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GENERAL NETWORK PATTERNS Danail Bonchev Center for the Study of Biological Complexity Virginia Commonwealth University Singapore, July 9-17, 2007 All Complex Dynamic Networks Have Similar Structure and Common Properties Hubs Scale-Freeness Small-Worldness Centrality

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General network patterns l.jpg

GENERAL NETWORKPATTERNS

Danail Bonchev

Center for the Study of Biological Complexity

Virginia Commonwealth University

Singapore, July 9-17, 2007


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All Complex Dynamic Networks

Have Similar Structure and

Common Properties

  • Hubs

  • Scale-Freeness

  • Small-Worldness

  • Centrality

  • Motifs

  • Modules


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Hubs – The Celebrities of

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


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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


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Gerstein’s Single- and Multiple

Interface Hubs

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



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Gerstein’s Single- and

Multiple Interface Hubs - 3


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Gerstein’s Single- and

Multiple Interface Hubs - 4


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Some More About Hubs

  • 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


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3

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


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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


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Clustering vs. Local Connectivity in the

Yeast Protein Interaction Network

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


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Scale-Freeness

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


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Dynamic evolutionary networks

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


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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)


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Longevity Gene/Protein Network

Power law Distribution

(T. Witten, D. Bonchev, 2007)


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increasing randomization

Small-Worldness

  • Stanley Milgram, 1967

  • Six Degrees of Separation, Broadway, early 1990s

  • Watts and Strogatz, Nature, 1998


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Small-Worldness vs Clustering

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?


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The Concept

of

Node Centrality


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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?


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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).


Network centrality l.jpg

1

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


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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!


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1 2 3 4

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?


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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


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X Y Z

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


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X

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


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S

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


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Software for Finding Network

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)


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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/


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Do You See Any Internal

Structure Here?


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