Abstract
In network analysis, one of the most important structures is the k-core: the maximal set of nodes such that each node in the k-core has at least k neighbors within the core. Recently, the notion of the skeletal k-core– a minimal subgraph that preserves the core structure of the graph– has attracted attention. However, the literature to date has contained only a biased greedy heuristic for sampling skeletal cores, which resulted in a skewed analysis of the network. In this work, we introduce a novel MCMC algorithm for sampling skeletal cores uniformly at random, as well as a novel algorithm for estimating the size of the space of skeletal k-cores, which, as we show, is important for understanding the core resilience of the network. With these algorithms, we demonstrate the relationship between resilience of the network and the core structure of the graph and suggest fast heuristics for evaluating graph structure from a skeletal cores perspective. We show that the normalized number of skeletal cores in the graph correlates with the resilience of k-core towards edge deletion attacks.
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Notes
- 1.
Note that this sometimes overestimates the desired value, as loss of an edge (u, v) can trigger reductions in core numbers of other nodes, and thus lower the core number of other neighbors of u; however, computing the exact value is more computationally intensive.
- 2.
Experimentally this bound proved to be very loose, which may reduce the rate of convergence.
- 3.
Extended version and source code are available at https://github.com/honcharov-danylo/extended_skeletal.
- 4.
Such convergence diagnostics for MCMC methods don‘t guarantee convergence, and should be seen as a type of statistical analysis [6].
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Acknowledgements
Honcharov and Soundarajan were supported by NSF Award #1908048. Sarıyüce was supported by NSF Award #1910063.
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Honcharov, D., Sarıyüce, A.E., Laishram, R., Soundarajan, S. (2023). Skeletal Cores and Graph Resilience. In: Koutra, D., Plant, C., Gomez Rodriguez, M., Baralis, E., Bonchi, F. (eds) Machine Learning and Knowledge Discovery in Databases: Research Track. ECML PKDD 2023. Lecture Notes in Computer Science(), vol 14171. Springer, Cham. https://doi.org/10.1007/978-3-031-43418-1_18
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