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Complex Contagions on Configuration Model Graphs with a Power-Law Degree Distribution

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Web and Internet Economics (WINE 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10123))

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Abstract

In this paper we analyze k-complex contagions (sometimes called bootstrap percolation) on configuration model graphs with a power-law distribution. Our main result is that if the power-law exponent \(\alpha \in (2, 3)\), then with high probability, the single seed of the highest degree node will infect a constant fraction of the graph within time \(O\left( \log ^{\frac{\alpha -2}{3-\alpha }}(n)\right) \). This complements the prior work which shows that for \(\alpha > 3\) boot strap percolation does not spread to a constant fraction of the graph unless a constant fraction of nodes are initially infected. This also establishes a threshold at \(\alpha = 3\).

The case where \(\alpha \in (2, 3)\) is especially interesting because it captures the exponent parameters often observed in social networks (with approximate power-law degree distribution). Thus, such networks will spread complex contagions even lacking any other structures.

We additionally show that our theorem implies that \(\omega (\left( n^{\frac{\alpha -2}{\alpha -1}}\right) \) random seeds will infect a constant fraction of the graph within time \(O\left( \log ^{\frac{\alpha -2}{3-\alpha }}(n)\right) \) with high probability. This complements prior work which shows that \(o\left( n^{\frac{\alpha -2}{\alpha -1}}\right) \) random seeds will have no effect with high probability, and this also establishes a threshold at \(n^{\frac{\alpha -2}{\alpha -1}}\).

G. Schoenebeck—Supported by National Science Foundation Career Award #1452915

F.-Y. Yu—Supported by National Science Foundation Algorithms in the Field Award #1535912.

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Authors and Affiliations

  1. University of Michigan, Ann Arbor, USA

    Grant Schoenebeck & Fang-Yi Yu

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  1. Grant Schoenebeck
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  2. Fang-Yi Yu
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Correspondence to Fang-Yi Yu .

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Editors and Affiliations

  1. School of Computer Science, McGill University, Montreal, Québec, Canada

    Yang Cai

  2. Department of Mathematics and Statistics, and School of Computer Science, McGill University, Montreal, Québec, Canada

    Adrian Vetta

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Schoenebeck, G., Yu, FY. (2016). Complex Contagions on Configuration Model Graphs with a Power-Law Degree Distribution. In: Cai, Y., Vetta, A. (eds) Web and Internet Economics. WINE 2016. Lecture Notes in Computer Science(), vol 10123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54110-4_32

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  • DOI: https://doi.org/10.1007/978-3-662-54110-4_32

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