Learning to maximize global influence from local observations

We study a family online influence maximization problems where in a sequence of rounds $t=1,\ldots,T$, a decision maker selects one from a large number of agents with the goal of maximizing influence. Upon choosing an agent, the decision maker shares a piece of information with the agent, which information then spreads in an unobserved network over which the agents communicate. The goal of the decision maker is to select the sequence of agents in a way that the total number of influenced nodes in the network. In this work, we consider a scenario where the networks are generated independently for each $t$ according to some fixed but unknown distribution, so that the set of influenced nodes corresponds to the connected component of the random graph containing the vertex corresponding to the selected agent. Furthermore, we assume that the decision maker only has access to limited feedback: instead of making the unrealistic assumption that the entire network is observable, we suppose that the available feedback is generated based on a small neighborhood of the selected vertex. Our results show that such partial local observations can be sufficient for maximizing global influence. We model the underlying random graph as a sparse inhomogeneous Erd\H{o}s--R\'enyi graph, and study three specific families of random graph models in detail: stochastic block models, Chung--Lu models and Kronecker random graphs. We show that in these cases one may learn to maximize influence by merely observing the degree of the selected vertex in the generated random graph. We propose sequential learning algorithms that aim at maximizing influence, and provide their theoretical analysis in both the subcritical and supercritical regimes of all considered models.