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GENERAL NETWORK PATTERNS

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### GENERAL NETWORKPATTERNS

Danail Bonchev

Center for the Study of Biological Complexity

Virginia Commonwealth University

Singapore, July 9-17, 2007

Have Similar Structure and

Common Properties

- Hubs

- Scale-Freeness

- Small-Worldness

- Centrality

- Motifs

- Modules

Network World

- Definition: Highly connected nodes

- Mits and Realityof hubs connectivity

- Mark Vidal: “party” proteins and “date” ones

- Mark Gerstein:Which of the multiple interactions
occur simultaneously, and which are mutually exclusive

due to overlapping binding surfaces?

multi-interface ( “party”) and single-interface (“date”) domains

Mark Vidal’s “Party” and “Date” Hubs

J. D. Han et al. Nature 2004, 430, 88.

- The “party” hubs form stable complexes; they are conserved

- The “date” hubs evolve across species

Gerstein’s Single- and Multiple

Interface Hubs

P. M. Kim, L. J. Lu, Y. Xia, M. B. Gerstein Science 2006, 314, 1938

Multiple Interface Hubs - 3

Multiple Interface Hubs - 4

- The good news and the bad news

- Essentiality/Lethality

- Spreading of epidemics

- Side effects (medicines; gene engineering)

- The future of drug design and patient treatment

2

3

2

1

1

Positive assortativeness

Negative assortativeness

Can Two Celebrities Work in a Team?

Assortativeness

Protein interaction networks have negative assortitativeness

Hubs connect with high correlation to low connectivity nodes

Can Supporting Actors Work Together?

Definition:

0 ≤ Ci ≤ 1

Clustering Coefficient

- The larger the node clustering coefficient, the higher the local complexity

E 3 3 4 5 6

Conn 0.5 0.5 0.667 0.833 1

Ci 0 0 0 1 ; 0.67 1

- The average clustering coefficient of dynamic networks is much higher than
that of random networks

Cprot (yeast) = 0.142

Crand = 0.00139

Clustering vs. Local Connectivity in the

Yeast Protein Interaction Network

(AW Rives & T Galitski, PNAS, 100(2003)1128-1133)

What is scale-free?

- Self-similarity, both globally and locally.

- Topological invariance of a network structure, no
- matter how coarsely it is viewed.

- The presence of hubs irrespective of the scale of the network

- Barabasi, Albert, 1999: A network with a power-law degree
- distribution. (Price, 1965)

Preferential attachment

- Other mathematical laws: Dorogovtsev et al (2000),
- (exponential, polynomial,…)

o Sole et al. (2002), Vazquez et al. (2003) –

gene duplication generates power law distribution

o Kuznetsov (2006): Not all networks are scale-free

Dynamic evolutionary networks

Random networks

Power Law Distribution

Log/log Presentation

Poisson distribution

Example: λ=2.1, x=4, p=0.099

for x = 0, 1, 2, …

The Power Law

B.S. cerevisiae with N=3280, λ = 2.43±0.10

C. C. elegans with N=3228, λ = 2.37±0.10

A. Node degree distribution in the protein

network of S. cerevisiae vs distribution in a

random graph with the same number of

vertices and edges.

The Power Law In

Intra-Cellular Networks

(P. Fernandez, R.V. Solé, in Complexity in Chemistry, Biology, and Ecology, D. Bonchev abd D.H. Rouvray, Eds. Springer, New York, 2005, p. 171)

Small-Worldness

- Stanley Milgram, 1967

- Six Degrees of Separation, Broadway, early 1990s

- Watts and Strogatz, Nature, 1998

The normalized cluster coefficient and the normalized network

radius as a function of the probability of rewiring node-node links.

- The small-world effect is manifested with both small
network radius and high clustering coefficient.

- Why is the network small-worldness important?

ei = dij (max) = min

6

Vertex 1: 4x1, 2x2; Vertex 2: 3x1, 3x2; d(max) = 2

Vertex 3: 2x1, 3x2, 1x3; Vertex 4: 2x1, 2x2, 2x3 d(max) = 3

Vertex 5: 1x1, 3x2, 2x3; Vertex 6: 1x1, 3x2, 2x3d(max) = 3

Vertex 7: 1x1, 2x2, 3x3 d(max) = 3

2

1

7

5

4

3

e1 = e2 = min (d(max)) = 2

How to Define the Center of a Graph?

- Classical definition: The graph center is the vertex(es) having the
lowest eccentricity

(F. Harary, Graph Theory, Addison-Wesley, 1969)

Centric vertex ordering: (1,2), (2,3,4,5,6,7)

Is this definition sufficient?

ei = ej ; di < dj

13

9

8

14

13

12

11

D. Bonchev, A. T. Balaban and O. Mekenyan, Generalization of the Graph Center Concept, and Derived Topological Indexes. J. Chem. Inf. Comput. Sci. 20(1980)106‑113.

