Structural Stability in Networks

1
Structural Stability in Networks

• John Van Hemert
• April 2007

2
Summary

• Literature Prill et. al.
• Recreate results
• Apply GeneNetSim software
• Discrete
• Continuous
• Expand from Prill et. al.

3
Literature

• Dynamic Properties of Network Motifs Contribute
  to Biological Network Organization. Prill RJ,
  Iglesias PA, Levchenko A. PLoS Biology Vol. 3,
  No. 11, e343 doi10.1371/journal.pbio.0030343
• Is there something about inherent structure that
  guides Natures design of biological networks?

4
Literature SSS
Prill et al. PLoS Biology 311
5
Literature edges

• Motifs with more edges would obviously be easier
  to find
• But some motifs do not follow the trend

6
Recreate results

• Maple script
• For each of the 13 motifs
• Create a Jacobian matrix with XU(-1,0) on
  diagonal and XU(-1,1) for existing edges
• Use eigenvalues to determine stability
• Repeat 100 times
• View distribution of probability of stability

7
Probability of Stability
8
GeneNetSim

• Linear gene network model
• Simulate random network
• Hierarchical parameters
• number of level n clusters, connectivity between
  level n cluster nodes,number of nodes at lowest
  level,connectivity between nodes at lowest level
• Ie, 2,0,3,0.2,4,1
• 2 clusters with minimum connectivity, within each
  of those are 3 clusters with 20 max
  connectivity, and within each of those are 4
  nodes connected at maximum

9
Example heirarchical cluster parameters
10
Apply GeneNetSim force equilibrium

• Discrete model
• Equilibrium is when a nodes value is the sum of
  its incoming values
• Run each trial 1000 times with parameters
• 3,0
• 3,1
• 2,0,3,0,4,0
• 2,0,3,3,4,4
• Use Maple script from before on Jacobian matrices
• Only 3,1 had nonzero number of stable equilibria
• Only looked at one motif the fully connected one
• 12/1000 were stable
• Why different from before?
• not all have only negatives on diagonal

11
Discrete 3,1
12
Apply GeneNetSim force equilibrium

• Continuous model
• Equilibrium is when for each node, Zero is the
  sum of its incoming values
• Run each trial 1000 times with parameters
• 3,0
• 3,1
• 2,0,3,0,4,0
• 2,0,3,3,4,4
• Use Maple script from before on Jacobian matrices
• Only 3,1 had nonzero number of stable equilibria
• Only looked at one motif the fully connected one
• 321/1000 were stable using leading eigenvalue
• 261/1000 were stable using only real eigenvalues
• Why different from before?
• diagonals (and others?) are not restricted to
  (-1,0)

13
Continuous 3,1
14
Expand Combine 3-node motifs

• Of the 13 3-node motifs, look at every possible
  pair
• Combine the two subgraphs with one link
• Try all links from one subgraph to the other
• Calculate SSS for each using only real
  eigenvalues (like Prill et al does)
• Is the SSS interesting?

15
Example
1
4
3
6
2
Still stable? No longer stable?
5
A combination matrix
Number of stable J-matrices when motifs 4 (left)
and 5 (right) are combined by an edge from row to
column.
16
Combo Pairs
Any NonZero values in combination matrix?
17
Results

• Always stable Always stable Always stable
• Depending on where the link is, combinations can
  have higher SSS than subgraphs or lower SSS than
  subgraphs
• Some result in 0/100 stable trials no matter
  where link is placed

18
Future work

• Implement Maple scripts in C so that these and
  more complex examples can be computed much faster
  (this is very parallelizable)
• Investigate exactly how linking two motifs
  affects SSS
• Use GeneNetSim and MFinder1.1 (U. Alon group,
  www.weizmann.ac.il/mcb/UriAlon) for randomization
  test of motif distribution in networks.

19
End

• Thanks, Dr. Proulx and classmates.
• http//www.public.iastate.edu/jlv