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