week 3 - generative models

Question 1

what is generative modelling?

Answer 1

building or randomizing a brain network, with various constraints or rules

Question 2

what is the general pipeline for generative modelling

Answer 2

• question/hyptohesis
• build a model
• Validate the model to empirical data
• Iteritively refine the model to capture previously unexplained variance in empirical data or to capture mor parsiminous explantions of existing data

Question 3

What is the Erdos Renyi random graph model?

Answer 3

For a given number of nodes, add new edges with fixed probability p

(So for each pair of nodes, theres a 0.3 chance that a new edge is made)

The new edges are all independent from each other so the network is random

The degree dsitrbution (which is the distribution of the degree for each node) follows a gaussian distribution

This is not a very realistic model of real networks

Question 4

What is the Barabasi-Albert preferential attachment model

Answer 4

networks based on preferential attachment

This means that more highly connected nodes are more likely to recieve another edge (rich get richer)

You start with a seed network
The probability of connecting a new node to an existing node is the degree of the node/ the sum of the degree of all nodes

You then repeat this until you reach the desired number of nodes

If you iterate this loads and loads of times, you get a log log power law degree distribution between degree and frequency. This is a scale free network, meaning it contains disproportionately well connected hub nodes
Degree distributions of scale free networks follow a power law (straight line on a log log plot). A power law is the only distribution that is the same whatever scale we look at it from

Question 5

What is evidence that brain networsk are scale free?

Answer 5

The degree distribution of functional brain networks follows a power law

This means there are many nodes with only a few links, and a few nodes with many links

Question 6

what else follows a power law in brains?

Answer 6

• distance between nodes and edge weight (makes sense cus proximal nodes are more likely to be connected)
• This can be explained because it is the sweet spot in the cost efficiency trade off. Long connections are metabolically expensive but topographically advantageous

Question 7

what is the spatial constraint for modelling brain networks?

Answer 7

Proximal nodes are more likely to be connected

Question 8

Describe the spatial growth model

Answer 8

spatial growth model

Each node has a spatial coordinate. When you add a new node, the probability of connecting to each existing node is equal to beta (the density, or overall probability of edge formation) multiplied by an exponential decay of the scaled (with alpha) euclidean distance between nodes.

This means that as the distance between nodes increases, the probability of an edge beinig connected decreases exponentially. Alpha determines how fast this happens

If the new node does not establish any connections it is removed

Interestingly, exploring the alpha and beta parameter space generates:
- some random networks
- some scale free networks
- some small world networks
- some virtually unlimited spatial networks

If you fiddle with your parameters enough you can generate something that looks like empirical data in terms of small worldedness, clustering etc. HOWEVER, be careful of overfitting

Question 9

describe the exponential decay with distance model of functional brain networks

Answer 9

the probabiltiy of a new edge is equal to the exponential decay of the negative of the ecludian distance between nodes, multiplied by a distance penalty

Question 10

describe the economical preferential attachment model of functional connectivity

Answer 10

The probablity of a new edge is equal to (the product of the degree of each node, to the power of a preferential attachment parameter) multiplied by the (euclidean distance between each node, to the power of an exponential decay parameter)

This model combines both distance penalty and degree advantage

Question 11

What is the economical clustering model of functional connectivity?

Answer 11

The probability of a new edge is equal to the (number of common neighbours between each node, to the power of a preferential attachment parameter) multiplied by the (distance between each node, to the power of a exponential decay parameter).

This combines a neighbour advantage with a distance penalty. It does a good job of fitting the clustering and modularity of empirical networks

Question 12

which model fits empirical data the best?

Answer 12

The economical clustering model

A study calculated topographies such as clustering, modularity etc. and found hte economical clustering model fitted FC data the best

Question 13

what has empirical graph theory work found about empirical brain networks in schizophrenia?

Answer 13

They show slightly lower clustering and slightly shorter path length

This is equivalent to higher efficiency and a subtle randomization

Question 14

what can applying generative models to structural data from people with schizophrenia tell us?

Answer 14

• It can tell us about the difference in parameters in health and disease
• E.g economomical clustering model found that the optimal model for ppl with schizophrenia showed a lower distance parameter and a higher neighbour advantage

Question 15

go through the step by step of generative modelling

Answer 15

1. You have an initial seed matrix
2. You take the distance between pairs of nodes in the seed matrix and you raise it to the power of eta (distance penalty)
3. You then multiply it by the topological matrix (different types of these, e.g preferential attachment)
4. Remove edges already connected in the multpilied matrix
5. The final matrix gives you the relative probability of adding a new edge for every pair of nodes
6. sample from the final matrix, weighted by probability

Question 16

how can you compare the fit of generative model?

Answer 16

Compute and energy statistic

This combines the Kolmonov-Smirnov statistic for degree, clustering, betweenness and edge length

Question 17

Describe models of network growth in worms

Answer 17

The probability of adding a new edge is equal to the exponential decay of the changes in the distance between two nodes, over time

Created due to modelling worm networks. As the worm grows, the distance between nodes increased

interestingly they found a phase transition, where up until 200 nodes there was a quadratic relationsip between n nodes and n edges, and then suddenly it transitions into a linear relationship. This suggests that as the body of the worm gets longer, the cost of long distance connections is increasingly penalized

You can do this in humans too, with models of fetal gestation. They found that the growth model fits fetal FC development better than the static model

Question 18

5 oveview steps to model brain networks

Answer 18

1. Make a list of stylized facts
2. Come up with the simplist mechanism guiding this structure
3. Fit the model to the data, compare to other models
4. Validate the model on empirical data
5. Think of what the model doesn’t capture

Question 19

What is the degree distribution of most real world networks?

Answer 19

• Most real world networks follow a power law degree distribution, meaning that most nodes have a very low degree, but some nodes have a very high degree and they are the hubs of the neworks

Question 20

name facts about real world brain networks

Answer 20

• They are modular, and the modules are functionally relavent
• All the networks have hubs which tend to be organized in rich clubs
• They are small worlded, and small worldedness matters in terms of information processing
• They are spatially constrained

Question 21

why is small worlded networks useful in the brain?

Answer 21

• it allows for a shorter path length which allows for more efficient information processing

Question 22

How is path lenght related to IQ

Answer 22

• People with high IQ tend to have shorter average path lengths

Question 23

What is the wiring cost path length trade off

Answer 23

Networks are spatially constrained, and longer edges come at a high wiring cost, but they are also useful for decreasing path length

Question 24

Describe the economical clustering model in schizophrenia

Answer 24

• By tuning eta and gamma in the model, they were able to replicate the alterations in clustering commonly seen in schizophrenia
• When optimising the model on empirical data, they found scz patients had slightly different eta and gamma parameters.
• They then found that optimal eta and gamma paramaters actually correlate with polygenic risk scores for scz

Question 25

what do a lot of the generative models not capture?

Answer 25

- Topography of nodes and hubs (where they exist in space, not topology) - Cytoarchitecture - Local gene expression. This could be included by adding a term that includes similarity in gene expression between regions