1. Bioinformatics Center Institute for Chemical Research Kyoto University
九大数理集中講義 Comparison, Analysis, and Control of Biological Networks (1) Scale-free Networks
Tatsuya Akutsu
Bioinformatics Center
Institute for Chemical Research
Kyoto University

2. Contents of Course Scale-free Networks
Transformation of Scale-free Networks
Domain-based Mathematical Models for Protein Evolution
Boolean Networks: Attractor Detection and Control
Probabilistic Boolean Networks
Control of Complex Networks （数理談話会）
Boolean and Flux Balance Analyses of Metabolic Networks
Comparison of Chemical Graphs

3. Contents of Lecture (1) Background Graphs and Networks Small World
Scale-free Network
Models of Scale-free Networks
Preferential Attachment
Deterministic Model
Network Motif

4. Back Ground Systems Biology Network Biology
Understanding of cells/organisms as systems
Inference of networks and interactions
Computer simulation of cells and organisms
Stability analysis and control of biological systems
Experimental verifications
Network Biology
Small world (1998)
Scale-free network (1999)
Network motif (2002)
Analysis of structural features
Analysis of dynamical features

5. Graphs and Networks

6. Graphs and Networks Graph Network
Fundamental concept in discrete mathematics and computer science
Consisting of nodes and edges
node ⇔ object　（e.g., chemical compound）
edge ⇔ relation between two objects　（e.g., chemical reaction）
Undirected graph： edge does not have direction
Directed graph： edge has direction
Network
Edges with meaning and/or weights
We do not distinguish graphs from networks in this lecture
Undirected graph
Directed graph

7. Graphs and Biological Networks
Metabolic network (KEGG)
Graph
・nodes and edges

8. Graphs and Real Networks
Metabolic network
node ⇔ chemical compound,　edge ⇔ reaction　
Protein-protein interaction (PPI) network
node ⇔ protein,　edge ⇔ interaction
Genetic network
node ⇔ gene,　edge ⇔ gene regulation
WWW
node ⇔ WEB page, edge ⇔ link
Researchers’ network
node ⇔ researcher,　edge ⇔ existence of joint paper

9. Small World

10. Distance between Nodes
Path
Sequence of edges connecting two nodes
Length of path
#edges
Distance between nodes
Length of the shortest path between two nodes
Examples of paths between A and E
Path 1: (A,G), (G,B), (B,F) ,(F,E) ⇒ length=4
Path 2: (A,G), (G,F), (F,E)　　　　　 ⇒ length=3
Path 3: (A,B), (B,E)　　　　　　　　 　 ⇒ length=2
dist(A,E)=2　(dist(A,I)=3, dist(C,H)=3)

11. Cluster Coefficient i i mi ：#edges among neighboring nodes of node i
ki ： degree of node i
Measure of modularity
Ci ≒ 1 　⇔　like Clique
mi is at most i
Ci = 1 i
Ci = 0

12. Small World
Graph with short average distance （O(log n)以下） and large average clustering coefficient
It is reported that many real networks have small-world property
Average distance of WWW ⇒ around 19 (Albert al., Nature, 1999)
Ave. dist ≦ ３

13. Scale-free Network

14. Scale-free Network: Definition
Degree of node
#edges connecting to node
P(k)
Degree distribution
Frequency of nodes with degree k
Scale-free network
P(k) follows (approximately) a power-law
degree=5
degree=3
degree=2

15. Degree Distribution: ExampleC D F G H I J E
Degree
Degree 1： J
Degree 2： B, C, D, F, G, H
Degree 3： A, E, I
Degree distibution: P(k)
P(1)=0.1, P(2)=0.6, P(3)=0.3, P(4)=P(5)=P(6)=…=0

16. Degree Distribution in Scale-free Network
#nodes
#nodes ∝ (degree)-3
degree

17. Features of Scale-free Network
Def.: P(k) follows a power law ( )
Big difference from random (Erdos-Renyi) graph (with Poisson distribution: e-λλk/k!)
Existence of hubs (nodes with large degree)
Hubs often play important roles
k –γ in real networks
PPI：　γ≒2.2 (depending on organisms)
Metabolic：　γ≒2.24 (depending on organisms)
Movie stars：γ≒2.3
WWW：γ≒2.1
Power grid： γ≒4 (or, not scale-free ?)

