[Paper Review] Learning Influence Functions from Incomplete Observations

Xinran He, Ke Xu+2 more|arXiv (Cornell University)|Nov 7, 2016

Bayesian Modeling and Causal Inference19 citations

TL;DR

This paper proposes a framework for learning influence functions in social networks from incomplete cascade observations, where node activations are randomly missing. By modeling missing data as a transformed graph and using reachability-based feature parametrization with a modified loss function, the method achieves proper and improper PAC learnability for DIC, DLT, and CIC diffusion models, reducing estimation error by nearly 20% on real-world data despite significant missingness.

ABSTRACT

We study the problem of learning influence functions under incomplete observations of node activations. Incomplete observations are a major concern as most (online and real-world) social networks are not fully observable. We establish both proper and improper PAC learnability of influence functions under randomly missing observations. Proper PAC learnability under the Discrete-Time Linear Threshold (DLT) and Discrete-Time Independent Cascade (DIC) models is established by reducing incomplete observations to complete observations in a modified graph. Our improper PAC learnability result applies for the DLT and DIC models as well as the Continuous-Time Independent Cascade (CIC) model. It is based on a parametrization in terms of reachability features, and also gives rise to an efficient and practical heuristic. Experiments on synthetic and real-world datasets demonstrate the ability of our method to compensate even for a fairly large fraction of missing observations.

Motivation & Objective

• Address the challenge of learning influence functions when node activations in cascades are incompletely observed, a common issue in real-world social networks.
• Establish theoretical PAC learnability of influence functions under incomplete observations for widely used diffusion models like DIC and DLT.
• Design an efficient, practical learning algorithm that compensates for missing activation data without requiring complete observations.
• Extend theoretical guarantees to continuous-time models (CIC) and robustness to uncertainty in retention rates.
• Demonstrate empirical effectiveness on synthetic and real-world datasets, showing significant improvement over baseline methods.

Proposed method

• Model incomplete observations as complete observations in a modified graph, where edge weights are adjusted based on the retention rate r to preserve expected influence propagation.
• Use a reachability feature-based parametrization of influence functions, inspired by Du et al. [3], to represent influence as a function of reachable nodes from seed sets.
• Optimize a modified loss function based on Natarajan et al. [17] that accounts for missing activations by downweighting or adjusting contributions from unobserved nodes.
• Prove proper PAC learnability for DIC and DLT models by reducing incomplete observation learning to complete observation learning in a transformed graph.
• Establish improper PAC learnability for DIC, DLT, and CIC models via the reachability feature approach, even when marginalizing over hidden variables is computationally infeasible.
• Provide sample complexity bounds that scale moderately with the inverse of the retention rate r, showing that missing data only moderately increases required sample size.

Experimental results

Research questions

• RQ1 Can influence functions be properly PAC-learned under incomplete observations where node activations are randomly missing?
• RQ2 Does the sample complexity of learning influence functions increase significantly when observations are incomplete?
• RQ3 Can an efficient, practical learning algorithm be designed for influence function learning under incomplete observations, especially for continuous-time models like CIC?
• RQ4 How robust is the method to uncertainty in the retention rate r, especially when r is not known exactly?
• RQ5 To what extent can the method compensate for large fractions of missing activations in real-world cascades?

Key findings

• The paper establishes proper PAC learnability of influence functions under the DIC and DLT models with sample complexity ˜O(¯r²n³m/ε²), showing that incomplete observations only moderately increase required sample size.
• For the CIC model, improper PAC learnability is achieved via reachability feature parametrization and a modified loss function, extending theoretical guarantees beyond discrete-time models.
• The method reduces estimation error by nearly 20% compared to the best baseline on the MemeTracker real-world dataset, even with substantial missing data.
• The approach remains robust to misestimation of the retention rate r, with performance stable under moderate uncertainty (e.g., η ≤ 0.2).
• Theoretical results extend to cases where the true retention rate lies within a known interval I = [¯r(1−η), ¯r(1+η)], with an additive error term depending on η, which remains small for small uncertainty.
• Empirical results confirm that performance is not significantly degraded when the true retention rate per node is independently perturbed around the estimated mean rate.