the dentoform with an intraoral digital impression scanner. The digital impression files were used to fabricate 20 sets of stereolithographic casts, 10 definitive casts for the single ceramic crown, and 10 definitive casts for the FDP. Furthermore, 20 stone casts were made by the conventional method using polyvinyl siloxane impression material with a custom tray. Each definitive cast for stereolithographic cast and stone cast consisted of removable die-sectioned casts (DC) and nonsectioned solid casts (SC). Measurements of interproximal distance of each cast were made using CAD software to provide mean ±standard deviation (SD) values. Data were first analyzed by repeated measures analysis of variance (ANOVA), using different methods of cast fabrication (stone and stereolithography) as one within subject factor and different cast types (DC and SC) as another within subject factor. Post hoc analyses were performed to investigate the differences between stone and stereolithographic casts depending upon the results from the repeated measures ANOVA (α=.05). Results Analysis of interproximal distances showed the mean ±SD value of the single ceramic crown group was 31.2 ±24.5 μm for stone casts and 261.0 ±116.1 μm for stereolithographic casts, whereas the mean ±SD value for the FDP group was 46.0 ±35.0 μm for stone casts and 292.8 ±216.6 μm for stereolithographic casts. For both the single ceramic crown and the FDP groups, there were significant differences in interproximal distances between stereolithographic casts and stone casts (P\u3c.001). In addition, the comparisons of DC with SC of stone and stereolithographic casts for the single ceramic crown and FDP groups demonstrated there was statistically significant differences among interproximal distances between DC stereolithographic casts and SC stereolithographic casts only for the FDP group (P\u3c.001). Conclusions For both the single ceramic crown and the FDP groups, the stereolithographic cast group showed significantly larger interproximal distances than the stone cast group. In terms of the comparison between DC and SC, DC stereolithographic casts for the FDP group only showed significantly larger interproximal values than those of the SC stereolithographic casts for the FDP group

Surface Brightness Fluctuations as Primary and Secondary Distance Indicators

The surface brightness fluctuations (SBF) method measures the variance in a galaxy's light distribution arising from fluctuations in the numbers and luminosities of individual stars per resolution element. Once calibrated for stellar population effects, SBF measurements with HST provide distances to early-type galaxies with unrivaled precision. Optical SBF data from HST for the Virgo and Fornax clusters give the relative distances of these nearby fiducial clusters with 2% precision and constrain their internal structures. Observations in hand will allow us to tie the Coma cluster, the standard of comparison for distant cluster studies, into the same precise relative distance scale. The SBF method can be calibrated in an absolute sense either empirically from Cepheids or theoretically from stellar population models. The agreement between the model and empirical zero points has improved dramatically, providing an independent confirmation of the Cepheid distance scale. SBF is still brighter in the near-IR, and an ongoing program to calibrate the method for the F110W and F160W passbands of the WFC3 IR channel will enable accurate distance derivation whenever a large early-type galaxy or bulge is observed in these passbands at distances reaching well out into the Hubble flow.Comment: 8 pages, invited review at the conference "The Fundamental Cosmic Distance Scale: State of the Art and Gaia Perspective", to appear in Astrophysics and Space Scienc

The performance of a combined distance between time series

This paper presents the comparison of a proposed measure of dissimilarity between time series (COMB) with three baseline measures. COMB is a convex combination of Euclidean distance, a Pearson correlation based distance, a Periodogram based measure and a distance between estimated autocorrelation structures. The comparison resorts to 1-Nearest Neighbour classifier (1NN) since the effectiveness of the dissimilarity measures is directly reflected on the performance of 1NN. Data considered is available in the University of California Riverside (UCR) Time-Series Archive which includes data sets from a wide variety of application domains and have been used in similar studies. The COMB measure shows promising results: a good trade-off performance-computation time when compared to the alternative distances considered.info:eu-repo/semantics/acceptedVersio