Abstract
We present a rigorous and precise analysis of the degree distribution in a dynamic graph model introduced by Solé et al. in which nodes are added according to a duplication-divergence mechanism, i.e. by iteratively copying a node and then randomly inserting and deleting some edges for a copied node. This graph model finds many applications since it well captures the growth of some real-world processes e.g. biological or social networks. However, there are only a handful of rigorous results concerning this model. In this paper we present rigorous results concerning the degree distribution.
We focus on two related problems: the expected value and large deviation for the degree of a fixed node through the evolution of the graph and the expected value and large deviation of the average degree in the graph. We present exact and asymptotic results showing that both quantities may decrease or increase over time depending on the model parameters. Our findings are a step towards a better understanding of the overall graph behaviors, especially, degree distribution, symmetry, and compression, important open problems in this area.
This work was supported by NSF Center for Science of Information (CSoI) Grant CCF-0939370, and in addition by NSF Grant CCF-1524312, and National Science Center, Poland, Grant 2018/31/B/ST6/01294. This work was also supported by NSF Grant DMS1661063.
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Notes
- 1.
We choose \(\lambda _t = \varepsilon _t \left( \frac{t}{t_0}\right) ^{-(2 p - 1) \left( 1 + O(\varepsilon _t))\right) }\) so that \(\lambda _{t_0} \le \varepsilon _t\).
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Frieze, A., Turowski, K., Szpankowski, W. (2020). Degree Distribution for Duplication-Divergence Graphs: Large Deviations. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_18
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DOI: https://doi.org/10.1007/978-3-030-60440-0_18
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