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# Using Graph and Vertex Entropy to Compare Empirical Graphs with Theoretical Graph Models†

1
Institute of Informatics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
2
Institute of Computing Science, Poznań University of Technology, Piotrowo 2, 60-965 Poznań, Poland
*
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This paper is an extended version of our paper published in 2nd International Electronic Conference on Entropy and Its Applications, 15–30 November 2015.
Academic Editors: Dawn E. Holmes and Kevin H. Knuth
Entropy 2016, 18(9), 320; https://doi.org/10.3390/e18090320
Received: 31 May 2016 / Revised: 19 July 2016 / Accepted: 24 August 2016 / Published: 5 September 2016
Over the years, several theoretical graph generation models have been proposed. Among the most prominent are: the Erdős–Renyi random graph model, Watts–Strogatz small world model, Albert–Barabási preferential attachment model, Price citation model, and many more. Often, researchers working with real-world data are interested in understanding the generative phenomena underlying their empirical graphs. They want to know which of the theoretical graph generation models would most probably generate a particular empirical graph. In other words, they expect some similarity assessment between the empirical graph and graphs artificially created from theoretical graph generation models. Usually, in order to assess the similarity of two graphs, centrality measure distributions are compared. For a theoretical graph model this means comparing the empirical graph to a single realization of a theoretical graph model, where the realization is generated from the given model using an arbitrary set of parameters. The similarity between centrality measure distributions can be measured using standard statistical tests, e.g., the Kolmogorov–Smirnov test of distances between cumulative distributions. However, this approach is both error-prone and leads to incorrect conclusions, as we show in our experiments. Therefore, we propose a new method for graph comparison and type classification by comparing the entropies of centrality measure distributions (degree centrality, betweenness centrality, closeness centrality). We demonstrate that our approach can help assign the empirical graph to the most similar theoretical model using a simple unsupervised learning method. View Full-Text
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Kajdanowicz, T.; Morzy, M. Using Graph and Vertex Entropy to Compare Empirical Graphs with Theoretical Graph Models. Entropy 2016, 18, 320.