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Connected Graphs with a Given Degree Sequence: Efficient Sampling, Correlations, Community Detection and Robustness

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Proceedings of NetSci-X 2020: Sixth International Winter School and Conference on Network Science (NetSci-X 2020)

Part of the book series: Springer Proceedings in Complexity ((SPCOM))

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Abstract

Random graph models can help us assess the significance of the structural properties of real complex systems. Given the value of a graph property and its value in a randomized ensemble, we can determine whether the property is explained by chance by comparing its real value to its value in the ensemble. The conclusions drawn with this approach obviously depend on the choice of randomization. We argue that keeping graphs in one connected piece, or component, is key for many applications where complex graphs are assumed to be connected either by definition (e.g. the Internet) or by construction (e.g. a crawled subset of the World-Wide Web obtained only by following hyperlinks). Using an heuristic to quickly sample the ensemble of small connected simple graphs with a fixed degree sequence, we investigate the significance of the structural patterns found in real connected graphs. We find that, in sparse networks, the connectedness constraint changes degree correlations, the outcome of community detection with modularity, and the predictions of percolation on the ensemble.

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Notes

  1. 1.

    Available at https://gitlab.com/jhring/connected_cm.

  2. 2.

    Many standard optimization methods are capable of fulfilling the role of optimizer: Spectral embedding [37, 39] and greedy maximization [40] are well-known examples.

  3. 3.

    This strategy works thanks to the regularization properties of the modularity: The partition in a single community (K = 1) has zero modularity due to the negative contributions of the null model, and so does the partition in K = n communities of 1 node since no two nodes share a community. It follows that there is an optimum K ∗ somewhere in between these two extremes.

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Acknowledgements

This work is supported by the James S. McDonnell Foundation (JGY) and Grant No. DMS-1622390 from the National Science Foundation (LHD). The authors contributed equally.

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Authors and Affiliations

  1. Vermont Complex Systems Center, University of Vermont, Burlington, VT, USA

    John H. Ring IV & Laurent Hébert-Dufresne

  2. Department of Computer Science, University of Vermont, Burlington, VT, USA

    John H. Ring IV & Laurent Hébert-Dufresne

  3. Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI, USA

    Jean-Gabriel Young

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  1. John H. Ring IV
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  3. Laurent Hébert-Dufresne
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Correspondence to Laurent Hébert-Dufresne .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University at Buffalo, State University of New York, Buffalo, NY, USA

    Naoki Masuda

  2. Department of Physics, Korea University, Seoul, Korea (Republic of)

    Kwang-Il Goh

  3. College of Computer and Information Science, Southwest University, Chongqing, China

    Tao Jia

  4. School of Commerce, Waseda University, Tokyo, Tokyo, Japan

    Junichi Yamanoi

  5. Department of Systems Science and Industrial Engineering, Binghamton University, State University of New York, Binghamton, NY, USA

    Hiroki Sayama

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Ring, J.H., Young, JG., Hébert-Dufresne, L. (2020). Connected Graphs with a Given Degree Sequence: Efficient Sampling, Correlations, Community Detection and Robustness. In: Masuda, N., Goh, KI., Jia, T., Yamanoi, J., Sayama, H. (eds) Proceedings of NetSci-X 2020: Sixth International Winter School and Conference on Network Science. NetSci-X 2020. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-38965-9_3

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