1/36
Motivation
Graph theory
Structural MRI connectivity analyses using graph
Graph invariants
Previous work
theory approaches to differentiate subjects across
Graph applications
Neuroscience
the spectrum of AD
Graph
creation
Data set
Creation
Computational
issues
DAVID P HILLIPS
Results
Summary
Conclusion
Department of Mathematics
United States Naval Academy
Joint work with Leah Jager, Anja Soldan (Johns Hopkins), Alec McGlaughlin,
and Dave Ruth (USNA)
April 21, 2014
2/36
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
Analysis technique:
• Obtain imaging data for a group of subjects.
• Derive graphs for the subjects.
• Measure some properties of the graphs to differentiate
across the spectrum of AD.
3/36
Occam’s/Ockham’s razor?
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
For every complex problem there is an answer that
is clear, simple, and wrong.
–H.L. Mencken
Data set
Creation
Computational
issues
Results
Summary
Conclusion
Problems worthy
of attack
prove their worth
by hitting back.
–Piet Hein
3/36
Occam’s/Ockham’s razor?
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
For every complex problem there is an answer that
is clear, simple, and wrong.
–H.L. Mencken
Data set
Creation
Computational
issues
Results
Summary
Conclusion
Problems worthy
of attack
prove their worth
by hitting back.
–Piet Hein
What graph creation methods and measures, if any, work
best to differentiate across the AD spectrum?
4/36
Motivation
Graph theory
Graph invariants
A graph G = (N, E) consists of a set N of nodes and a set
E of edges which indicate relationships between pairs of
nodes.
Previous work
Graph applications
Neuroscience
5
6
Graph
creation
Data set
7
Creation
Computational
issues
Results
3
Summary
4
Conclusion
1
2
5/36
Paths and connectivity
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
• A path is a sequence of nodes with edges between
consecutive nodes.
• Nodes are connected if there is a path between them.
• The path length is the number of edges in the path.
5
6
Creation
Computational
issues
7
Results
Summary
Conclusion
3
1
4
2
7 − 5 − 3 − 4 − 2 is a path of length four.
All nodes here are connected.
6/36
Analyzing graphs
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• The complexity of graphs increases exponentially as
nodes and edges increase.
6/36
Analyzing graphs
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
• The complexity of graphs increases exponentially as
nodes and edges increase.
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• Graph invariants are node label independent functions
that can measure features such as efficiency and
assortativity.
7/36
An efficiency graph invariant
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• The shortest path between two nodes is the path with
shortest length.
7/36
An efficiency graph invariant
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
• The shortest path between two nodes is the path with
shortest length.
Neuroscience
Graph
creation
5
6
Data set
Creation
7
Computational
issues
Results
3
Summary
4
Conclusion
1
• 1 − 2 − 4 − 6 − 5 is a path of length 4.
2
7/36
An efficiency graph invariant
Motivation
Graph theory
Graph invariants
Previous work
• The shortest path between two nodes is the path with
shortest length.
Graph applications
Neuroscience
5
6
Graph
creation
Data set
7
Creation
Computational
issues
Results
3
Summary
4
Conclusion
1
• 1 − 2 − 4 − 6 − 5 is a path of length 4.
• 1 − 3 − 5 is a shortest path of length 2.
2
7/36
An efficiency graph invariant
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
• The characteristic path length of a graph is the average
of the shortest path lengths over all pairs of nodes in
the graph.
5
6
Data set
Creation
Computational
issues
7
Results
Summary
3
Conclusion
1
4
2
• The characteristic path length of this graph is 1.9.
7/36
An efficiency graph invariant
Motivation
• The characteristic path length of a graph is the average
Graph theory
of the shortest path lengths over all pairs of nodes in
the graph.
• Node label independence: changing the node labels
does not change this invariant.
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
5
1
Results
3
Summary
Conclusion
2
4
6
7
• The characteristic path length of this graph is still 1.9.
8/36
Measuring assortativity
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
• The degree of a node is how many edges touch a
node.
Neuroscience
Graph
creation
5
6
Data set
Creation
7
Computational
issues
Results
3
Summary
4
Conclusion
1
• Degree of node 7 is 1, of node 4 is 3.
2
8/36
Measuring assortativity
Motivation
Graph theory
• The degree of a node is how many edges touch a node.
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
• Assortativity refers to how frequently nodes of similar
degree share an edge.
