Bioinformatics Center Institute for Chemical Research Kyoto University

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Bioinformatics Center Institute for Chemical Research Kyoto University 九大数理集中講義 Comparison, Analysis, and Control of Biological Networks (2) Projection of Bipartite Network Tatsuya Akutsu Bioinformatics Center Institute for Chemical Research Kyoto University

Bipartite Networks U E V Many real networks have bipartite graph structures G(U,V;E) Metabolic networks: U⇔reactions, V⇔compounds Movie stars：U⇔movies, V⇔actors/actresses Researchers：U⇔joint papers、V⇔researchers U Res. 1 Res. 2 Res. 3 Res. 4 Paper 2 Paper 3 Paper 4 Paper 5 Paper 1 E V

Projection from Bipartite Network Construct G’(V,E’) from G(U,V;E) by {v1,v2}∊E’ iff (∃u∊U)({u,v1}∊E and {u,v2}∊E and v1≠v2)) U Res. 1 Res. 2 Res. 3 Res. 4 Paper 2 Paper 3 Paper 4 Paper 5 Paper 1 E V Res. 2 Res. 1 Res. 4 Res. 3 E E’

Top Projection and Bottom Projection Top Projection: Use of top nodes Bottom Projection： Use of bottom nodes

Projection and Degree Distribution Three kinds of (bottom) projections

Theoretical Results P(k): distribution after bottom projection 　　Case of (ES) : P(k)∝k –γ2 　　Case of (SE) : P(k)∝k -γ1+1 　　Case of (SS): P(k)∝k max(-γ1+1,-γ2) ・Case of (ES) is known ([Guillaume & Latapy 2006] [Birmele 2009]) ・ Results on top projection also follows ・It is assume that γ1>2 in (ES), γ1 > 4 in (SE) and (SS) 　　（However, it seems that the results hold for smaller γ1 ）

Theoretical Analysis Probability that randomly selected bottom node u has degree k after projection is

Analysis of (SE) – Part 1 To derive approximation of Define f(h,k) by Use of known property on sum of power law where γ=γ1-1

Analysis of (SE)-Part 2 Analysis （SS)： Get P(k)∝k-γ1+1 by the following Analysis （SS)：　　　　　　Omitted　（Use of cumulative distribution）

Summary Estimation of degree distribution after bottom projection from bipartite networks Validation of theoretical estimation by computer simulation and database analysis J. C. Nacher and T. Akutsu, On the degree distribution of projected networks mapped from bipartite networks, Physica A, 390:4636-4651, 2011.