Graph Theory

Question 1

Different kind of measurments of the “complexe brain”

name, what, how many regions, what is the outcome

Answer 1

univariate measures - magnitude, power - single region – process
bivariate measures- functional connectivity - two region – interaction
multivariate measures - network analysis - multi regions – patterns

Question 2

How can we quantitatively asses patterns

Answer 2

(branch of mathematics) Graph theory and (sub-brach) Complex Network Theory

Question 3

composition of graphs

Answer 3

• nodes and edges.

- degree of node is equal to the number of edges

Question 4

relation of graphs and matrices

Answer 4

• graphs or networks can be represented by matrices
• graph/network is represented bei columns and rows of the matrix
• connection between two nodes i and j is represented by matrix element (i,j)
• binary networks: matrix elements are either 0 (no connection exists) or 1 ( connection exists)
• weighted networks: matrix elements are continuous values and can range from strong to weak

Question 5

complex network theory in neuroimaging

Answer 5

= modeling endeavor that provides a set of representational rules to describe the brain subcomponets(regions) and their relationship (white matter tracts/ functional connection)
- brain networks can be constructed in two ways: (1) structural connectivity (Diffusion Tractography (2) functional connectivity (fMRI, EEG, MEG)

Question 6

Preprocessing during network modeling and its problems

Answer 6

• lack of golden standard
• very sensitive and will reflect any change of network modeling in the topology
• comparability should be given with same modeling strategy
• not too severe for simple preprocessing (but regression of covariates and frequency range of bandbass filter)
• from Paper Neurosci. 2015: Evaluating the reliability….
  … a “ broad” frequency range results in more robust networks
  …order of preprocessing steps may impact results
  …. motion correction pipelines may impact results

Question 7

approaches to defining nodes

Answer 7

cytoarchtecture
probabilistic
chemoarchitecture
anatomcal
random
functional
data-driven
voxel-based
myeloarchitecture
multimodal (Glasser, Nature, 2016)

Question 8

building the graph: absolute vs. proportional weight threshold

Answer 8

applying an absolute or proportional threshold solves the problem of weak links (association matrix –> adjacency matrix)

absolute: everything < thr = 0
proportional: % of highest correlations are considered as links

Question 9

binary s. weigthed, directed vs. undirected networks and combinations

Answer 9

undirected networks: symmetrical outcome; datasets from: diffusion MRI, structural MRI, functional
directed networks: unsymmetrical, datasets from: tract tracing, inference of causality from functional data

undirected binary
undirected weighted
directed binary
undirected weighted

Question 10

problems with edge definition

Answer 10

1. thresholding can modify the network structure (low connectivity –> gain of links, high connectivity –> loss of links)
2. thresholding can easily result in different stages of “ network fragmentation” for each subject and thus in different numbers of nodes forming the network
3. low edge thresholds may result in “uniformity” of the networks –> no possibility to detect differences between subjects
4. results may uniquely dependent on the precise threshold value

–> one solution: construct each individual network over a range of thresholds

Question 11

Classification of network measurements

Answer 11

(can be characterized on a local basis and a global basis)

functional segregation
functional integration
centrality
resilience

Question 12

What is the clustering coefficient

Answer 12

• measure of local connectedness ( how many nearest neighbors of node i are connected to each other as well)
• all connected nodes with one node are neighbors

C(A)= 4/10=0.4 [meaning that 40% of connections between neighbors exist]

Question 13

on what are simple measures of segregation based ?

