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 $$\alpha \in 2, 3$$α∈2,3, then with high probability, the single seed of the highest degree node will infect a constant fraction of the graph within time $$O\left \log ^{\frac{\alpha -2}{3-\alpha }}n\right $$Ologα-23-αn. This complements the prior work which shows that for $$\alpha > 3$$α>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 $$\alpha = 3$$α=3. The case where $$\alpha \in 2, 3$$α∈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 $$\omega \left n^{\frac{\alpha -2}{\alpha -1}}\right $$ωnα-2α-1 random seeds will infect a constant fraction of the graph within time $$O\left \log ^{\frac{\alpha -2}{3-\alpha }}n\right $$Ologα-23-αn with high probability. This complements prior work which shows that $$o\left n^{\frac{\alpha -2}{\alpha -1}}\right $$onα-2α-1 random seeds will have no effect with high probability, and this also establishes a threshold at $$n^{\frac{\alpha -2}{\alpha -1}}$$nα-2α-1.
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