title: Clustering under the line graph transformation: application to reaction network.
abstract: BACKGROUND
  Many real networks can be understood as two complementary networks with two kind of nodes. This is the case of metabolic networks where the first network has chemical compounds as nodes and the second one has nodes as reactions. In general, the second network may be related to the first one by a technique called line graph transformation (i.e., edges in an initial network are transformed into nodes). Recently, the main topological properties of the metabolic networks have been properly described by means of a hierarchical model. While the chemical compound network has been classified as hierarchical network, a detailed study of the chemical reaction network had not been carried out.
  
  
  RESULTS
  We have applied the line graph transformation to a hierarchical network and the degree-dependent clustering coefficient C(k) is calculated for the transformed network. C(k) indicates the probability that two nearest neighbours of a vertex of degree k are connected to each other. While C(k) follows the scaling law C(k) approximately k(-1.1) for the initial hierarchical network, C(k) scales weakly as k0.08 for the transformed network. This theoretical prediction was compared with the experimental data of chemical reactions from the KEGG database finding a good agreement.
  
  
  CONCLUSIONS
  The weak scaling found for the transformed network indicates that the reaction network can be identified as a degree-independent clustering network. By using this result, the hierarchical classification of the reaction network is discussed.
authors.affiliation: Bioinformatics Center, Institute for Chemical Research, Kyoto University, Uji, 611-0011, Japan. nacher@kuicr.kyoto-u.ac.jp
authors.forename: Jose C
authors.initials: JC
authors.lastname: Nacher
authors.forename: Nobuhisa
authors.initials: N
authors.lastname: Ueda
authors.forename: Takuji
authors.initials: T
authors.lastname: Yamada
authors.forename: Minoru
authors.initials: M
authors.lastname: Kanehisa
authors.forename: Tatsuya
authors.initials: T
authors.lastname: Akutsu
mesh_terms: D000465:Algorithms; D016000:Cluster Analysis; D019295:Computational Biology / Q000379:methods*; D003196:Computer Graphics; D003198:Computer Simulation; D008956:Models, Chemical; D015233:Models, Statistical; D013269:Stochastic Processes; D049490:Systems Biology; D014584:User-Computer Interface