`Two Truths`

Phenomenon in Spectral Graph ClusteringJohns Hopkins University

Carey E. Priebe, Youngser Park, Joshua T. Vogelstein, John M. Conroy, Vince Lyzinskic, Minh Tang, Avanti Athreya, Joshua Cape, and Eric Bridgeford, “On a ‘Two Truths’ Phenomenon in Spectral Graph Clustering,”

Proceedings of National Academy of Science, submitted, 2018.

Clustering is concerned with coherently grouping observations without any explicit concept of true groupings. Spectral graph clustering – clustering the vertices of a graph based on their spectral embedding – is commonly approached via K-means (or, more generally, Gaussian mixture model) clustering composed with either Laplacian or Adjacency spectral embedding (LSE or ASE). Recent theoretical results provide new understanding of the problem and solutions, and lead us to a ‘Two Truths’ LSE vs. ASE spectral graph clustering phenomenon convincingly illustrated here via a diffusion MRI connectome data set: the different embedding methods yield different clustering results, with LSE capturing left hemisphere/right hemisphere affinity structure and ASE capturing gray matter/white matter core-periphery structure.

Keywords: Spectral Embedding, Spectral Clustering, Graph, Network, Connectome

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
```

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.

`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!
```

`library(TwoTruth)`