1. 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

2. 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

3. 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’

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

5. Projection and Degree Distribution
Three kinds of (bottom) projections

6. 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 ）

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

8. 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

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

10. 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: , 2011.