WINE 2016: Web and Internet Economics pp 459-472

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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10123)

## 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 $$n^{\frac{\alpha -2}{\alpha -1}}$$.

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