QUANTITATIVE LAWS June 13 -June 24

1. Technion
Israel Institute of Technology
Exploratory Adaptation in
Large Random Networks
Hallel Schreier and Naama Brenner
Network Biology Research Laboratories, Technion-IIT
Quantitative Laws II
Lake Como
June 2016

2. ⋅ 𝐽𝑖𝑗 (𝑡)𝑤𝑖𝑗 = 𝑇𝑖𝑗 𝑤𝑖𝑗𝜑 is an Element-wise
Sigmoidal Function
( )i ij j i
j
x w x x 
𝑦
Time
Time
Time
Microscopic
Dynamical
Network
( )x W x x 
r r r&
( ) ( )W t T J t o
Macroscopic
Phenotype
Global Demand
Mismatch
(Stress)
M
Exploration
Exploratory Adaptation Model
System
Reorganizes
to a Stable
State
Convergence
Stably
Satisfied
Random
walk in
Relaxes
ijJ
ix
M > 0
𝑦 = 𝑥 ⋅ 𝑏
𝑦∗
± 𝜀
𝒚∗
𝑀 𝑦 − 𝑦∗
> 0
𝑦∗
𝑦 ≈ 𝑦∗
𝑀 𝑦 − 𝑦∗
≈ 0
𝑴 ≈ 𝟎
Interactions Dissipation
Topological
Backbone
(Adjacency
matrix)
Interactions
Strengths ib
Random Walk in
interactions
strengths ijJ
?
𝑦
≈ 0
Element-wise
Product
-1
0
1
𝜑(𝑥𝑗)
𝑥𝑗
Constant: 0/1

3. 0
0.4
0.8
0 500 1000 1500
ConvergenceFraction
Network size
Some Results
Successful Adaptation Depends on Topology
0.72
0.60
0.14
0.50
0.03
0
0 0.4 0.8
SF SF
SF Exp
Exp SF
SF Binom
Binom SF
Binom Binom
Out In
Network degrees
distributions:
Hubs Play a key Role in Adaptation Process
0
0.4
0.8
0 2 4 6 8 10
ConvergenceFraction
# Deleted
Hub
Random
Node deletion
Take-Home Massages:
• Model demonstrates the feasibility of adaptation by exploration
• Exploratory adaptation strongly depends on network topology
and is most effective for out-going scale-free topology
• Hubs play a key role in adaptation process
0
0.4
0.8
0 200 400 600 800 1000
ConvergenceFraction
Out Degree of Largest Hub
Largest Out-Going Hub
in Network
Check it out on arXiv

4. Thank You!