Graph Theory Steffie Tomson UCLA NITP 2013
What is graph theory? Mathematical descriptions of relationships between regions Node Opsahl et al. 2010
Why use graphs? Nodal Measures: Participation Coefficient Clustering Coefficient Degree Local efficiency Global Measures: Global efficiency Modularity Hubs
Graph Pipeline Network organization Aleman Gomez et al. 2006 Functional MRI Structural MRI A B Brain Regions Brain Regions
Parcellation How do you divide up the brain? Anatomical atlas AAL Havard Oxford Talairach Functional regions Craddock et al. 2012 Power et al. 2012 Craddock et al. 2012
Graph Pipeline Network organization Aleman Gomez et al. 2006 Functional MRI Structural MRI A B Brain Regions Brain Regions
Structural connectivity Fractional anisotropy (FA) Measure of white matter diffusivity at each voxel Tractography Directional pattern of white matter tracts Structural CoCoMac
Functional connectivity Looooooong timeseries Resting state Listening to audio Watching video Acquire twice as many timepoints as nodes in your graph (Braun et al. 2012) Prepocessing Regress out motion Small smoothing kernel Functional MRI EEG MEG NIRS
Functional connectivity Functional MRI Full correlation Partial correlation Bayes nets LiNGAM EEG/MEG Coherence Granger Functional MRI EEG MEG NIRS Smith et al. 2011. Network modeling methods for fmri. NeuroImage.
Connectivity measures Functional MRI Correlation A B C Partial Correlation A B C All three regions appear to be correlated because they all covary. One region, however, could be driving activity in two other regions, inducing a false When the correlation between A and B is evaluated given C s activity, C is actually better at explaining the variance of A and B than either are of each other. C is the common driver. correlation. Marrelec et al. 2006, Smith et al. 2011
Graph Pipeline Network organization Aleman Gomez et al. 2006 Functional MRI Structural MRI A B Brain Regions Brain Regions
Graph pipeline Functional MRI A Structural MRI B zz Quantify relationship between nodes Brain Regions Brain Regions
Threshold 1 0.8 0.6? 0.4 0.2 Brain Regions 0-0.2-0.4-0.6-0.8 Brain Regions
Threshold Keep only 10, 20, 30% of connections Keep everything above a value Keep everything above an absolute value Discard values around zero Use range of thresholds
Network Metric Group 1 Group 2.125.225.325.425 Matrix Threshold
Stability selection 1 subject Dr. Genevera Allen Manjari Narayan λ 1 λ 2 λ n 1 λ 2-1 Average Average Average Sparse network best supported by data Liu et al., 2010.
MONET Markov Network Estimation Toolbox https://bitbucket.org/gastats/monet/overview Genevera Allen, Manjari Naryan, Jonathan Stewart at Rice University
Graph pipeline Aleman Gomez et al. 2006 Functional MRI Structural MRI A B Brain Regions Brain Regions
Types of graphs Weighted Binary
Rubinov and Sporns 2010
Graph pipeline Network organization Aleman Gomez et al. 2006 Functional MRI Structural MRI A B Brain Regions Brain Regions
NODE Brain Regions Brain Regions EDGE Node = Brain region Edge = Connection between brain regions
Graph theory metrics Rubinov and Sporns. Complex network measures of brain connectivity: uses and interpretations. 2010. Neuroimage.
Graph theory metrics Degree High degree Low degree Number of edges emanating from a single node
Graph theory metrics Clustering coefficient High Low How many of your nearest neighbors are connected to one another?
Graph theory metrics Local efficiency High Low Average shortest path connecting all neighbors of a given node
Graph theory metrics Characteristic path length Low High Average shortest path length between all node pairs
Graph theory metrics Global efficiency High Low Inverse of the average path length
Graph theory metrics Betweenness centrality Hubness High Low Number of shortest paths that pass through a given node
Graph theory metrics Hub Buckner et al. 2009
Graph theory metrics Modularity
Graph theory metrics Modularity Control Child onset schizophrenia Alexander Bloch et al. 2010
Brain connectivity toolbox https://sites.google.com/site/bctnet/
Testing for differences Statistical inference Bootstrap procedure Permutation tests Tomson et al. 2013 In Press.
Testing for differences Statistical inference Bootstrap procedure Permutation tests Tomson et al. 2013 In Press.
Multiple comparisons False discovery rate Benjamini Hochberg Family wise error rate Gaussian random field theory Bonferroni Choose graph metrics you care about Choose graph metrics least mathematically related
Visualization http://umcd.humanconnectomeproject.org/ http://www.nitrc.org/projects/bnv/ Connectome Viewer http://cmtk.org/viewer/datasets//
Resources Brain connectivity toolbox MONET UMCD Brain Net viewer ABIDE ADNI COINS Connectome Project Open fmri Rubinov and Sporns 2010 Smith et al. 2011 Bullmore and Sporns 2009. Nat Rev Neuro.
Acknowledgements Bookheimer lab Dappretto lab Bearden lab Dr. Genevera Allen Manjari Narayan NITP 2011 Neurobehavioral Genetics grant
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