5
6
Data set
Creation
Computational
issues
7
Results
Summary
3
Conclusion
4
1
• Degree of node 7 is 1, of node 4 is 3.
2
8/36
Measuring assortativity
Motivation
Graph theory
Graph invariants
• The Randić index is R(G) =
P
ij∈G
di · dj .
Previous work
Graph applications
Neuroscience
5
6
Graph
creation
7
Data set
Creation
Computational
issues
3
Results
4
Summary
Conclusion
1
2
• Degree of node 7 is 1, of node 4 is 3.
• R(G) = 2·2+2·3+2·3+3·3+3·3+3·2+3·2+3·1 = 49.
8/36
Measuring assortativity
Motivation
Graph theory
• The Randić index is R(G) =
Graph invariants
Previous work
P
ij∈G
di · dj .
• R(G) has node label independence.
Graph applications
Neuroscience
Graph
creation
5
1
Data set
3
Creation
Computational
issues
Results
2
Summary
6
Conclusion
4
• The Randić index is still 49.
7
9/36
G1
Motivation
G2
5
5
6
6
7
Graph theory
7
Graph invariants
Previous work
Graph applications
3
Neuroscience
3
4
4
Graph
creation
Data set
Creation
1
2
1
2
Computational
issues
Results
Degree sequence: d = (2, 2, 3, 3, 3, 2, 1)
Summary
Conclusion
I NVARIANT
C LUSTERING COEFFICIENT
G1
0
G2
0.57
C HARACTERISTIC PATH LENGTH
1.9
2.0
F IEDLER VALUE
0.61
0.34
0.37
0.17
49
49
NORMALIZED
F IEDLER VALUE
R ANDI Ć INDEX
10/36
Graph theory literature
Motivation
Graph theory
• Euler (1735): Bridges of Königsburg
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
• Applications include delivery (e.g., FedEX, UPS), air
transport (e.g., crew scheduling), telecommunications
(e.g., cell phone tower placement), and manufacturing
(e.g., circuit board design)
Creation
Computational
issues
Results
• Personal applications: Google/Yahoo/Apple maps and
directions, Facebook friend networks
Summary
Conclusion
• Some relevant literature for graph invariants:
• Fiedler (1973): Algebraic connectivity of graphs
• Randić (1976): Characterization of molecular
branching.
• Watts and Strogatz (1998): Collective dynamics of
‘small-world’ networks
• A couple of surveys: Kincaid & Phillips (2011),
Chandrasekaran, Parrilo, and Willsky (2012)
11/36
Graphs and neuroscience
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• Alzheimer’s disease (e.g., He et al., 2008; Yao et al., 2010;
Lo et al., 2010, Shu et al., 2012, Tijms, et al., 2014)
• Schizophrenia (e.g., Bassett et al., 2008; Zhang et al., 2012;
Skudlarski et al., 2010; van den Heuvel et al., 2010; Wang et
al., 2012)
• Multiple Sclerosis (Shu et al., 2011; Li et al., 2012; Gamboa
et al., 2013)
• Epilepsy (Wilke et al., 2011; van Diessen et al., 2013;
Bonilha et al., 2014; Gong et al., 2014; Wang et al., 2014)
• Autism–spectrum disorders (e.g., Dennis et al., 2011; Keown
et al., 2013; Maximo et al., 2013; Redcay et al., 2013; You et
al., 2013)
• Normal Aging (Micheloyannis, et al., 2009; Zhu et al., 2012;
Petti et al., 2013; Fisher et al., 2014)
• Development during infancy and childhood (Alexander-Bloch
et al., 2013; Huang et al., 2013; Chen et al., 2013; Batalle et
al., 2013; Duan et al., 2014)
12/36
Imaging modalities
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• Structural MRI (cortical thickness, volume)
• Diffusion Tensor Imaging (DTI)
• Resting state functional MRI (rsfMRI)
• Task–related fMRI
• SPECT cerebral blood flow
• FDG–PET (brain metabolism)
• EEG
• MEG
13/36
AD results, various modalities
Motivation
Graph theory
Authors
Previous work
Graph applications
Neuroscience
Graph
creation
Summary
Conclusion
L
El
CC
HC 30, AD 25
&
%
n.s.
n.s.
Bai et al. (‘12)
HC 30, aMCI 38
&
%
n.s.
n.s.
