Review of Graph Invariants for Quantitative Analysis of Structure Dynamics

In this work we review graph invariants used for quantitative analysis of evolving graphs. Focusing on graph datasets derived from structural pattern recognition and complex networks fields, we demonstrate how to capture relevant topological features of networks. In an experimental setup, we study structural properties of graphs representing rotating 3D objects and show how they are related to characteritics of undelying images. We present how evolving strucure of Autonomous Systems (ASs) network is reflected by non-trivial changes in scalar graph descriptors. We also inspect characteristics of growing tumor vascular networks, obtained from a simulation. Additionally, the overview of currently used graph invariants with several possible groupings is provided.

[1]  Francisco Escolano,et al.  Graph-Based Representations in Pattern Recognition, 6th IAPR-TC-15 International Workshop, GbRPR 2007, Alicante, Spain, June 11-13, 2007, Proceedings , 2007, GbRPR.

[2]  L. da F. Costa,et al.  Characterization of complex networks: A survey of measurements , 2005, cond-mat/0505185.

[3]  J W Baish,et al.  Fractals and cancer. , 2000, Cancer research.

[4]  Witold Dzwinel,et al.  Exploring Complex Networks with Graph Investigator Research Application , 2011, Comput. Informatics.

[5]  Christos Faloutsos,et al.  Graph evolution: Densification and shrinking diameters , 2006, TKDD.

[6]  David A. Yuen,et al.  Efficient Graph Comparison and Visualization Using GPU , 2011, 2011 14th IEEE International Conference on Computational Science and Engineering.

[7]  Alberto Prieto,et al.  Bio-inspired systems: Computational and ambient intelligence , 2011, Neurocomputing.

[8]  Wojciech W. Czech.  Graph Descriptors from B-Matrix Representation , 2011, GbRPR.

[9]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[10]  Mark E. J. Newman.  A measure of betweenness centrality based on random walks , 2005, Soc. Networks.

[11]  David J. Marchette,et al.  Random Graphs for Statistical Pattern Recognition: Marchette/Statistical Pattern Recognition , 2004 .

[12]  Christopher G. Harris,et al.  A Combined Corner and Edge Detector , 1988, Alvey Vision Conference.

[13]  Conformation-networks of two-dimensional lattice homopolymers , 2005, cond-mat/0507182.

[14]  Ernesto Estrada,et al.  Communicability betweenness in complex networks , 2009, 0905.4102.

[15]  Stanley Wasserman,et al.  Social Network Analysis: Methods and Applications , 1994 .

[16]  A. Barabasi,et al.  Network biology: understanding the cell's functional organization , 2004, Nature Reviews Genetics.

[17]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[18]  H Rieger,et al.  Physical determinants of vascular network remodeling during tumor growth , 2010, The European physical journal. E, Soft matter.

[19]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[20]  M. Gordon,et al.  Non-random polycondensation : statistical theory of the substitution effect , 1964 .

[21]  Berk,et al.  Scale-invariant behavior and vascular network formation in normal and tumor tissue. , 1995, Physical review letters.

[22]  D. Marchette.  Random Graphs for Statistical Pattern Recognition , 2004 .

[23]  Pawel Topa,et al.  Dynamically Reorganising Vascular Networks Modelled Using Cellular Automata Approach , 2008, ACRI.

[24]  Stefan Burr,et al.  The Mathematics of networks , 1982 .

[25]  John R. Gilbert,et al.  Solving path problems on the GPU , 2010, Parallel Comput..

[26]  S. Havlin,et al.  Self-similarity of complex networks , 2005, Nature.

[27]  Sameer A. Nene,et al.  Columbia Object Image Library (COIL100) , 1996 .

[28]  Edwin R. Hancock,et al.  Pattern Vectors from Algebraic Graph Theory , 2005, IEEE Trans. Pattern Anal. Mach. Intell..

[29]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[30]  P. Carmeliet,et al.  Angiogenesis in cancer and other diseases , 2000, Nature.

[31]  Francisco Escolano,et al.  Graph-Based Representations in Pattern Recognition and Computational Intelligence , 2009, IWANN.

[32]  Edwin R. Hancock,et al.  Graph characteristics from the heat kernel trace , 2009, Pattern Recognit..

[33]  Ulrik Brandes,et al.  Biological Networks , 2013, Handbook of Graph Drawing and Visualization.

[34]  Leonard M. Freeman,et al.  A set of measures of centrality based upon betweenness , 1977 .

[35]  Ernesto Estrada.  Generalized walks-based centrality measures for complex biological networks. , 2010, Journal of theoretical biology.

[36]  Edwin R. Hancock,et al.  Spectral embedding of graphs , 2003, Pattern Recognit..

[37]  Mario Vento,et al.  Thirty Years Of Graph Matching In Pattern Recognition , 2004, Int. J. Pattern Recognit. Artif. Intell..

[38]  Edwin R. Hancock,et al.  Clustering and Embedding Using Commute Times , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[39]  Rajeev Motwani,et al.  The PageRank Citation Ranking : Bringing Order to the Web , 1999, WWW 1999.

[40]  P. Rios,et al.  Complex network analysis of free-energy landscapes , 2007, Proceedings of the National Academy of Sciences.

[41]  Desmond J. Higham,et al.  Network Properties Revealed through Matrix Functions , 2010, SIAM Rev..

[42]  Witold Dzwinel,et al.  Interactive Visualization Tool for Planning Cancer Treatment , 2013 .

[43]  John R. Platt,et al.  Influence of Neighbor Bonds on Additive Bond Properties in Paraffins , 1947 .

[44]  Almerima Jamakovic,et al.  On the relationships between topological measures in real-world networks , 2008, Networks Heterog. Media.

[45]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

[46]  L. Freeman.  Centrality in social networks conceptual clarification , 1978 .

[47]  Edwin R. Hancock,et al.  A generative model for graph matching and embedding , 2009, Comput. Vis. Image Underst..

[48]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[49]  David A. Yuen,et al.  A 3-D model of tumor progression based on complex automata driven by particle dynamics , 2009, Journal of Molecular Modeling.