Maximizing profit of multiple adoptions in social networks with a martingale approach

Information propagation plays an important role in social network, which helps shaping consumer’s purchasing decisions. Most of existing works focus on maximizing the influence of one product. But in our reality life, the majority of the companies produce various products for meeting customer needs. So it is important to learn about how to distribute the limited budget to maximize the companies profits. In this paper, we use the martingale technique to handle the Profit Maximization with Multiple Adoptions ( $$PM^{2}A$$ P M 2 A ) problem, which aims to identify a seed set for each product with overall activation cost at most B such that the expected total profit is maximized. We design a $$PM^{2}AM$$ P M 2 A M algorithm which returns a $$(\frac{1}{2}(1-\frac{1}{e^{2}})-\varepsilon )$$ ( 1 2 ( 1 - 1 e 2 ) - ε ) -approximate solution and runs in $$O\left( (k^{*}+\ell )(m+n)nqp_{max}\ln nq/ (\varepsilon ^{2}\cdot p_{min})\right) $$ O ( k ∗ + ℓ ) ( m + n ) n q p max ln n q / ( ε 2 · p min ) expected time.