Shu et al. (‘12)
HC 36, aMCI 28
&
%
–
–
n.s.
–
&
–
–
&
%
n.s.
–
–
%
–
–
n.s.
%
&
rsfMRI
Creation
Results
Eg
Lo et al. (‘10)
Data set
Computational
issues
Population
DTI
Graph invariants
Superkar et al., (‘08)
Sanz-Arigita et al., (‘10)
Zhao et al., (‘12)
Brier et al., (‘14)
HC18, AD 21
HC 21, AD 18
HC 20 , AD 33
HC 205, CDR>0 121
Stam et al., 2007
HC 13, AD 15
–
%
–
n.s.
De Haan et al., 2009
Stam et al., 2009
HC 23, AD 20
HC 18, AD 18
–
–
–
%
–
–
–
&
EEG
%means increasing with more AD, &means decreasing with more AD
n.s.means not significant, – means not measured
14/36
Structural MRI results for AD
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Study
Population
L
CC
graphs for each diagnostic category
He et al., 2008
HC 97, AD 92
%
%
Yao et al., 2010
HC 98, MCI 113
%
%
MCI-s 37, MCI-p 39, AD 37
–
&
HC 38, AD 38
&
&
Creation
graphs for each subject
Computational
issues
Results
Li et al., 2012
Summary
Conclusion
Tijms et al., 2013
%means increasing with more AD,
&means decreasing with more AD
n.s.means not significant,
– means not measured
15/36
ADNI Data set
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• Structural MRI on a 1.5 T magnet
• Cortical Thickness (CT) measurements calculated
across 68 regions via FreeSurfer.
15/36
ADNI Data set
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• Structural MRI on a 1.5 T magnet
• Cortical Thickness (CT) measurements calculated
across 68 regions via FreeSurfer.
• Four diagnostic categories:
• Normal (N): Cognitively normal at baseline and
remained for at least 3 years
• Mild cognitive impairment–stable (MCI-S): Diagnosed
MCI at baseline and remained so for at least 3 years
• Mild cognitive impairment–progressive (MCI-P):
Diagnosed MCI at baseline, progressed to dementia
within 3 years
• Dementia (D): Diagnosed dementia due to AD at
baseline
• 86 excluded for not satisfying this categorization
16/36
ADNI Data set
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
n
age
mean (sd):
range:
F/M:
N
127
MCI-S
104
MCI-P
106
D
108
80.8 (4.8)
65–96
63/64
79.7 (7.9)
61–94
37/67
79.5 (6.9)
61–94
47/61
80.0 (7.7)
63–97
57/49
16/36
ADNI Data set
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
n
age
mean (sd):
range:
F/M:
N
127
MCI-S
104
MCI-P
106
D
108
80.8 (4.8)
65–96
63/64
79.7 (7.9)
61–94
37/67
79.5 (6.9)
61–94
47/61
80.0 (7.7)
63–97
57/49
Regression used to remove the effect of age, gender, and
education level from each CT measurement.
17/36
Graph creation
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• One graph per diagnostic group
17/36
Graph creation
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• One graph per diagnostic group
• Correlations between CT in each region (two choices)
• Gong, et. al. (2012): Roughly half of significant positive correlations
correspond to fiber tracks
• Partial versus Pearson’s
17/36
Graph creation
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• One graph per diagnostic group
• Correlations between CT in each region (two choices)
• Gong, et. al. (2012): Roughly half of significant positive correlations
correspond to fiber tracks
• Partial versus Pearson’s
• Edge between regions if correlation significant (three choices)
• False discovery rate with q = .05
• All correlations, positives only, negatives only
17/36
Graph creation
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• One graph per diagnostic group
• Correlations between CT in each region (two choices)
• Gong, et. al. (2012): Roughly half of significant positive correlations
correspond to fiber tracks
• Partial versus Pearson’s
• Edge between regions if correlation significant (three choices)
• False discovery rate with q = .05
• All correlations, positives only, negatives only
• Edge weighting (four choices)
• None used, i.e., 1 or 0
• Correlation coefficient
• CT weighting: weight scaled by
(scaled mean CT in i) · (scaled mean CT in j)
17/36
Graph creation
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• One graph per diagnostic group
• Correlations between CT in each region (two choices)
• Gong, et. al. (2012): Roughly half of significant positive correlations
correspond to fiber tracks
• Partial versus Pearson’s
• Edge between regions if correlation significant (three choices)
• False discovery rate with q = .05
• All correlations, positives only, negatives only
• Edge weighting (four choices)
• None used, i.e., 1 or 0
• Correlation coefficient
• CT weighting: weight scaled by
(scaled mean CT in i) · (scaled mean CT in j)
• Graph density
• Only a fixed percentage of possible edges used
• Largest connected subgraph measured
17/36
Graph creation
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• One graph per diagnostic group
• Correlations between CT in each region (two choices)
• Gong, et. al. (2012): Roughly half of significant positive correlations
correspond to fiber tracks
• Partial versus Pearson’s
• Edge between regions if correlation significant (three choices)
• False discovery rate with q = .05
• All correlations, positives only, negatives only
• Edge weighting (four choices)
• None used, i.e., 1 or 0
• Correlation coefficient
• CT weighting: weight scaled by
(scaled mean CT in i) · (scaled mean CT in j)
• Graph density
• Only a fixed percentage of possible edges used
• Largest connected subgraph measured
Twenty four possible graphs for each diagnostic group and sparsity level.