Answer 13

number of triangles in the network (high number of triangles imply segregation)

• -> describe presence of densely interconnected groups
• -> more sophisticated measures can also find the exact size and composition of these individual groups

Question 14

what is functional segregation in the brain

Answer 14

• the ability for specialized processing of densly interconnected goups of brain regions
• quantifies the presence of such groups (clusters or modules) within the network
• suggests an organization of brain regions indicative of segregated neural processing

Question 15

relations between segregation and (A) getting older (B) memory performance

Answer 15

A) the older you get the stronger your executie functions become, this is directly related to increased modularity

B) the ability to reconfigure your network lead to greater memory performance

Question 16

Modular structure

Answer 16

subdivining the network into groups of nodes, with a amximally possible number of withingroup links, and minimally possible number of between-group links

Question 17

What is integration of measured brain networks

Answer 17

• ability to rapidly combine specialized infomation from distributed regions
• commonly based on the concept of a path (sequences of links between distinct nodes)
• represent potential routse of information flow between pairs of brain regions

(IQ Study: n= 20: The shorter the path lengths the higher the IQ; replicated study: n=1096 with different thresholds -> not the same findings)

Question 18

what does small-worldness mean in this context of network measuring?

Answer 18

= optimal balane of functional integration and segregation

• compared to social network: high clustering ( different groups of people in the network are organized in cliquish manner) and a relatively short distance to travel from one person to another person (short pathlength) –> small worldnes = high clustering and low pathlenght –> not in schizophrenia (random; low clustering an pathlength)
• brain is also organized as “ small-world”
• -> high global and local effiency of parallel infomation processing faciliates rapid adaptive reconfiguration of neuronal assamblies in support of changing cognitive states

Question 19

measurements of centrality

Answer 19

degree
betweenness
connection between modules

Question 20

hypothesis of the “rich-club”

Answer 20

“rich people only hang out with rich people”–> overlaping of the hubs
the rich club may serve to link different functionsl modules in the brain, through partial overlap with several resting-state network

Question 21

What are hubs?

Answer 21

important brain regions

• 6 % of all connections connect 90% of the brain
• highest edge betweenness centrality
• highest functional connecivity (FC)
• above avarage FC distance
• organize in a “rich-club”

Question 22

meaning of centrality?

Answer 22

important brain regions (hubs) often interact with many other regions and facilitate functional integration

Question 23

What is the concept of degeneracy?

Answer 23

• capacity of structurally distinct elements of a system to carry out the same
function
• describes the ability of distinct neuronal systems to make overlapping
contributions to the same output, offering both functional adaptability and
robustness to damage

Question 24

resilience of the brain

Answer 24

• Anatomical brain connectivity influences the capacity of neuropathological
lesions to affect functional brain activity
e.g.: The extent of functional dysfunction is heavily determined by the
affected anatomical region in a stroke

Question 25

graph model and statistics

Answer 25

?????? help

I ll give it a try on slides 28 and 29, but not very deep and no guarantees (jurena)

Question 26

what not to do with network measures

Answer 26

Comparing Brain networks of different size and connectivity density using graph theory

Question 27

pipeline of network measurement

Answer 27

[not sure if it is right]

1. Data
2. Preprocessing
3. defining nodes (segmentation?)
4. defining edges
5. building the graph
6. network analysis [functional segregation, functional integration, centrality, resilience]

Question 28

building a graph (or did that come up before???)

Answer 28

define nodes 
(intrinsic consistency, extrinsic differentiation and spatially constrained) also don t take in too many: ~25% of connections and ~200 nodes are sufficient to not loose granularity

Define or rather find edges
by statistical dependence measure (which node correlates with whom i guess) but be aware of the correlation coeeficient problem-> only because there is correlation you don t know the direction of the info flow -> take partial correlations

build the graph by applying threshold
-> get rid of weak links for sparsity (more next slide)

Question 29

NBS network based statistics

what to do?

Answer 29

use a matrix to compare every region/voxel with every other region
-> many comparisons (200 x 200 = 19 000 comparisons - yes, that makes sense because you need every comparison only once and no need to find connection between a region and itself..) -> low power
control significance level /threshold for every comparison
-> threshold oc effects sensitivity
- liberal p < .05 also weak links are taken into the graph
- conservative p < .001 only strong links
then for all links look at the size(nr of nodes), extend (nr of connections) and intensity (i guess how it relates to the rest of the network)

anything missing / does that make sense?