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Data Mining When Each Data Point is a Network

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Patterns of Dynamics (PaDy 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 205))

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Abstract

We discuss the problem of extending data mining approaches to cases in which data points arise in the form of individual graphs. Being able to find the intrinsic low-dimensionality in ensembles of graphs can be useful in a variety of modeling contexts, especially when coarse-graining the detailed graph information is of interest. One of the main challenges in mining graph data is the definition of a suitable pairwise similarity metric in the space of graphs. We explore two practical solutions to solving this problem: one based on finding subgraph densities, and one using spectral information. The approach is illustrated on three test data sets (ensembles of graphs); two of these are obtained from standard literature graph generating algorithms, while the graphs in the third example are sampled as dynamic snapshots from an evolving network simulation. We further combine these approaches with equation free techniques, demonstrating how such data mining can enhance scientific computation of network evolution dynamics.

To Bernold Fiedler, with admiration for his choice of research problems in mathematics and modeling, and for what he has taught us about them.

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Notes

  1. 1.

    Note that an alternative equivalent way to define the similarity measure would be to directly compare the contribution of the different eigenvectors to \(S_i\) instead of summing the contributions and then using different values of \(\lambda \). However, it is difficult to generalize this approach to cases where there are graphs of varying sizes.

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Acknowledgements

The work of IGK was partially supported by the US National Science Foundation, as well as by AFOSR (Dr. Darema) and DARPA contract HR0011-16-C-0016.

Author information

Authors and Affiliations

  1. Department of Chemical & Biological Engineering, Princeton University, Present Address: 731 Lexington Avenue, New York, NY, 10022, USA

    Karthikeyan Rajendran

  2. Theory Group, Department of Computer Science, University of Toronto, 10 King’s College Road, Toronto, ON, M5S 3G4, Canada

    Assimakis Kattis

  3. Department of Chemical & Biological Engineering, Princeton University, 41 Olden Street, Princeton, NJ, 08544, USA

    Alexander Holiday

  4. Machine Learning Group, Computer Science & Statistics, University of Chicago, Ryerson 257B, 1100 E. 58th Street, Chicago, Il, 60637, USA

    Risi Kondor

  5. A319 Engineering Quad, Department of Chemical & Biological Engineering & Program in Applied & Computational Mathematics, Princeton University Technische Universität München - Institute for Advanced Study; Zuse Institut Berlin, 41 Olden Street, Princeton, NJ, 08544, USA

    Ioannis G. Kevrekidis

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  1. Karthikeyan Rajendran
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  2. Assimakis Kattis
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  3. Alexander Holiday
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  5. Ioannis G. Kevrekidis
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Correspondence to Karthikeyan Rajendran .

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Editors and Affiliations

  1. Mathematical Institute, Free University of Berlin, Berlin, Germany

    Pavel Gurevich

  2. Mathematical Institute, Free University of Berlin, Berlin, Germany

    Juliette Hell

  3. Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA

    Björn Sandstede

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Arnd Scheel

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Rajendran, K., Kattis, A., Holiday, A., Kondor, R., Kevrekidis, I.G. (2017). Data Mining When Each Data Point is a Network. In: Gurevich, P., Hell, J., Sandstede, B., Scheel, A. (eds) Patterns of Dynamics. PaDy 2016. Springer Proceedings in Mathematics & Statistics, vol 205. Springer, Cham. https://doi.org/10.1007/978-3-319-64173-7_17

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