[Paper Review] Online Influence Maximization under Linear Threshold Model

Shuai Li, Fang Kong+3 more|arXiv (Cornell University)|Nov 12, 2020

Advanced Bandit Algorithms ResearchDecision Sciences44 references21 citations

TL;DR

This paper proposes the first online influence maximization algorithm under the Linear Threshold (LT) model with node-level feedback, introducing the ${\tt LT\mathchar 45\relax LinUCB}$ algorithm that leverages linear bandit principles to handle aggregated, group-based observations. It establishes a regret bound of $\tilde{O}(\mathrm{poly}(m)\sqrt{T})$, the first such theoretical result for OIM under the LT model, and further presents a simpler ${\tt OIM\mathchar 45\relax ETC}$ algorithm with $O(\mathrm{poly}(m)T^{2/3})$ regret.

ABSTRACT

Online influence maximization (OIM) is a popular problem in social networks to learn influence propagation model parameters and maximize the influence spread at the same time. Most previous studies focus on the independent cascade (IC) model under the edge-level feedback. In this paper, we address OIM in the linear threshold (LT) model. Because node activations in the LT model are due to the aggregated effect of all active neighbors, it is more natural to model OIM with the node-level feedback. And this brings new challenge in online learning since we only observe aggregated effect from groups of nodes and the groups are also random. Based on the linear structure in node activations, we incorporate ideas from linear bandits and design an algorithm LT-LinUCB that is consistent with the observed feedback. By proving group observation modulated (GOM) bounded smoothness property, a novel result of the influence difference in terms of the random observations, we provide a regret of order $ ilde{O}(\mathrm{poly}(m)\sqrt{T})$, where $m$ is the number of edges and $T$ is the number of rounds. This is the first theoretical result in such order for OIM under the LT model. In the end, we also provide an algorithm OIM-ETC with regret bound $O(\mathrm{poly}(m)\ T^{2/3})$, which is model-independent, simple and has less requirement on online feedback and offline computation.

Motivation & Objective

• To address online influence maximization (OIM) under the Linear Threshold (LT) model, which better captures herd behavior in social networks than the independent cascade model.
• To model feedback at the node level, reflecting real-world scenarios where only overall node activations are observable, not individual edge influences.
• To overcome the challenge of observing only aggregated, random group effects on node activations in the LT model.
• To develop an online learning algorithm that adapts to unknown influence weights while maximizing influence spread over time.
• To establish the first theoretical regret bound of order $\tilde{O}(\mathrm{poly}(m)\sqrt{T})$ for OIM under the LT model with node-level feedback.

Proposed method

• Propose the ${\tt LT\mathchar 45\relax LinUCB}$ algorithm, which integrates linear bandit techniques to model influence weights as linear functions of observed node-level feedback.
• Introduce the group observation modulated (GOM) bounded smoothness property to characterize the influence difference between two weight vectors under random, aggregated node observations.
• Use a linear structure in the LT model to derive confidence intervals for influence weights based on cumulative node activation feedback.
• Design a UCB-style exploration strategy that balances uncertainty in weight estimates with influence maximization.
• Provide a model-agnostic ${\tt OIM\mathchar 45\relax ETC}$ algorithm that requires less feedback and offline computation, using a fixed exploration phase followed by exploitation.
• Prove regret bounds by relating influence spread differences to the $\ell_2$-norm of weight vector differences through the GOM property.

Experimental results

Research questions

• RQ1 Can online influence maximization be effectively formulated under the Linear Threshold model with only node-level feedback?
• RQ2 How can aggregated, group-based observations of node activations be leveraged to learn influence weights in a principled way?
• RQ3 What is the theoretical regret bound achievable for online influence maximization under the LT model with node-level feedback?
• RQ4 Can a simpler, model-agnostic algorithm achieve competitive regret with reduced feedback and computational requirements?
• RQ5 How does the GOM bounded smoothness property relate to influence spread differences under different weight vectors?

Key findings

• The ${\tt LT\mathchar 45\relax LinUCB}$ algorithm achieves a regret bound of $\tilde{O}(\mathrm{poly}(m)\sqrt{T})$, where $m$ is the number of edges and $T$ is the number of rounds, representing the first such theoretical result for OIM under the LT model.
• The group observation modulated (GOM) bounded smoothness property is established as a key technical contribution, enabling the regret analysis under node-level feedback.
• The ${\tt OIM\mathchar 45\relax ETC}$ algorithm achieves a regret bound of $O(\mathrm{poly}(m)T^{2/3})$, which is model-independent and requires less feedback and offline computation than ${\tt LT\mathchar 45\relax LinUCB}$.
• The regret bound $O(\mathrm{poly}(m)\log(T)/\Delta^{2})$ is also established for ${\tt OIM\mathchar 45\relax ETC}$, where $\Delta$ is the gap between the best and second-best seed set.
• The theoretical analysis confirms that the influence spread is Lipschitz continuous with respect to the $\ell_2$-norm of the weight vector, with a bound proportional to $mn\|w - w'\|_2$.
• The work generalizes online learning techniques from edge-level feedback in the IC model to node-level feedback in the LT model, enabling more realistic modeling of herd behavior.