ABSTRACT
In this paper, we proposed a network comparison method based on the mathematical theory of diffusion over manifolds using random walks over graphs. We show that our method not only distinguishes between graphs with different degree distributions, but also different graphs with the same degree distributions. We compare the undirected power law graphs generated by Barabasi-Albert model and directed power law graphs generated by Krapivsky's model to the random graphs generated by Erdos-Renyi model. We also compare power law graphs generated by four different generative models with the same degree distribution.
- M. Vandenberg and P. B. Gilkey. Heat content asymptotics of a riemannian manifold with boundary. Journal of Functional Analysis, 120:48--71, 1994.Google ScholarCross Ref
- S. Butler. Interlacing for weighted graphs using the normalized laplacian. Electronic Journal of Linear Algebra, 16:90--98, 2007.Google ScholarCross Ref
- F. Chung. Spectral Graph Theory. American Mathematical Society, 1997.Google Scholar
- F. Chung, L. Lu, and V. Vu. Spectra of random graphs with given expected degrees. Proceedings of the National Academy of Sciences of the United States of Amherica, 100(11):6313--6318, 2003.Google ScholarCross Ref
- W. Gong. Can one hear the shape of a concept? In Proceedings of the 31st Chinese Control Conference (Plenary Lecture), pages 22--26, Hefei, China, July 2012.Google Scholar
- W. Gong. Transient response functions for graph structure addressable memory. In Proceedings of the 52th IEEE Conference on Decision and Control, Florence, Italy, December 2013.Google ScholarCross Ref
- J.H.H.Grisi-Filho, R. Ossada, F. Ferreira, and M. Amaku. Scale-free networks with the same degree distribution: Different structural properties. Physics Research International, 2013:234180(1--9), 2013.Google ScholarCross Ref
- T. Kalisky, R. Cohen, D. Ben-Avraham, and S. Havlin. Tomography and stability of complex networks. Complex Networks, 650:3--34, 2004.Google ScholarCross Ref
- D. Koutra, A. Parikh, A. Ramdas, and J. Xiang, "Algorithms for graph similarity and subgraph matching," 2011, Available at https://www.cs.cmu.edu/ jingx/docs/DBreport.pdf.Google Scholar
- P. L. Krapivsky, G. J. Rodgers, and S. Redner. Degree distributions of growing networks. Physical Review Letters, 86:5401--5404, 2001.Google ScholarCross Ref
- P. Mcdonald and R. Meyers. Diffusion on graphs, poisson problems and spectral geometry. Transaction of the American Mathematical Society, 354(12):5111--5136, 2002.Google ScholarCross Ref
- P. Mcdonald and R. Meyers. Isospectral polygons, planar graphs and heat content. Proceedings of the American Mathematical Society, 131(11):3589--3599, 2003.Google ScholarCross Ref
- A. Mohaisen, A. Yun, and Y. Kim. Measuring the mixing time of social graphs. Internet Measurement Conference, pages 383--389, 2010. Google ScholarDigital Library
- M. Molloy and B. Reed. The size of the giant component of a random graph with a given degree sequence. Combinatorics, Probability and Computing, 7(03):295--305, 1998. Google ScholarDigital Library
- J. Park and K. Kim. The heat energy content of a riemannian manifold. Trends in Mathematics, Information Center for Mathematical Sciences, 5(2):125--129, 2002.Google Scholar
Index Terms
- Complex network comparison using random walks
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