Scalable Lattice Influence Maximization

Influence maximization is the task of finding <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> seed nodes in a social network such that the expected number of activated nodes in the network (under certain influence propagation model), referred to as the influence spread, is maximized. Lattice influence maximization (LIM) generalizes influence maximization such that, instead of selecting <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> seed nodes, one selects a vector <inline-formula> <tex-math notation="LaTeX">$\mathbf {x}=(x_{1}, \ldots, x_{d})$ </tex-math></inline-formula> from a discrete space <inline-formula> <tex-math notation="LaTeX">$\mathcal {X}$ </tex-math></inline-formula> called a lattice, where <inline-formula> <tex-math notation="LaTeX">$x_{j}$ </tex-math></inline-formula> corresponds to the <inline-formula> <tex-math notation="LaTeX">$j$ </tex-math></inline-formula>th marketing strategy and <inline-formula> <tex-math notation="LaTeX">$\mathbf {x}$ </tex-math></inline-formula> represents a marketing strategy mix. Each strategy mix <inline-formula> <tex-math notation="LaTeX">$\mathbf {x}$ </tex-math></inline-formula> has probability <inline-formula> <tex-math notation="LaTeX">$h_{u}(\mathbf {x})$ </tex-math></inline-formula> to activate a node <inline-formula> <tex-math notation="LaTeX">$u$ </tex-math></inline-formula> as a seed. LIM is the task of finding a strategy mix under the constraint <inline-formula> <tex-math notation="LaTeX">$\sum _{j} x_{j} \le k$ </tex-math></inline-formula> such that its influence spread is maximized. We adopt the reverse influence sampling (RIS) approach and design scalable algorithms for LIM. We explore two complementary design choices: one algorithm <italic>IMM-PRR</italic> is based on partial coverage on reverse-reachable sets, and the other <italic>IMM-VSN</italic> is based on incorporating virtual strategy nodes. <italic>IMM-PRR</italic> can be applied as a general solution to LIM, and we further improve its efficiency for a large family of models where each strategy independently activates seed nodes. <italic>IMM-VSN</italic> is explicitly designed for the case of independent strategy activation, and it uses virtual nodes to represent strategies to reduce LIM back to the original influence maximization problem. We prove that both <italic>IMM-PRR</italic> and <italic>IMM-VSN</italic> guarantee <inline-formula> <tex-math notation="LaTeX">$1-1/e-\varepsilon $ </tex-math></inline-formula> approximation for small <inline-formula> <tex-math notation="LaTeX">$\varepsilon > 0$ </tex-math></inline-formula>. We further extend LIM to the partitioned budget case where strategies are partitioned into groups, each of which has a separate budget, and show that a minor variation of our algorithms would achieve <inline-formula> <tex-math notation="LaTeX">$1/2 -\varepsilon $ </tex-math></inline-formula> approximation ratio with the same time complexity. Empirically, through extensive tests, we demonstrate that <italic>IMM-VSN</italic> runs faster than <italic>IMM-PRR</italic> and much faster than other baseline algorithms while providing the same level of influence spread. We conclude that <italic>IMM-VSN</italic> is the best one for models with independent strategy activations, while <italic>IMM-PRR</italic> works for general modes without this assumption.