D. Bonchev, The Concept for the Center of a Chemical Structure and Its Applications, Theochem 185, 1989, 155‑168.

Graph Center - 2

- Hierarchical definition 2:If several vertices have the same
eccentricity ei,the center is the vertex having the lowest vertex

distance di.

Centric vertex ordering:

(1,2), (2,3,4,5,6,7) {1},{2},{3},{4},{5,6},{7}

Other Hierarchical Criteria

The network vertices are thus be characterized by their centrality,

and ordered in concentric circles around the central vertex(es).

2

Network Centrality- Vertex Centrality,(Bonchev et al., 1980)

Defined according to a set of hierarchically ordered criteria –

eccentricity, vertex distance, DDS,…

- Closeness Centrality,(Freeman, 1978)

Contradictions:

# 1: d1 = 4x1 + 1x2 + 1x3 = 9

CC(1) = 6/9 = 0.667

# 2: d2 = 2x1 + 4x2 = 10 > d1

CC(2) = 6/10 = 0.600

Node 1 is more central than node 2

However, e1 = 3

e2 = 2 < e1

Node 2 is more central than node 1

Np (1) = 12 {2-5,2-6,2-7,3-5,3-6,3-7,4-5,4-6.,4-7,5-6,5-7,6-7}

1

2

7

Np (2) = 8 {1-3,1-4,3-5,3-6,3-7,4-5,4-6,4-7}

3

4

Np (3) = 5 {1-4,2-4,4-5,4-6,4-7}

Np = 25

6

5

BC(1) = 12/25 = 0.48

BC(2) = 8/25 = 0.32

Bc(3) = 5/25 = 0.20

B(4) = B(5) = B(6) = B(7) = 0

Network Centrality - 2

- Betweenness Centrality(Freeman, 1978)

The shortest paths are used only!

1

2

3

4

X 1 0 0

1 X 1 0

0 1 0 1

0 0 1 x

Example:

= 0 x4 - 3x2 + 1 = 0

aj

2 3 3 2

λ1 = 1.618; λ2 = 0.618

1

2

3

4

Extended connectivity

Network Centrality - 3

- Eigenvector Centrality(Bonacich, 1972)

How to calculate the principal eigenvalue λ?

Eigenvector centralities are computed from the values

of the first eigenvector of the graph adjacency matrix

Why is centrality important?

Motifs– The Simple Building

Blocks of Complex Networks

(R. Milo et al., Science, 298, 2002, 824-827)

Definition: Subgraphs occurring in complex networks

at frequencies much higher than those in randomized

networks

All 13 types of connected subgraphs of three nodes

Three chain food webs

X Y Z

Feed-forward loopprotein, neuron, electronic

X

Feedback loop gene regulatory, electronic

Y

Z

X Y

By-fan protein, neuron, electronic

Z W

X

Fully connected triad World Wide Web

Y

Z

Network Motifs - 2

Type of Motif

Name Abundance in different kind of networks

Network Nodes Edges NrealNrand ± SD NrealNrand ± SD

Gene regulation

(transcription)

Y

Feed-

Forward

Motif

By-Fan

Motif

X Y

Z W

Z

E. coli424 519 40 7 ± 3 203 47 ± 12

S. Cerevisiae 685 1,052 70 11 ± 4 1812 300 ± 40

Network Motifs As

Species Fingerprints

T

S

T

S

T

FFB

4187±13

2

FFA

1

3

S

T

1

2

3800±16

3

6287±16

FFC

4542±15

Network Motifs and Dynamics

Search for motifs with the fastest dynamics

A. Apte, D. Bonchev, S. Fong (2007)

Synthetic Biology

Motifs

MFinder 1.2

http://www.weizmann.ac.il/mcb/UriAlon/

Also there: Motif dictionary

FANMOD

http://www.minet.uni-jena.de/~wernicke/motifs/

(by S. Wernicke and F. Rasche)

MAVisto

http://mavisto.ipk-gatersleben.de/

(by F. Schreiber and H. Schwobbermeyer)

Useful Software for Visualization

and Manipulation of Networks

Pajek - http://vlado.fmf.uni-lj.si/pub/networks/pajek/

default.htm

Cytoscape - http://www.cytoscape.org/

Pathway Studio 5.0 (Ariadnegenomics.com)

Ingenuity Patway Analysis – IPA 5.0 (Ingenuity.com)

NetworkBlast - http://chianti.ucsd.edu/nct/

Structure Here?

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