He, et al., and Yao, et al. reported using one of the possible methods.
18/36
Partials, permutations
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• Partial correlations:
• For two given regions, correlations can be affected by other regions.
• Partial correlations can be used to remove the effects of other regions.
• For this data, ≈ 2200 regressions required!
18/36
Partials, permutations
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
• Partial correlations:
• For two given regions, correlations can be affected by other regions.
• Partial correlations can be used to remove the effects of other regions.
• For this data, ≈ 2200 regressions required!
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• Permutation testing:
• Tests the significance of pairwise difference in measures.
• In theory: if permutating the group categorization for each subject
results in a smaller difference, than success, else failure.
• (# failures)/(# permutations) = significance level.
• Too many permutations! Used 10, 000 random permutations.
• 10, 000 · 2200 = 22, 000, 000 regressions! Each partial correlation
run required ≈ 24 hours on 20 processors working in parallel.
Thanks to HPC at USNA!
19/36
Edge density
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
• He et al. and Yao et al. both control the number of edges by only keeping a
fixed percentage of possible edges.
• Both studies focused on lower edge density levels.
• We examined density up to the highest percentage found by FDR.
• Ordinary correlations: ≈ 80%, Partial correlations ≈ 9%.
20/36
Fiedler value, partial correlations
Motivation
Graph theory
N
MCI–s
D
MCI–p
Graph invariants
Graph applications
Neuroscience
Graph
creation
Data set
Creation
not CT-weighted
Previous work
0.20
0.60
0.40
0.10
0.20
Computational
issues
Results
CT-weighted
Summary
Conclusion
0.15
0.06
0.10
0.04
0.05
0.02
4
5
6
7
not r -weighted
8
9
4
5
6
7
r -weighted
8
9
21/36
Fiedler value, ordinary correlations
Motivation
N
MCI–s
MCI–p
D
Graph theory
Graph applications
Neuroscience
Graph
creation
Data set
not CT-weighted
Graph invariants
Previous work
1.5
4.0
1.0
*
2.0
0.5
*
Creation
0.0
Computational
issues
0.0
Results
CT-weighted
Conclusion
0.6
1.5
Summary
0.4
1.0
#
0.5
0.2
#
0.0
0.0
7
17
40 47 55 62 70
7
not r -weighted
17
40 47 55 62 70
r -weighted
#p < 0.1
*p ≤ 0.05
22/36
Normalized Fiedler, partial correlations
Motivation
N
MCI–s
D
MCI–p
Graph theory
Graph invariants
0.2
Graph applications
Neuroscience
Graph
creation
Data set
not CT-weighted
0.2
Previous work
#
#
0.1
0.1
0.0
0.0
0.2
0.2
0.1
0.1
Creation
Computational
issues
Results
CT-weighted
Summary
Conclusion
0.0
0.0
4
5
6
7
8
9
4
not r -weighted
5
6
7
r -weighted
#p < 0.1
8
9
23/36
Normalized Fiedler, ordinary correlations
Motivation
N
Graph theory
Graph applications
Neuroscience
Graph
creation
Data set
not CT-weighted
Graph invariants
Previous work
MCI–s
D
MCI–p
1.0
1.0
*
0.8
*
*
*
*
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
Creation
Computational
issues
Summary
*
0.8
CT-weighted
Conclusion
1.0
1.0
Results
*
*
#
*
*
#
#
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
7
17
40 47 55 62 70
7
not r -weighted
17
40 47 55 62 70
r -weighted
#p < 0.1
*p ≤ 0.05
24/36
Clustering coef., partial correlations
Motivation
N
MCI–s
D
