Lossless Representation of Graphs using Distributions

We consider complete graphs with edge weights and/or node weights taking values in some set. In the first part of this paper, we show that a large number of graphs are completely determined, up to isomorphism, by the distribution of their sub-triangles. In the second part, we propose graph representations in terms of one-dimensional distributions (e.g., distribution of the node weights, sum of adjacent weights, etc.). For the case when the weights of the graph are real-valued vectors, we show that all graphs, except for a set of measure zero, are uniquely determined, up to isomorphism, from these distributions. The motivating application for this paper is the problem of browsing through large sets of graphs.

[1]  Leon W. Cohen,et al.  Conference Board of the Mathematical Sciences , 1963 .

[2]  King-Sun Fu,et al.  A graph distance measure for image analysis , 1984, IEEE Transactions on Systems, Man, and Cybernetics.

[3]  Radu Horaud,et al.  Symbolic image matching by simulated annealing , 1990, BMVC.

[4]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[5]  Jin Hyung Kim,et al.  Automatic Input of Logic Diagrams by Recognizing Loop-Symbols and Rectilinear Connections , 1994, Int. J. Pattern Recognit. Artif. Intell..

[6]  Horst Bunke,et al.  Efficient Subgraph Isomorphism Detection: A Decomposition Approach , 2000, IEEE Trans. Knowl. Data Eng..

[7]  S. Ulam.  A collection of mathematical problems , 1960 .

[8]  Mireille Boutin,et al.  On reconstructing n-point configurations from the distribution of distances or areas , 2003, Adv. Appl. Math..

[9]  David G. Kirkpatrick,et al.  A Theoretical Analysis of Various Heuristics for the Graph Isomorphism Problem , 1980, SIAM J. Comput..

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

[11]  Brian Falkenhainer,et al.  The Structure-Mapping Engine: Algorithm and Examples , 1989, Artif. Intell..

[12]  William J. Christmas,et al.  PROBABILISTIC RELAXATION FOR MATCHING OF SYMBOLIC STRUCTURES , 1993 .

[13]  Aurelio López-López,et al.  Conceptual graph matching: a flexible algorithm and experiments , 1992, J. Exp. Theor. Artif. Intell..

[14]  Georg Gati,et al.  Further annotated bibliography on the isomorphism disease , 1979, J. Graph Theory.

[15]  Ronen Feldman,et al.  The Data Mining and Knowledge Discovery Handbook , 2005 .

[16]  Steven Skiena,et al.  Implementing discrete mathematics - combinatorics and graph theory with Mathematica , 1990 .

[17]  Raghu Ramakrishnan,et al.  When Is Nearest Neighbors Indexable? , 2005, ICDT.

[18]  Julian R. Ullmann,et al.  An Algorithm for Subgraph Isomorphism , 1976, J. ACM.

[19]  Edward K. Wong.  THREE-DIMENSIONAL OBJECT RECOGNITION BY ATTRIBUTED GRAPHS , 1990 .

[20]  Mireille Boutin,et al.  Which Point Configurations Are Determined by the Distribution of their Pairwise Distances? , 2003, Int. J. Comput. Geom. Appl..

[21]  King-Sun Fu,et al.  A distance measure between attributed relational graphs for pattern recognition , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[22]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[23]  PetrouMaria,et al.  Structural Matching in Computer Vision Using Probabilistic Relaxation , 1995 .

[25]  Radu Horaud,et al.  Stereo Correspondence Through Feature Grouping and Maximal Cliques , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[26]  Brendan D. McKay,et al.  Small graphs are reconstructible , 1997, Australas. J Comb..

[27]  L. Kitchen.  Discrete Relaxation for Matching Relational Structures , 1978 .

[28]  William J. Christmas,et al.  Structural Matching in Computer Vision Using Probabilistic Relaxation , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[29]  King-Sun Fu,et al.  Error-Correcting Isomorphisms of Attributed Relational Graphs for Pattern Analysis , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[30]  D. Corneil,et al.  An Efficient Algorithm for Graph Isomorphism , 1970, JACM.

[31]  Jonathan Goldstein,et al.  When Is ''Nearest Neighbor'' Meaningful? , 1999, ICDT.

[32]  Shinji Umeyama,et al.  An Eigendecomposition Approach to Weighted Graph Matching Problems , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[33]  Robert M. Haralick,et al.  Structural Descriptions and Inexact Matching , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[34]  Horst Bunke,et al.  A decision tree approach to graph and subgraph isomorphism detection , 1999, Pattern Recognit..

[35]  Derek G. Corneil,et al.  The graph isomorphism disease , 1977, J. Graph Theory.

[36]  Salih O. Duffuaa,et al.  A Linear Programming Approach for the Weighted Graph Matching Problem , 1993, IEEE Trans. Pattern Anal. Mach. Intell..