Information-Theoretic Lower Bounds for Recovery of Diffusion Network Structures

TL;DR

This paper establishes fundamental lower bounds on the sample complexity for recovering diffusion network structures, demonstrating the optimality of existing algorithms in discrete time and raising questions for continuous time.

Contribution

It introduces new lower bounds for both discrete and continuous-time diffusion models, highlighting the optimality of certain algorithms and posing open questions.

Findings

Lower bound of order Ω(k log p) for discrete-time models

Lower bound of order Ω(k log p) for continuous-time models

Existing algorithms are statistically optimal in the discrete-time setting

Abstract

We study the information-theoretic lower bound of the sample complexity of the correct recovery of diffusion network structures. We introduce a discrete-time diffusion model based on the Independent Cascade model for which we obtain a lower bound of order $\Omega(k \log p)$, for directed graphs of $p$ nodes, and at most $k$ parents per node. Next, we introduce a continuous-time diffusion model, for which a similar lower bound of order $\Omega(k \log p)$ is obtained. Our results show that the algorithm of Pouget-Abadie et al. is statistically optimal for the discrete-time regime. Our work also opens the question of whether it is possible to devise an optimal algorithm for the continuous-time regime.