MCI–p
Graph theory
Graph invariants
Graph applications
Neuroscience
Graph
creation
Data set
not CT-weighted
*
Previous work
**
**
*
#
*
*
**
**
*
*
*
0.4
0.4
0.2
0.2
0.0
0.0
Creation
Computational
issues
Results
Summary
0.4
CT-weighted
Conclusion
*
*
**
*
#
#
*
**
**
#
#
*
0.2
0.4
0.2
0.0
0.0
4
5
6
7
8
9
not r -weighted
#p < 0.1
4
5
6
7
r -weighted
*p ≤ 0.05
**p ≤ 0.01
8
9
25/36
Clustering coef., ordinary correlations
Motivation
N
MCI–s
MCI–p
D
Graph theory
Graph invariants
Graph applications
Neuroscience
Graph
creation
Data set
not CT-weighted
1.0
Previous work
Creation
***
*
*
*
**
*
*
*
** ***
1.0
0.8
0.8
0.6
0.6
0.4
0.4
Computational
issues
Results
1.0
CT-weighted
Summary
Conclusion
***
*
**
*
*
***
*
*
** **
1.0
0.8
0.8
0.6
0.6
0.4
0.4
7
17
40 47 55 62 70
#p < 0.1
7
40 47 55 62 70
17
r -weighted
not r -weighted
*p ≤ 0.05
**p ≤ 0.01
26/36
Char. path length, partial correlations
Motivation
N
Graph theory
MCI–s
D
MCI–p
4.0
Graph invariants
10.0
Graph applications
Neuroscience
Graph
creation
Data set
not CT-weighted
#
Previous work
Creation
***
3.5
*
*
*
8.0
3.0
6.0
2.5
Computational
issues
Results
12.0
CT-weighted
Summary
Conclusion
20.0
#
10.0
15.0
8.0
4
5
6
7
8
9
4
not r -weighted
#p < 0.1
*p ≤ 0.05
5
6
7
r -weighted
**p ≤ 0.01
***p ≤ 0.005
8
9
27/36
Char. path length, ordinary correlations
Motivation
N
Graph theory
Graph applications
Neuroscience
Graph
creation
Data set
not CT-weighted
Graph invariants
Previous work
MCI–s
D
MCI–p
5.0
3.0
***
*
*
**
2.5
***
#
*
*
*
**
4.0
2.0
1.5
3.0
Creation
Computational
issues
10.0
Results
***
Conclusion
CT-weighted
Summary
**
*
#
** *** ***
***
#
14.0
8.0
12.0
6.0
10.0
4.0
8.0
7
17
40 47 55 62 70
7
not r -weighted
#p < 0.1
*p ≤ 0.05
17
40 47 55 62 70
r -weighted
**p ≤ 0.01
***p ≤ 0.005
28/36
Assortativity, partial correlations
Motivation
N
MCI–s
D
MCI–p
Graph theory
***
Graph applications
Neuroscience
Graph
creation
Data set
***
***
***
***
#
#
#
6
7
8
#
***
***
***
***
***
#
*
*
*
*
*
*
4
5
6
7
8
9
not CT-weighted
Graph invariants
Previous work
Creation
Computational
issues
Results
Conclusion
CT-weighted
Summary
4
5
9
not r -weighted
#p < 0.1
*p ≤ 0.05
r -weighted
**p ≤ 0.01
***p ≤ 0.005
29/36
Assortativity, ordinary correlations
Motivation
N
MCI–s
MCI–p
Graph theory
*
Graph applications
Neuroscience
Graph
creation
Data set
*
*** *** *** ** ***
not CT-weighted
Graph invariants
Previous work
***
**
***
**
7
17
Creation
Computational
issues
Results
*** ***
*** *** *** *** ***
Conclusion
CT-weighted
Summary
7
40 47 55 62 70
17
not r -weighted
#p < 0.1
*p ≤ 0.05
40 47 55 62 70
r -weighted
**p ≤ 0.01
***p ≤ 0.005
D
30/36
Discriminating N and MCI–s
Motivation
Graph theory
Assortativity, ordinary correlations, binary edges
Graph invariants
N
MCI–s
Previous work
Graph applications
***
Neuroscience
Graph
creation
*
Data set
Creation
Computational
issues
**
Results
***
Summary
***
Conclusion
*
7
#p < 0.1
#
17
40
*p ≤ 0.05
47
55
**p ≤ 0.01
62
70
***p ≤ 0.005
31/36
Discriminating N and MCI–s
Motivation
Graph theory
Clustering coefficient, ordinary correlations, binary edges
Graph invariants
MCI–s
Previous work
Graph applications
Neuroscience
**
Graph
creation
*
Data set
Creation
*
Computational
issues
Results
#
#
Summary
Conclusion
*
7
#p < 0.1
17
40
*p ≤ 0.05
47
55
**p ≤ 0.01
62
70
***p ≤ 0.005
N
