Topological Distances Between Brain Networks

Many existing brain network distances are based on matrix norms. The element-wise differences may fail to capture underlying topological differences. Further, matrix norms are sensitive to outliers. A few extreme edge weights may severely affect the distance. Thus it is necessary to develop network distances that recognize topology. In this paper, we introduce Gromov-Hausdorff (GH) and Kolmogorov-Smirnov (KS) distances. GH-distance is often used in persistent homology based brain network models. The superior performance of KS-distance is contrasted against matrix norms and GH-distance in random network simulations with the ground truths. The KS-distance is then applied in characterizing the multimodal MRI and DTI study of maltreated children.

[1]  Dinggang Shen,et al.  Matrix-Similarity Based Loss Function and Feature Selection for Alzheimer's Disease Diagnosis , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[2]  Hyekyoung Lee,et al.  Integrated multimodal network approach to PET and MRI based on multidimensional persistent homology , 2014, Human brain mapping.

[3]  Guido Gerig,et al.  Unbiased diffeomorphic atlas construction for computational anatomy , 2004, NeuroImage.

[4]  Moo K. Chung,et al.  Persistent Homology in Sparse Regression and Its Application to Brain Morphometry , 2014, IEEE Transactions on Medical Imaging.

[5]  Michael F. Bonner,et al.  Gray Matter Density of Auditory Association Cortex Relates to Knowledge of Sound Concepts in Primary Progressive Aphasia , 2012, The Journal of Neuroscience.

[6]  P. Jezzard,et al.  Sources of distortion in functional MRI data , 1999, Human brain mapping.

[7]  Moo K. Chung,et al.  Computational Neuroanatomy: The Methods , 2012 .

[8]  Alan C. Evans,et al.  Structural Insights into Aberrant Topological Patterns of Large-Scale Cortical Networks in Alzheimer's Disease , 2008, The Journal of Neuroscience.

[9]  Jean Dickinson Gibbons,et al.  Nonparametric Statistical Inference , 1972, International Encyclopedia of Statistical Science.

[10]  O. Sporns,et al.  Complex brain networks: graph theoretical analysis of structural and functional systems , 2009, Nature Reviews Neuroscience.

[11]  Brian B. Avants,et al.  Symmetric diffeomorphic image registration with cross-correlation: Evaluating automated labeling of elderly and neurodegenerative brain , 2008, Medical Image Anal..

[12]  Moo K. Chung,et al.  Exact Topological Inference for Paired Brain Networks via Persistent Homology , 2017, bioRxiv.

[13]  Andreas Daffertshofer,et al.  Comparing Brain Networks of Different Size and Connectivity Density Using Graph Theory , 2010, PloS one.

[14]  Daniel C. Alexander,et al.  Camino: Open-Source Diffusion-MRI Reconstruction and Processing , 2006 .

[15]  Olaf Sporns,et al.  Complex network measures of brain connectivity: Uses and interpretations , 2010, NeuroImage.

[16]  Bung-Nyun Kim,et al.  Persistent Brain Network Homology From the Perspective of Dendrogram , 2012, IEEE Transactions on Medical Imaging.

[17]  Moo K. Chung,et al.  Computing the Shape of Brain Networks Using Graph Filtration and Gromov-Hausdorff Metric , 2011, MICCAI.

[18]  Facundo Mémoli,et al.  Characterization, Stability and Convergence of Hierarchical Clustering Methods , 2010, J. Mach. Learn. Res..

[19]  Kathleen M. Carley,et al.  Metric inference for social networks , 1994 .

[20]  C. Stam,et al.  Small‐world properties of nonlinear brain activity in schizophrenia , 2009, Human brain mapping.

[21]  Alexey A. Tuzhilin,et al.  Who Invented the Gromov-Hausdorff Distance? , 2016, 1612.00728.

[22]  E. Bullmore,et al.  Neurophysiological architecture of functional magnetic resonance images of human brain. , 2005, Cerebral cortex.