Graph characterizations from von Neumann entropy

In this paper we explore how the von Neumann entropy can be used as a measure of graph complexity. We also develop a simplified form for the von Neumann entropy of a graph that can be computed in terms of node degree statistics. We compare the resulting complexity with Estrada's heterogeneity index which measures the heterogeneity of the node degree across a graph and reveal a new link between Estrada's index and the commute time on a graph. Finally, we explore how the von Neumann entropy can be used in conjunction with thermodynamic depth. This measure has been shown to overcome problems associated with iso-spectrality encountered when using complexity measures based on spectral graph theory. Our experimental evaluation of the simplified von Neumann entropy explores (a) the accuracy of the underlying approximation, (b) a comparison with alternative graph characterizations, and (c) the application of the entropy-based thermodynamic depth to characterize protein-protein interaction networks.

[1]  Desmond J. Higham,et al.  Network Science - Complexity in Nature and Technology , 2010, Network Science.

[2]  G. Bianconi,et al.  Shannon and von Neumann entropy of random networks with heterogeneous expected degree. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Edwin R. Hancock,et al.  Learning Generative Graph Prototypes Using Simplified von Neumann Entropy , 2011, GbRPR.

[4]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[5]  F. Hohn,et al.  Elementary Matrix Algebra , 1959 .

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

[7]  Pablo Suau,et al.  Bayesian optimization of the scale saliency filter , 2008, Image Vis. Comput..

[8]  Ernest Valveny,et al.  Exact Median Graph Computation Via Graph Embedding , 2008, SSPR/SPR.

[9]  Edwin R. Hancock,et al.  Spectral Embedding of Feature Hypergraphs , 2008, SSPR/SPR.

[10]  Edwin R. Hancock,et al.  Discovering Shape Classes using Tree Edit-Distance and Pairwise Clustering , 2007, International Journal of Computer Vision.

[11]  Xueliang Li,et al.  A Survey on the Randic Index , 2008 .

[12]  Edwin R. Hancock,et al.  Graph Clustering Using the Jensen-Shannon Kernel , 2011, CAIP.

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

[14]  Ernesto Estrada.  Quantifying network heterogeneity. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Simone Severini,et al.  The von Neumann Entropy of Networks , 2008 .

[16]  N. Biggs.  Algebraic Graph Theory , 1974 .

[17]  Ernest Valveny,et al.  On the Relation Between the Median and the Maximum Common Subgraph of a Set of Graphs , 2007, GbRPR.

[18]  Kaspar Riesen,et al.  Graph Embedding in Vector Spaces by Means of Prototype Selection , 2007, GbRPR.

[19]  Edwin R. Hancock,et al.  An Information Theoretic Approach to Learning Generative Graph Prototypes , 2011, SIMBAD.

[20]  Edwin R. Hancock,et al.  Learning shape-classes using a mixture of tree-unions , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[21]  Frank Nielsen,et al.  Fitting the Smallest Enclosing Bregman Ball , 2005, ECML.

[22]  Edwin R. Hancock,et al.  A spectral approach to learning structural variations in graphs , 2003, Pattern Recognit..

[23]  Edwin R. Hancock,et al.  What Is the Complexity of a Network? The Heat Flow-Thermodynamic Depth Approach , 2010, SSPR/SPR.

[24]  Ulrike von Luxburg,et al.  Influence of graph construction on graph-based clustering measures , 2008, NIPS.

[25]  Edwin R. Hancock,et al.  A Riemannian approach to graph embedding , 2007, Pattern Recognit..

[26]  S. Severini,et al.  The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States , 2004, quant-ph/0406165.

[27]  Miguel Cazorla,et al.  Region and constellations based categorization of images with unsupervised graph learning , 2009, Image Vis. Comput..

[28]  J. Crutchfield,et al.  Measures of statistical complexity: Why? , 1998 .

[29]  M. Randic.  Characterization of molecular branching , 1975 .

[30]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[31]  Edwin R. Hancock,et al.  Graph embedding using tree edit-union , 2007, Pattern Recognit..