Graph theory for TLE lateralization: Supporting Information 1
APPENDIX A. fMRI measures of brain graph topology.
1. Local Clustering Coefficient (CC) and Normalized Clustering Coefficient (γ)
Normalized clustering coefficient (γ) provides a measure of the level of “cliquishness,” or
local interconnectedness, of a network (1). The local clustering coefficient of node i (CC)
was calculated for each node as the ratio of the number of existing to the number of
possible connections in the Gi, the subgraph of node i:
CCi 
Ei
,
K i ( K i  1) / 2
[1]
where Ei denotes the number of edges in subgraph Gi and Ki denotes the degree of node i.
The absolute clustering coefficient of the network (C) was then calculated as the mean
value of Ci, averaged over all nodes (2). To avoid the influence of other network
characteristics, γ was calculated as the ratio of C to Crandom, the averaged clustering
coefficient over 500 randomly rewired null models (3).
2. Normalized Characteristic Path Length (λ)
The characteristic path length (L) is defined as the average shortest path length between
all pairs of nodes, and provides a measure of the global efficiency of information transfer
in a network (1). To account for infinite path lengths between disconnected nodes, L was
calculated as the harmonic mean of the shortest path lengths:
L
N ( N  1)
1

d i, j
i  jG
,
[2]
where di,j denotes the shortest path length between nodes i and j, and N denotes the
number of nodes in the graph (2). To avoid the influence of other network characteristics,
λ was then calculated as the ratio of L to Lrandom, the averaged characteristic path length
over 500 randomly rewired null models (3).
Graph theory for TLE lateralization: Supporting Information 2
3. Small-World Index (σ)
Small-world index (σ) is a measure of the network’s balance between network
segregation and integration (4), and was calculated as the ratio of clustering coefficient to
path length:


.

[3]
Small-world networks typically have γ>1, λ≈1, σ>1 (1). Because noise reduction and
better class separation can potentially be achieved through the inclusion of redundant
variables (5), small-world index was included along with clustering coefficient and path
length as a feature input.
4. Local efficiency (LE) and Global Efficiency (GE)
The local efficiency (LE) of node i was calculated as the mean of the inverse shortest path
length from node i to all other nodes:
LEi 
1
N

jG: j i
1
di, j
,
[4]
where di,j denotes the shortest path length between nodes i and j and N denotes the
number of nodes in the graph. Global efficiency (GE), which measures the average level
of efficiency in interregional communication, was calculated as the average local
efficiency over all nodes:
GE 
1
N
 LE
iG
i
.
[5]
4. Connectivity Strength (CS)
Connectivity strength provides a measure of the network’s average level of connectivity,
and was calculated as the mean of all pairwise correlations (6):
Graph theory for TLE lateralization: Supporting Information 3
CS 
1
N
r
i  jG
ij
,
[6]
where rij denotes the pairwise Pearson correlation between nodes i and j and N denotes
the number of nodes in the graph.
5. Connectivity Diversity (CD)
Connectivity diversity provides a measure of the heterogeneity of the network’s
connectivity. The connectivity diversity of node i (CDi) is defined as the unbiased sample
variance of all pairwise correlations with node i:
CDi 
1
( rij  ri ) 2

N  1 jG: j i
[7]
where ri denotes the mean pairwise Pearson correlation between node i and all other
nodes in graph G, rij denotes the pairwise Pearson correlation between nodes i and j, and
N denotes the number of nodes in the graph. The connectivity diversity of a network was
then calculated as the mean value of CDi across all nodes (6):
CD 
1
N
 CD .
iG
i
[8]
6. Betweenness Centrality (BC)
The betweenness centrality of a node describes the number of shortest paths that pass
through the node. It provides a measure of the node’s importance, with higher levels of
betweenness centrality indicating nodes located on highly traveled paths. Because
hippocampal sclerosis is the most common finding in TLE (7), betweenness centrality of
the left hippocampus (BC(L)) and right hippocampus (BC(R)) were calculated as the
fraction of shortest paths in the network containing the respective hippocampal node:
Graph theory for TLE lateralization: Supporting Information 4
BC ( L) 
BC ( R ) 

M ij ( LH )
i  j  LH
M ij

M ij ( RH )
i  j  RH
M ij
[9]
,
[10]
where RH denotes the right hippocampus, LH denotes the left hippocampus, Mij(RH) and
Mij(LH) denote the number of shortest paths in the network from node i to node j which
pass through the right and left hippocampus, respectively, and Mij denotes the number of
shortest paths in the network from node i to node j.
All unweighted measures were averaged across the non-random connection density range, where
the nonrandom connection density range was chosen as the range of connection densities that
retained small-world topology, such that (1) more than 99% of nodes were connected, and (2)
small-world index (σ) > 1. Graph construction and feature definitions were performed using
Matlab (Mathworks, Inc., Natick, MA, USA). Graph theory metrics were calculated using the
Brain Connectivity Toolbox (2).
Graph theory for TLE lateralization: Supporting Information 5
APPENDIX B. Quadratic discriminant analysis and leave-one-out cross-validation.
QDA is a classification procedure used to estimate the decision boundary between classes, based
on a nonlinear generalization of Fisher discriminant analysis. This method assumes a
multivariate Gaussian distribution for each class density allowing unequal covariance matrices,
resulting in quadratic decision boundaries. QDA has been found to perform well on many
diverse classification tasks, due to the stability of estimates provided by Gaussian models (8).
For further reference, we refer the reader to (8,9).
Cross-validation is a widely used statistical method for estimating prediction error. It is
especially useful for moderate sample sizes, where it may not be feasible to set aside a validation
set to assess predictive performance (10). It has been shown that the LOO-CV estimator for
prediction error is unbiased for the true prediction error (11,12). In LOO-CV, the discriminant
function is trained on a training set that includes all subjects except one. The excluded subject is
then tested using the discriminant function built using the other subjects. This is repeated until
each subject has been excluded once, thereby allowing an average prediction accuracy measure
to be assessed based on the number of correctly classified subjects. Since, for each crossvalidation fold, the “test set” (the excluded subject) is kept apart from the “training set” (all
subjects except the excluded subject), this method allows for feature selection and assessment of
classification performance which is not biased by the test sample.
Graph theory for TLE lateralization: Supporting Information 6
APPENDIX C. Fisher separability criterion and backward stepwise variable importance.
The Fisher separability criterion (FSC) is defined as
  L, j   R , j


j





2
[11]
(13), where  L, j is the mean value of the jth fMRI feature among left TLE patients,  R , j is the
mean value of the jth fMRI feature among right TLE patients, and  j is the standard deviation of
the jth fMRI feature among all TLE patients. As can be seen from Equation 11, FSC indicates
greater separability when the means are further apart and when the variance is lower. FSC has
been observed by (13) to be a natural unit measuring univariate separability of clusters.
The backward stepwise variable importance (BSVI) of a feature was defined as the increase in
LOO-CV error upon removing the respective feature from the optimal feature set. First, the
optimal feature subset was defined as the subset of fMRI features which minimized LOO-CV
error based on a quadratic discriminant function. Next, the contribution to the multivariate
discriminatory power of each fMRI feature in the optimal feature subset was estimated, by
computing the increase in LOO-CV error when the respective feature was removed.
Graph theory for TLE lateralization: Supporting Information 7
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