2.1 Data

Here we make available the connectome data used in our paper: 114 graphs, and for each graph every vertex has a {Left,Right} label and a {Gray,White} label.

NB: This is not meant to be a finding of neurscientific significance; rather, this is an illustration of our ‘Two Truths’ phenomenon. As such, we consider binarized versions of the largest connected component of the graphs. (The original diffusion MRI connectomes are symmetric, hollow, and weighted.)

The \(m=114\) adjacency matrices on \(n \approx 40,000\) vertices and associated vertex label attributes can be downloaded as an R object. NB: 8GB!

print(load(url("http://www.cis.jhu.edu/~parky/TT/Data/TT-glist114-binary.rda"))) 
# ~8GB, may take several minutes to load.
# This may fail with "Error: vector memory exhaused (limit reached?)" if there is not enough memory on your computer!
#[1] "glist"
length(glist)
#[1] 114

library(igraph)
summary(glist[[1]])
#IGRAPH b40d5fe UNW- 40813 2224492 -- sub-0025864_ses-1_dwi_DS72784
#+ attr: name (g/c), name (v/c), hemisphere (v/c), tissue (v/c), Y
#| (v/c), weight (e/n)

table(V(glist[[1]])$hemisphere)
# left right 
#20412 20401 

table(V(glist[[1]])$tissue)
# gray white 
#19353 21460 

table(V(glist[[1]])$Y)
#   LG    LW    RG    RW 
# 9664 10748  9689 10712 

Figure 2.1: Figure S1. Summary of data: number of vertices per graph by hemisphere/tissue type.

Note that there is one bad graph in the data set: image processing failed for left hemisphere of subject 50 scan 2. (This anomaly is shown in the bar plot, not shown in the box plot.) We have left this graph in, as is.

2.2 Code

R code for the reproducing the experiemntal results presented in the manuscript is available in the demo folder at github.

require(devtools)
devtools::install_github("youngser/TwoTruth")
# WARNING: This may take a while to install all the required packages!
# Also, depending on the enviroment, many packages need to be reinstalled/updated manually!

2.3 Figures

library(TwoTruth)

Figure 2.2: Figure S2. Spectral embedding dimension \(\widehat{d}\) via Zhu & Ghodsi for our 114 connectomes.

Figure 2.3: Figure S3. Number of clusters \(\widehat{K}\) via Mclust BIC for our 114 connectomes.

Figure 2.4: Figure 6. Results of the (\(\widehat{d},\widehat{K}\)) model selection for spectral graph clustering for each of our 114 connectomes. For LSE we see \(\widehat{d} \in \{30,\dots,60\}\) and \(\widehat{K} \in \{2,\dots,20\}\); for ASE we see \(\widehat{d} \in \{2,\dots,20\}\) and \(\widehat{K} \in \{10,\dots,50\}\). The color-coding represents clustering performance in terms of ARI for each of LSE and ASE against each of the two truths {Left,Right} and {Gray,White}, and shows that LSE clustering identifies \(\{\mbox{Left},\mbox{Right}\}\) better than \(\{\mbox{Gray},\mbox{White}\}\) and ASE identifies \(\{\mbox{Gray},\mbox{White}\}\) better than \(\{\mbox{Left},\mbox{Right}\}\). Our ‘Two Truths’ phenomenon is conclusively demonstrated: LSE finds \(\{\mbox{Left},\mbox{Right}\}\) (affinity) while ASE finds \(\{\mbox{Gray},\mbox{White}\}\) (core-periphery).

Figure 2.5: Figure 7. Spectral graph clustering assessment via ARI. For each of our 114 connectomes, we plot the difference in ARI for the \(\{\mbox{Left},\mbox{Right}\}\) truth against the difference in ARI for the \(\{\mbox{Gray},\mbox{White}\}\) truth for the clusterings produced by each of LSE and ASE: x = ARI(LSE,LR) \(-\) ARI(LSE,GW) vs. y = ARI(ASE,LR) \(-\) ARI(ASE,GW). A point in the \((+,-)\) quadrant indicates that for that connectome the LSE clustering identified \(\{\mbox{Left},\mbox{Right}\}\) better than \(\{\mbox{Gray},\mbox{White}\}\) and ASE identified \(\{\mbox{Gray},\mbox{White}\}\) better than \(\{\mbox{Left},\mbox{Right}\}\). Marginal histograms are provided. Our ‘Two Truths’ phenomenon is conclusively demonstrated: LSE identifies \(\{\mbox{Left},\mbox{Right}\}\) (affinity) while ASE identifies \(\{\mbox{Gray},\mbox{White}\}\) (core-periphery).

Generating figures from pre-calculated spectral clustering results:

• Figure S2 (spectral embedding dimension \(\widehat{d}\) via Zhu & Ghodsi): demo(doFigS1)
  
• Figure S3 (number of clusters \(\widehat{K}\) via Mclust BIC): demo(doFigS2)
  
• Figure 6 (\(\widehat{d}\), \(\widehat{K}\), and ARI against {Left, Right} and against {Gray, White}): demo(doFig6)
  
• Figure 7 (ARI comparison demonstrating the ‘Two Truths’ phenomenon): demo(doFig7)

2.4 One Graph

Generating spectral clustering results from the data may take tens of hours.
Here is an example to run our spectral clustering on one graph:

library(TwoTruth)
library(igraph)
library(tidyverse)

load(url("http://www.cis.jhu.edu/~parky/TT/Data/g-s1s1.Rbin"))
summary(g)
#IGRAPH aa0dcd3 UNW- 40813 2224492 -- sub-0025864_ses-1_dwi_DS72784
#+ attr: name (g/c), name (v/c), hemisphere (v/c), tissue (v/c), Y (v/c), weight (e/n)

# Embed into 'optimal' dimension and cluster into 'optimal' number of groups
out.ase <- sclust(g, embed="ASE", dmax=100, Kmax=50, clustering="mclust")
out.lse <- sclust(g, embed="LSE", dmax=100, Kmax=50, clustering="mclust") 

df <- data.frame(embed=c("ASE","LSE"), round(rbind(out.ase$mout$df[,-5], 
                                                   out.lse$mout$df[,-5]),2))
df
#  embed dhat Khat   LR   GW
#1   ASE   15   39 0.01 0.04
#2   LSE   46   10 0.03 0.01

2.5 Simulation

To run a simple simulation using the synthetic data based on the real 4 x 4 B matrix, please do

demo(simulation)
#    ASE-K1 ASE-K2 LSE-K1 LSE-K2
# LG      1    278      0    279
# LW    219      0      0    219
# RG      2    280    282      0
# RW    219      0    219      0

Figure 2.6: Histograms from the results of 100 repeat of the simulation. It is apparent that ASE separates {Gray,White} better and LSE separates {Left, Right} better.

A simulation demo writeen in Python is also available in the demo folder as a Jupyter notebook.