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 Scale-Freeness Small-Worldness Centrality Motifs Hubs Modules Hubs The Celebrities of Network World Definition: Highly connected nodes Mits and Reality of 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 Vidals 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 Gersteins Single- and Multiple Interface Hubs P. M. Kim, L. J. Lu, Y. Xia, M. B. Gerstein Science 2006, 314, 1938 Gersteins Single- and Multiple Interface Hubs - 2 Gersteins Single- and Multiple Interface Hubs - 3 Gersteins Single- and Multiple Interface Hubs - 4 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 Can Two Celebrities Work in a Team? Assortativeness 1 23 Positive assortativeness Negative assortativeness Protein interaction networks have negative assortitativeness Hubs connect with high correlation to low connectivity nodes Clustering Coefficient The larger the node clustering coefficient, the higher the local complexity E Conn C i ; The average clustering coefficient of dynamic networks is much higher than that of random networks C prot (yeast) = C rand = Definition : 0 C i 1 Can Supporting Actors Work Together? Clustering vs. Local Connectivity in the Yeast Protein Interaction Network (AW Rives & T Galitski, PNAS, 100(2003) ) Scale-Freeness What is scale-free? o Self-similarity, both globally and locally. o The presence of hubs irrespective of the scale of the network o Topological invariance of a network structure, no matter how coarsely it is viewed. o Barabasi, Albert, 1999: A network with a power-law degree distribution. (Price, 1965) o 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 Preferential attachment The Power Law Random networks Poisson distribution for x = 0, 1, 2, Example: =2.1, x=4, p=0.099 Dynamic evolutionary networks Power Law Distribution Log/log Presentation Dynamic evolutionary networks The Power Law In Intra-Cellular Networks (P. Fernandez, R.V. Sol, in Complexity in Chemistry, Biology, and Ecology, D. Bonchev abd D.H. Rouvray, Eds. Springer, New York, 2005, p. 171) Longevity Gene/Protein Network Power law Distribution (T. Witten, D. Bonchev, 2007) Small-Worldness Stanley Milgram, 1967 Six Degrees of Separation, Broadway, early 1990s Watts and Strogatz, Nature, 1998 increasing randomization Small-Worldness vs Clustering Why is the network small-worldness important? 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. The Concept of Node Centrality 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) e i = d ij (max) = min 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, 2x3 d(max) = 3 Vertex 7: 1x1, 2x2, 3x3 d(max) = 3 e 1 = e 2 = min (d(max)) = 2 Is this definition sufficient? Centric vertex ordering: (1,2), (2,3,4,5,6,7) Hierarchical definition 2: If several vertices have the same eccentricity e i,the center is the vertex having the lowest vertex distance d i. Other Hierarchical Criteria The network vertices are thus be characterized by their centrality, and ordered in concentric circles around the central vertex(es). Graph Center - 2 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 e i = e j ; d i < d j Centric vertex ordering: (1,2), (2,3,4,5,6,7) {1},{2},{3},{4},{5,6},{7} Network Centrality Closeness Centrality, (Freeman, 1978) Vertex Centrality, (Bonchev et al., 1980) Defined according to a set of hierarchically ordered criteria eccentricity, vertex distance, DDS, Contradictions: 1 2 # 1: d 1 = 4x1 + 1x2 + 1x3 = 9 CC(1) = 6/9 = # 2: d 2 = 2x1 + 4x2 = 10 > d 1 CC(2) = 6/10 = Node 1 is more central than node 2 However, e 1 = 3 e 2 = 2 < e 1 Node 2 is more central than node 1 Betweenness Centrality (Freeman, 1978) Network Centrality N p (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} N p (2) = 8 {1-3,1-4,3-5,3-6,3-7,4-5,4-6,4-7} N p (3) = 5 {1-4,2-4,4-5,4-6,4-7} N p = 25 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 The shortest paths are used only! Network Centrality - 3 Eigenvector Centrality (Bonacich, 1972) How to calculate the principal eigenvalue ? X X x = 0 x 4 - 3x = 0 1 = 1.618; 2 = Example: ajaj Extended connectivity Eigenvector centralities are computed from the values of the first eigenvector of the graph adjacency matrix Why is centrality important? All 13 types of connected subgraphs of three nodes Motifs The Simple Building Blocks of Complex Networks Definition: Subgraphs occurring in complex networks at frequencies much higher than those in randomized networks (R. Milo et al., Science, 298, 2002, ) Network Motifs - 2 X Y Z Feed-forward loop protein, neuron, electronic X Y Z Three chain food webs X Y Z Feedback loop gene regulatory, electronic By-fan protein, neuron, electronic X Y Z W X Y Z Fully connected triad World Wide Web Type of Motif Name Abundance in different kind of networks Network Motifs As Species Fingerprints Network Nodes Edges N real N rand SD N real N rand SD Gene regulation (transcription) E. coli 12 S. Cerevisiae 685 1, 40 Feed- Forward Motif X Y Z X Y Z W By-Fan Motif Size-4 Motifs in Mus musculus ID Motif Frequency [Original] Mean- Freq [Random] Standard-Dev [Random] Z-Scorep-Value % % % % % % % % Network Motifs and Dynamics T ST TS TS 3800 15 FF A FF C FF B 6287 S Search for motifs with the fastest dynamics A. Apte, D. Bonchev, S. Fong (2007) Synthetic Biology FANMOD Software for Finding Network Motifs MFinder 1.2Also there: Motif dictionary (by S. Wernicke and F. Rasche ) MAVisto (by F. Schreiber and H. Schwobbermeyer) Useful Software for Visualization and Manipulation of Networks Pajek -default.htm Cytoscape -Pathway Studio 5.0 (Ariadnegenomics.com) Ingenuity Patway Analysis IPA 5.0 (Ingenuity.com) NetworkBlast - Do You See Any Internal Structure Here?