Abstract
Binomial random intersection graphs can be used as parsimonious statistical models of large and sparse networks, with one parameter for the average degree and another for transitivity, the tendency of neighbours of a node to be connected. This paper discusses the estimation of these parameters from a single observed instance of the graph, using moment estimators based on observed degrees and frequencies of 2-stars and triangles. The observed data set is assumed to be a subgraph induced by a set of \(n_0\) nodes sampled from the full set of n nodes. We prove the consistency of the proposed estimators by showing that the relative estimation error is small with high probability for \(n_0 \gg n^{2/3} \gg 1\). As a byproduct, our analysis confirms that the empirical transitivity coefficient of the graph is with high probability close to the theoretical clustering coefficient of the model.
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Notes
- 1.
For number sequences \(f=f_\nu \) and \(g=g_\nu \) indexed by integers \(\nu \ge 1\), we denote \(f \sim g\) if \(f_\nu /g_\nu \rightarrow 1\) and \(f \ll g\) if \(f_\nu /g_\nu \rightarrow 0\) as \(\nu \rightarrow \infty \). The scale parameter is usually omitted.
- 2.
subgraphs isomorphic to the graph \(K_3\) with \(V(K_3) = \{1,2,3\}\) and \(E(K_3) = \{12,13,23\}\).
- 3.
subgraphs isomorphic to the graph \(S_2\) with \(V(S_2) = \{1,2,3\}\) and \(E(S_2) = \{12,13\}\).
- 4.
For clarity, we write 12 and 123 as shorthands of the sets \(\{1,2\}\) and \(\{1,2,3\}\).
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Acknowledgments
Part of this work has been financially supported by the Emil Aaltonen Foundation, Finland. We thank Mindaugas Bloznelis for helpful discussions, and the two anonymous reviewers for helpful comments.
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Karjalainen, J., Leskelä, L. (2017). Moment-Based Parameter Estimation in Binomial Random Intersection Graph Models. In: Bonato, A., Chung Graham, F., Prałat, P. (eds) Algorithms and Models for the Web Graph. WAW 2017. Lecture Notes in Computer Science(), vol 10519. Springer, Cham. https://doi.org/10.1007/978-3-319-67810-8_1
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