GENERAL NETWORK PATTERNS

1. GENERAL NETWORKPATTERNS Danail Bonchev Center for the Study of Biological Complexity Virginia Commonwealth University Singapore, July 9-17, 2007

2. All Complex Dynamic Networks Have Similar Structure and Common Properties • Hubs • Scale-Freeness • Small-Worldness • Centrality • Motifs • Modules

3. 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

4. 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

5. Gerstein’s Single- and Multiple Interface Hubs P. M. Kim, L. J. Lu, Y. Xia, M. B. Gerstein Science 2006, 314, 1938

6. Gerstein’s Single- and Multiple Interface Hubs - 2

7. Gerstein’s Single- and Multiple Interface Hubs - 3

8. Gerstein’s Single- and Multiple Interface Hubs - 4

9. 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

10. 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

11. 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

12. Clustering vs. Local Connectivity in the Yeast Protein Interaction Network (AW Rives & T Galitski, PNAS, 100(2003)1128-1133)

13. 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

14. 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

15. 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)

16. Longevity Gene/Protein Network Power law Distribution (T. Witten, D. Bonchev, 2007)

17. increasing randomization Small-Worldness • Stanley Milgram, 1967 • Six Degrees of Separation, Broadway, early 1990s • Watts and Strogatz, Nature, 1998

18. 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?

19. The Concept of Node Centrality

20. 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?

21. 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).

22. 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

23. 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!

24. 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?

25. 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

26. 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

27. 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

29. 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

30. 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)

31. 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/

32. Do You See Any Internal Structure Here?