Network Properties

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

1) Degree Distribution G

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

3) Average Diameter u • Distance between a pair of nodes u and v: • Du,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 {Du,v for all pairs of nodes {u,v} in G} • Spectrum of the shortest path lengths G v E.g. (not for G)

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

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  - Zi is statistical significance of subgraph i, SPi 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  - Zi is statistical significance of subgraph i, SPi 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) http://www.weizmann.ac.il/mcb/UriAlon/

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

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

2.2) Graphlet Degree Distributions Generalize node degree

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

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

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.

Signature Similarity Measure between nodes u and v T. Milenkovic and N. Przulj, “Uncovering Biological Network Function via Graphlet Degree Signatures”, Cancer Informatics, vol. 4, pg. 257-273, 2008.

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

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.

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 http://www.ics.uci.edu/~bio-nets/graphcrunch/

Network models

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. 711-716, 2006.

Network Properties