Scalable Lattice Influence Maximization

Abstract

Influence maximization is the task of finding k 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 k seed nodes, one selects a vector x = (x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⋯</sub> x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> ) from a discrete space X called a lattice, where xj corresponds to the jth marketing strategy and x represents a marketing strategy mix. Each strategy mix x has probability h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> (x) to activate a node u as a seed. LIM is the task of finding a strategy mix under the constraint Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> ≤ k 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 IMM-PRR is based on partial coverage on reverse-reachable sets, and the other IMM-VSN is based on incorporating virtual strategy nodes. IMM-PRR 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. IMM-VSN 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 IMM-PRR and IMM-VSN guarantee 1 - 1/e - ε approximation for small ε > 0. 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 1/2 - ε approximation ratio with the same time complexity. Empirically, through extensive tests, we demonstrate that IMM-VSN runs faster than IMM-PRR and much faster than other baseline algorithms while providing the same level of influence spread. We conclude that IMM-VSN is the best one for models with independent strategy activations, while IMM-PRR works for general modes without this assumption.