The Infinity Mirror Test for Analyzing the Robustness of Graph Generators

Graph generators learn a model from a source graph in order to generate a new graph that has many of the same properties. The learned models each have implicit and explicit biases built in, and its important to understand the assumptions that are made when generating a new graph. Of course, the differences between the new graph and the original graph, as compared by any number of graph properties, are important indicators of the biases inherent in any modelling task. But these critical differences are subtle and not immediately apparent using standard performance metrics. Therefore, we introduce the infinity mirror test for the analysis of graph generator performance and robustness. This stress test operates by repeatedly, recursively fitting a model to itself. A perfect graph generator would have no deviation from the original or ideal graph, however the implicit biases and assumptions that are cooked into the various models are exaggerated by the infinity mirror test allowing for new insights that were not available before. We show, via hundreds of experiments on 6 real world graphs, that several common graph generators do degenerate in interesting and informative ways. We believe that the observed degenerative patterns are clues to future development of better graph models.

[1]  Jennifer Neville,et al.  Fast Generation of Large Scale Social Networks While Incorporating Transitive Closures , 2012, 2012 International Conference on Privacy, Security, Risk and Trust and 2012 International Confernece on Social Computing.

[2]  Travis A. Jarrell,et al.  The Connectome of a Decision-Making Neural Network , 2012, Science.

[3]  Jon M. Kleinberg,et al.  Subgraph frequencies: mapping the empirical and extremal geography of large graph collections , 2013, WWW.

[4]  F. Chung,et al.  The average distances in random graphs with given expected degrees , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[5]  M. Newman,et al.  Mixing patterns in networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Ryan A. Rossi,et al.  Efficient Graphlet Counting for Large Networks , 2015, 2015 IEEE International Conference on Data Mining.

[7]  Christos Faloutsos,et al.  Kronecker Graphs: An Approach to Modeling Networks , 2008, J. Mach. Learn. Res..

[8]  Jennifer Neville,et al.  Incorporating Assortativity and Degree Dependence into Scalable Network Models , 2015, AAAI.

[9]  Tijana Milenkovic,et al.  Proper evaluation of alignment-free network comparison methods , 2015, Bioinform..

[10]  Yuval Shavitt,et al.  Efficient Counting of Network Motifs , 2010, 2010 IEEE 30th International Conference on Distributed Computing Systems Workshops.

[11]  Tamara G. Kolda,et al.  The Similarity Between Stochastic Kronecker and Chung-Lu Graph Models , 2011, SDM.

[12]  References , 1971 .

[13]  Tamara G. Kolda,et al.  A Scalable Generative Graph Model with Community Structure , 2013, SIAM J. Sci. Comput..

[14]  Christos Faloutsos,et al.  Graphs over time: densification laws, shrinking diameters and possible explanations , 2005, KDD '05.

[15]  Natasa Przulj,et al.  Biological network comparison using graphlet degree distribution , 2007, Bioinform..

[16]  F. Chung,et al.  Connected Components in Random Graphs with Given Expected Degree Sequences , 2002 .

[17]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

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

[19]  Jennifer Neville,et al.  Assortativity in Chung Lu Random Graph Models , 2014, SNAKDD'14.

[20]  Garry Robins,et al.  An introduction to exponential random graph (p*) models for social networks , 2007, Soc. Networks.

[21]  Yiming Yang,et al.  Introducing the Enron Corpus , 2004, CEAS.