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Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution 2)Clustering coefficient and spectrum 3)Average diameter 4)Centralities

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1) Degree Distribution G

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C v – Clustering coefficient of node v C A = 1/1 = 1 C B = 1/3 = 0.33 C C = 0 C D = 2/10 = 0.2 … C = Avg. clust. coefficient of the whole network = avg {C v over all nodes v of G} C(k) – Avg. clust. coefficient of all nodes of degree k E.g.: C(2) = (C A + C C )/2 = (1+0)/2 = 0.5 => Clustering spectrum E.g. (not for G ) 2) Clustering Coefficient and Spectrum G

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3) Average Diameter G u v E.g. (not for G) Distance between a pair of nodes u and v: D u, v = min {length of all paths between u and v} = min {3,4,3,2} = 2 = dist(u,v) Average diameter of the whole network: D = avg {D u,v for all pairs of nodes {u,v} in G} Spectrum of the shortest path lengths

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Network Properties 2. Local Network Properties ( Chapter 5 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Network motifs 2)Graphlets: 2.1) Relative Graphlet Frequence Distance between 2 networks 2.2) Graphlet Degree Distribution Agreement between 2 networks

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Small subgraphs that are overrepresented in a network when compared to randomized networks Network motifs: –Reflect the underlying evolutionary processes that generated the network –Carry functional information –Define superfamilies of networks - Z i is statistical significance of subgraph i, SP i is a vector of numbers in 0-1 But: –Functionally important but not statistically significant patterns could be missed –The choice of the appropriate null model is crucial, especially across “families” 1) Network motifs (Uri Alon’s group, ’02-’04)

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Small subgraphs that are overrepresented in a network when compared to randomized networks Network motifs: –Reflect the underlying evolutionary processes that generated the network –Carry functional information –Define superfamilies of networks - Z i is statistical significance of subgraph i, SP i is a vector of numbers in 0-1 But: –Functionally important but not statistically significant patterns could be missed –The choice of the appropriate null model is crucial, especially across “families” 1) Network motifs (Uri Alon’s group, ’02-’04)

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Small subgraphs that are overrepresented in a network when compared to randomized networks Network motifs: –Reflect the underlying evolutionary processes that generated the network –Carry functional information –Define superfamilies of networks - Z i is statistical significance of subgraph i, SP i is a vector of numbers in 0-1 Also – generation of random graphs is an issue: –Random graphs with the same degree in- & out- degree distribution as data constructed –But this might not be the best network null model 1) Network motifs (Uri Alon’s group, ’02-’04)

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

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N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free or Geometric?,” Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004. _____ Different from network motifs: Induced subgraphs Of any frequency 2) Graphlets (Przulj, ’04-’09)

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N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free or Geometric?,” Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004.

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N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free or Geometric?,” Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004.

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N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free or Geometric?,” Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004. 2.1) Relative Graphlet Frequency (RGF) distance between networks G and H:

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Generalize node degree 2.2) Graphlet Degree Distributions

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N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” ECCB, Bioinformatics, vol. 23, pg. e177-e183, 2007.

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T. Milenkovic and N. Przulj, “Uncovering Biological Network Function via Graphlet Degree Signatures”, Cancer Informatics, vol. 4, pg. 257-273, 2008. Network structure vs. biological function & disease Graphlet Degree (GD) vectors, or “node signatures”

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Similarity measure between “node signature” vectors T. Milenkovic and N. Przulj, “Uncovering Biological Network Function via Graphlet Degree Signatures”, Cancer Informatics, vol. 4, pg. 257-273, 2008.

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Signature Similarity Measure between nodes u and v

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Later we will see how to use this and other techniques to link network structure with biological function.

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N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” Bioinformatics, vol. 23, pg. e177-e183, 2007. Generalize Degree Distribution of a network The degree distribution measures: the number of nodes “touching” k edges for each value of k.

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N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” Bioinformatics, vol. 23, pg. e177-e183, 2007.

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/ sqrt(2) ( to make it between 0 and 1) This is called Graphlet Degree Distribution (GDD) Agreement netween networks G and H.

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Software that implements many of these network properties and compares networks with respect to them: GraphCrunch http://www.ics.uci.edu/~bio-nets/graphcrunch/

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Network models Degree distributionClustering coefficientDiameter Real-world (e.g., PPI) networksPower-lawHighSmall Erdos-Renyi graphsPoissonLowSmall Random graphs with the same degree distribution as the data Power-lawLowSmall Small-world networksPoissonHighSmall Scale-free networksPower-lawLowSmall Geometric random graphsPoissonHighSmall Stickiness network modelPower-lawHighSmall

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Network models

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Geometric Gene Duplication and Mutation Networks Intuitive “geometricity” of PPI networks: Genes exist in some bio-chemical space Gene duplications and mutations Natural selection = “evolutionary optimization” N. Przulj, O. Kuchaiev, A. Stevanovic, and W. Hayes “Geometric Evolutionary Dynamics of Protein Interaction Network”, Pacific Symposium on Biocomputing (PSB’10), Hawaii, 2010.

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Network models Stickiness-index-based model (“STICKY”) N. Przulj and D. Higham “Modelling protein-protein interaction networks via a stickiness indes”, Journal of the Royal Society Interface 3, pp. 711-716, 2006.

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