Steering Macro-Scale Network Community Structure by Micro-Scale Features

Network science plays an increasingly important role to model complex data in many scientific disciplines. One notable feature of network organization is community structure, which refers to clusters of tightly interconnected nodes. A prominent problem is how to investigate the relationship between macro-scale modules that are retrieved by optimizing global network measures, and micro-scale structure that are defined by specific queries of the analysis (e.g., nodal features). By generalizing fundamental concepts of joint space-frequency localization to network theory, here we propose a flexible framework to study interactions between micro- and macro-structure. Similar to pointing and focusing a magnifying glass, the analysis can be directed to specific micro-scale structure, while the degree of interaction with the macro-scale community structure can be seamlessly controlled. In addition, the method is computationally efficient as a result of the underlying low-dimensional optimization problem.

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

[2]  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.

[3]  Jelena Kovacevic,et al.  Discrete Signal Processing on Graphs: Sampling Theory , 2015, IEEE Transactions on Signal Processing.

[4]  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.

[5]  M. Fiedler Laplacian of graphs and algebraic connectivity , 1989 .

[6]  D. Slepian Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case , 1978, The Bell System Technical Journal.

[7]  Renaud Lambiotte,et al.  Uncovering space-independent communities in spatial networks , 2010, Proceedings of the National Academy of Sciences.

[8]  Pascal Frossard,et al.  The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains , 2012, IEEE Signal Processing Magazine.

[9]  Sune Lehmann,et al.  Link communities reveal multiscale complexity in networks , 2009, Nature.

[10]  Pierre Vandergheynst,et al.  Wavelets on Graphs via Spectral Graph Theory , 2009, ArXiv.

[11]  Dimitri Van De Ville,et al.  Tight Wavelet Frames on Multislice Graphs , 2013, IEEE Transactions on Signal Processing.

[12]  Danny C. Sorensen,et al.  Deflation Techniques for an Implicitly Restarted Arnoldi Iteration , 1996, SIAM J. Matrix Anal. Appl..

[13]  D. Slepian,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — II , 1961 .

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

[15]  Yue M. Lu,et al.  A Spectral Graph Uncertainty Principle , 2012, IEEE Transactions on Information Theory.

[16]  Pierre Borgnat,et al.  Graph Wavelets for Multiscale Community Mining , 2014, IEEE Transactions on Signal Processing.

[17]  Alfred O. Hero,et al.  Deep Community Detection , 2014, IEEE Transactions on Signal Processing.

[18]  Mark A. Wieczorek,et al.  Spatiospectral Concentration on a Sphere , 2004, SIAM Rev..

[19]  Jukka-Pekka Onnela,et al.  Community Structure in Time-Dependent, Multiscale, and Multiplex Networks , 2009, Science.

[20]  M. Newman,et al.  Finding community structure in networks using the eigenvectors of matrices. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Stefan Harmeling Exploring model selection techniques for nonlinear dimensionality reduction , 2007 .

[22]  A. Barabasi,et al.  Hierarchical Organization of Modularity in Metabolic Networks , 2002, Science.

[23]  Jeff Irion,et al.  Hierarchical graph Laplacian eigen transforms , 2014, JSIAM Lett..

[24]  Michael I. Jordan,et al.  On Spectral Clustering: Analysis and an algorithm , 2001, NIPS.

[25]  H. Pollak,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — III: The dimension of the space of essentially time- and band-limited signals , 1962 .