Optimal map of the modular structure of complex networks

The modular structure is pervasive in many complex networks of interactions observed in natural, social and technological sciences. Its study sheds light on the relation between the structure and the function of complex systems. Generally speaking, modules are islands of highly connected nodes separated by a relatively small number of links. Every module can have the contributions of links from any node in the network. The challenge is to disentangle these contributions to understand how the modular structure is built. The main problem is that the analysis of a certain partition into modules involves, in principle, as much data as the number of modules times the number of nodes. To confront this challenge, here we first define the contribution matrix, the mathematical object containing all the information about the partition of interest, and then we use truncated singular value decomposition to extract the best representation of this matrix in a plane. The analysis of this projection allows us to scrutinize the skeleton of the modular structure, revealing the structure of individual modules and their interrelations.

[1]  Alex Arenas,et al.  Analysis of the structure of complex networks at different resolution levels , 2007, physics/0703218.

[2]  M. Rosenblum,et al.  Chapter 9 Phase synchronization: From theory to data analysis , 2001 .

[3]  D. Botstein,et al.  Singular value decomposition for genome-wide expression data processing and modeling. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[4]  R. Guimerà,et al.  Classes of complex networks defined by role-to-role connectivity profiles. , 2007, Nature physics.

[5]  Gene H. Golub,et al.  Matrix computations , 1983 .

[6]  R. Guimerà,et al.  The worldwide air transportation network: Anomalous centrality, community structure, and cities' global roles , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Peter Langfelder,et al.  Eigengene networks for studying the relationships between co-expression modules , 2007, BMC Systems Biology.

[8]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Dmitri V. Krioukov,et al.  AS relationships: inference and validation , 2006, CCRV.

[10]  R. Bro,et al.  Resolving the sign ambiguity in the singular value decomposition , 2008 .

[11]  T. Landauer,et al.  A Solution to Plato's Problem: The Latent Semantic Analysis Theory of Acquisition, Induction, and Representation of Knowledge. , 1997 .

[12]  Leon Danon,et al.  Comparing community structure identification , 2005, cond-mat/0505245.

[13]  R. Guimerà,et al.  Functional cartography of complex metabolic networks , 2005, Nature.

[14]  A. Arenas,et al.  Synchronization processes in complex networks , 2006, nlin/0610057.

[15]  Sergio Gómez,et al.  Multiple resolution of the modular structure of complex networks , 2007, ArXiv.

[16]  J. Kurths,et al.  Phase synchronization: from theory to data analysis , 2003 .

[17]  G. Golub,et al.  Inverse Eigenvalue Problems: Theory, Algorithms, and Applications , 2005 .

[18]  M E J Newman,et al.  Finding and evaluating community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  W. Zachary,et al.  An Information Flow Model for Conflict and Fission in Small Groups , 1977, Journal of Anthropological Research.

[20]  T. Vicsek,et al.  Uncovering the overlapping community structure of complex networks in nature and society , 2005, Nature.

[21]  Dmitri V. Krioukov,et al.  Revealing the Autonomous System Taxonomy: The Machine Learning Approach , 2006, ArXiv.

[22]  Susan T. Dumais,et al.  Using Linear Algebra for Intelligent Information Retrieval , 1995, SIAM Rev..

[23]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[24]  A. Arenas,et al.  Community detection in complex networks using extremal optimization. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Alessandro Vespignani,et al.  Evolution thinks modular , 2003, Nature Genetics.