32/36
Discriminating MCI–s and MCI–p
Motivation
Assortativity, partial correlations
Graph theory
MCI–s
Graph invariants
MCI–p
Previous work
Graph applications
*
*
Neuroscience
Graph
creation
*
*
Data set
*
Creation
**
Computational
issues
*
*
Results
*
Summary
*
Conclusion
*
*
4
5
6
7
8
9
4
not r -weighted
5
6
7
r -weighted
*p ≤ 0.05
**p ≤ 0.01
8
9
33/36
Discriminating MCI–s and MCI–p
Motivation
Char. path length, ordinary correlations
Graph theory
MCI–s
Graph invariants
MCI–p
Previous work
Graph applications
Neuroscience
Graph
creation
*
Data set
#
Creation
Computational
issues
*
***
#
Results
*
*
*
62
70
78
#
*
Summary
Conclusion
*
*
40
47
55
62
70
78
40
not r -weighted
47
55
r -weighted
*p ≤ 0.05
**p ≤ 0.01
34/36
Method of graph creation matters!
Motivation
Graph theory
•
Partial vs. ordinary Pearson correlation
• E.g., assortativity and clustering increase (partial) or
Graph invariants
decrease (ordinary) with symptom severity
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
•
CT-weighted vs. unweighted
• Path length has better group discrimination with
CT-weighted edges (ordinary graphs) than binary or
r-weighted edges.
• For assortativity, better group discrimination without CT
weighting (partial graphs).
34/36
Method of graph creation matters!
Motivation
Graph theory
•
Partial vs. ordinary Pearson correlation
• E.g., assortativity and clustering increase (partial) or
Graph invariants
decrease (ordinary) with symptom severity
Previous work
Graph applications
Neuroscience
Graph
creation
Data set
Creation
Computational
issues
Results
Summary
Conclusion
•
CT-weighted vs. unweighted
• Path length has better group discrimination with
CT-weighted edges (ordinary graphs) than binary or
r-weighted edges.
• For assortativity, better group discrimination without CT
weighting (partial graphs).
Findings caution against directly interpreting changes
in network properties in terms of underlying neurobiological processes.
35/36
Motivation
Graph theory
Graph invariants
Previous work
Graph applications
Symptom sensitivity of graph measures.
• Fiedler and normalized Fiedler values discriminate
poorly across groups.
Neuroscience
• Suggests that global network connectivity does not
Graph
creation
change significantly with AD.
Data set
Creation
Computational
issues
Results
Summary
Conclusion
•
Assortativity, clustering, and path length change
across the AD spectrum, with assortativity being
most sensitive to diagnostic category.
• Suggests that the degree to which brain regions with
similar connectivity connect to each other progressively
changes with AD.
36/36
Conclusions
Motivation
•
Graph theory
Graph invariants
Previous work
Some graph measures may be useful for
predicting disease progression and/or symptom
development.
• Relax node label independence?
Graph applications
Neuroscience
Graph
creation
•
Data set
Creation
More research needed to understand biological
basis of changes in graph measures.
• Compare graph measures across imaging modalities in
Computational
issues
the same subjects.
Results
Summary
Conclusion
•
Need to develop methods to create
individual-subject networks using sMRI.
•
Need better methods for comparing graph
measures across groups.
• Is density correction the right thing?
• What level of density is best?