week 2 - graph theory

Question 1

what is network topology?

Answer 1

properties of a network that are unrelated to where the nodes are in space, instead it just matters the connection between nodes and edges

Question 2

What are the different types of graph theoretical measures?

Answer 2

Local - properties of each node e.g degree

Meso-scale - properties of sub networks

Global network properties

Question 3

what is edge density?

Answer 3

num actual edges/ num possible edges

Question 4

how to calculate number of possible edges?

Answer 4

N*(N-1)/2
(removes the diagonal and the lower triangular part)

Question 5

what are components in the network?

Answer 5

sets of nodes that aren’t connected to any other components

Question 6

what is average edge weight?

Answer 6

the average of existing edges within the upper triangular

Question 7

what is degree?

Answer 7

number of edges connecting a node

Question 8

what is the strength of a node?

Answer 8

sum of edge weights connected to a node?

Question 9

What is eigenvector centrality?

Answer 9

To what extent is the node connected to other nodes with high eigenvector centrality

Its basically a measure of how important the nodes you are connected to are

Question 10

what is path length?

Answer 10

the number of edges between two nodes

Question 11

what is characteristic path length?

Answer 11

the average path length between all node pairs

Question 12

what is global efficiency?

Answer 12

average inverse shortest path length

Question 13

what is betweenness centrality?

Answer 13

the fraction of shortest paths in an entire network that pass through a given node

Question 14

describe the spectrum of communication processes

Answer 14

from diffusion to routing

In diffusion, the structure of the network is unknown and the walk through the network is random

In routing you take the shortest path, this requires full knowledge of the network topology

Question 15

what is the clustering coefficient?

Answer 15

the fraction of a nodes neighbours that are connected to each other

Question 16

what is the global clustering coefficient?

Answer 16

the average nodal clustering coefficient

Question 17

what is transitivity?

Answer 17

the ratio of triangles to triplets in the network

or the overall probability that two nodes connected to a common node are also connected to each other across the entire network.
It’s basically the global version of the clustering coefficient.

Question 18

what is a module? what about in fmri

Answer 18

subnetworks of nodes that are very densely connected with each other and sparesly connected to other modules

in fMRI this is the resting state networks e.g DMN, DAN

Question 19

what are hub nodes?

Answer 19

Important nodes that are very highly connected

connector hubs = connections with nodes in different communities

Provincial hubs = have many connections with different communities

Question 20

what is rich club?

Answer 20

A set of nodes that have high degree that are highly connected to each other

rich clubs tend to integrate modules together

Question 21

what is segregation? and measures related

Answer 21

locally, densely connected clusters of nodes

clustering, transivity and modularity

Question 22

what is integration? and measures related

Answer 22

the ability of any node to be reached in a few steps

path length, efficiency, rich club

Question 23

what is a small world network?

Describe the Watts-Strogatz model

Answer 23

A network that has both high integration and high segregation

For example the Watts-Strogatz model, that is used to generate small world networks. You start with a regular lattice, and then you rewire some of the edges with a given probability to introduce shortcuts, this makes the efficiency higher

Question 24

describe the inter-dependence of network features

Answer 24

• density and average degree are positively correlated
• density and path length are negatively correlated
• density and transivity are positively correlated
• This shows that edge density is a confound

Question 25

what is wrong with proportional thresholding when binarizing networks based on correlation coefficient?

Answer 25

When you build functional connectivity networks, you often threshold them (e.g., keep only the top X% of edges). If a subject (or a group) has higher mean functional connectivity (i.e., stronger average correlations across the brain), more edges will survive thresholding — or the edges will be stronger overall. Stronger connections → shorter paths between nodes. Shorter paths → higher global efficiency. So even if the underlying network structure (wiring, organisation) hasn't changed, just having higher average correlations will mechanically lead to higher global efficiency.

Question 26

describe Null models in network neuroscience

Answer 26

you take a feature of interest, e.g average clustering you then build a null network where you preserve some attributes (e.g edge density) and then randomise others you then recalculate the network p number of times to create a Null distribution. You can then benchmark your empirical network against the null distribution If your empirical network falls in the middle of the null distribution, that would suggest that the clustering arises simply as a byproduct of the edge density. If it falls outside of the distribution then the clustering isn't simply due to the edge density, it is actually related to the topology

Question 27

what is the small world index?

Answer 27

the ratio of clustering to path length

Question 28

What is the rich club coefficient?

Answer 28

the tendency for high degree nodes to be highly connected to each other In other words: Are the most highly connected nodes more densely interconnected among themselves than would be expected by chance?

Question 29

how are rich club coefficient and small world index normalized?

Answer 29

They are normalized to random networks, according to a null model

Question 30

strengths of graph theory

Answer 30

it is independent of spatial and temporal scales, species and imaging modalities It is a good way to reduce the dimensionality of data, as N nodes is often much less than N voxels