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

1) Degree Distribution G

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

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

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

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)

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)

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)

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

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

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

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

Generalize node degree 2.2) Graphlet Degree Distributions

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

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

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

Signature Similarity Measure between nodes u and v

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

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

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

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

Software that implements many of these network properties and compares networks with respect to them: GraphCrunch

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

Network models

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.

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 , 2006.