18. Poisson Distribution vs. Power-law Distribution
（Random graph）
Power-law
（Scale-free graph）
P　(k) k
log(k)
log P　(k)

19. Analysis of PPI Network (Yeast)
PPI (protein-protein interaction) network follows power-law
node： protein
edge： interaction
Nodes with degree ≦ 5　（93%）
Around 21％ are essential (lethal)
Nodes with degree ≧16　（0.7％）
Around 62％ are essential
Referred as Hubs many of which play important roles
[Jeong et al., Nature 411:41-42, 2001]

20. Models of Scale-free Networks

21. Growth and Preferential Attachment Model
Growth and preferential attachment [Barabasi & Albert 1999]
Also referred as Rich-get-richer Model
Method（yielding a network with P(k) ∝ k -3 ）
Construct a complete graph with m0 nodes
Repeat the following
Add a new node v to current graph
Add edges between v and m nodes in current graph, where each node is selected with probability proportional its degree (i.e., deg(vi)/(Σj deg(vj)) )
c.f.: construction of random graph
Create all N nodes
Add an edge between randomly chosen two nodes
　(or, Connect two nodes with uniform probability p)

22. Random Network vs. Scale-free Network
2/6
3/10
2/10
4/14
2/14
Random Network
Scale-free Network

23. Analysis by Mean-field Approximation
ki(t): degree of node i (created at time ti） at time t
#edges at time t ≒ mt
Prob. that degree of node i increases at time t
By solving this diff. eq. with ki(ti)=m
Suppose network is completed at time tn .
From ki(tn)=k, creation time of node i with
degree k at time tn is given by
Change of ti according to change of k is estimated
　　by differentiate the above term
⇒Creeation time changes by 2tnm2k -3 with unit change of k
⇒ #nodes with degree k is approximately 2tnm2k -3

24. Illustration of Analysis
ki(t): degree of node i at time t
Add m edges
at time t
t=0
t=1
t=2
t=3
k1(t) 2 3 4 5
k2(t)
k3(t)
k4(t) -
k5(t)
Sum of degrees

26. Analysis by Master Equation

27. Evolution of Biological Networks
Preferential attachment mode is reasonable for web graphs, but not for biological networks
⇒　Duplication and divergence model
Gene duplication (copy of node) + mutation (loss of edge)

28. Deterministic Scale-free Networks

29. Hierarchical Scale-Free Network
Hierarchical Scale-Free Network [Ravasz, Barabasi et al. 2002]
Also called as：Deterministic Scale-Free Network
Recursive
construction
Like fractal
For L gons,
P(k)∝
k -1-(ln(L+1)/ln(L))

30. Analysis of Construction of Hierarchical Network
Hub of level i=1
Hub of level i=2

31. L+M Model Extension of Hierarchical model
Able to construct networks with arbitrary γ (>2)　　
(vs. γ<2.58 for Hierarchical model )
(L=2)
(M=2)
Nacher et al.,
Physical Review E, 2005

32. Analysis of L+M Model

33. Relation between Two Deterministic Models
Hierarchical model corresponds to
　　the case of M=1 in L+M model
L=3, M=1

34. Network Motif

35. Network Motif Sequence Motif Network Motif Examples
Pattern appearing in sequences with common feature
E.g.,　L-x(6)-L-x(6)-L-x(6)-L (Leucine Zipper Motif)
Network Motif
Frequently appearing network pattern in given network(s), compared with randomized networks
Network patterns are usually given by subgraphs
Randomized networks are constructed via random exchanges of edge pairs
Examples
Feed-forward Loop
Single Input Module
Dense Overlapping regulons

36. Example of Sequence Motifs
Zing finger motif
C-x(2,4)-C-x(3)-[LIVMFYWC]-x(8)-H-x(3,5)-H
Leucine zipper motif
L-x(6)-L-x(6)-L-x(6)-L

37. Example of Network Motif (1)

38. Summary Graph/Network Small World Scale-free Network
Defined by nodes and edges
Small World
Short average distance
Scale-free Network
Degree distribution follows a power-law
Models of scale-free networks
Growth and preferential attachment model
Deterministic model
Network Motif
Frequently appearing small subgraphs