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We study the $r$-complex contagion influence maximization problem. In theinfluence maximization problem, one chooses a fixed number of initial seeds ina social network to maximize the spread of their influence. In the $r$-complexcontagion model, each uninfected vertex in the network becomes infected if ithas at least $r$ infected neighbors. In this paper, we focus on a random graphmodel named the stochastic hierarchical blockmodel, which is a special case ofthe well-studied stochastic blockmodel. When the graph is not exceptionallysparse, in particular, when each edge appears with probability$\omega(n^{-(1+1/r)})$, under certain mild assumptions, we prove that theoptimal seeding strategy is to put all the seeds in a single community. Thismatches the intuition that in a nonsubmodular cascade model placing seeds neareach other creates synergy. However, it sharply contrasts with the intuitionfor submodular cascade models (e.g., the independent cascade model and thelinear threshold model) in which nearby seeds tend to erode each others'effects. Our key technique is a novel time-asynchronized coupling of fourcascade processes. Finally, we show that this observation yields a polynomialtime dynamic programming algorithm which outputs optimal seeds if each edgeappears with a probability either in $\omega(n^{-(1+1/r)})$ or in $o(n^{-2})$.

Think Globally, Act Locally: On the Optimal Seeding for Nonsubmodular Influence Maximization