[Paper Review] On the Convexity of Latent Social Network Inference

Seth A. Myers, Jure Leskovec|arXiv (Cornell University)|Oct 26, 2010

Complex Network Analysis TechniquesPhysics and Astronomy27 references174 citations

TL;DR

This paper proposes a convex optimization framework for inferring latent social networks from diffusion data—where only infection times are observed—by modeling contagion spread as a probabilistic process and using l1-regularized maximum likelihood estimation. The method achieves near-perfect network recovery and scales efficiently to thousands of nodes, outperforming prior approaches in accuracy and optimality guarantees.

ABSTRACT

In many real-world scenarios, it is nearly impossible to collect explicit social network data. In such cases, whole networks must be inferred from underlying observations. Here, we formulate the problem of inferring latent social networks based on network diffusion or disease propagation data. We consider contagions propagating over the edges of an unobserved social network, where we only observe the times when nodes became infected, but not who infected them. Given such node infection times, we then identify the optimal network that best explains the observed data. We present a maximum likelihood approach based on convex programming with a l1-like penalty term that encourages sparsity. Experiments on real and synthetic data reveal that our method near-perfectly recovers the underlying network structure as well as the parameters of the contagion propagation model. Moreover, our approach scales well as it can infer optimal networks of thousands of nodes in a matter of minutes.

Motivation & Objective

• To address the challenge of inferring unobserved social networks when only node infection times are available, without observing who infected whom.
• To develop a general, optimal, and scalable method for latent network inference that does not assume homogeneous edge transmission probabilities.
• To overcome limitations of prior heuristic or threshold-based methods by formulating the problem as a convex optimization task with guaranteed optimality.
• To enable accurate recovery of both network topology and edge transmission parameters from real-world diffusion cascades.

Proposed method

• Formulates a generative probabilistic model of contagion spread on a latent network, where infection times are observed but transmission paths are hidden.
• Derives the maximum likelihood estimation problem for the latent network structure and edge transmission parameters.
• Transforms the non-convex likelihood maximization into a convex optimization problem via a series of mathematical reformulations.
• Introduces an l1-like penalty term to encourage sparsity in the inferred network, promoting realistic, low-density structures.
• Solves the resulting convex program efficiently using standard convex optimization techniques, enabling scalability to large networks.
• Employs a power-law or Weibull distribution model for transmission times, depending on the data type, to parameterize the diffusion process.

Experimental results

Research questions

• RQ1 Can we infer the underlying social network structure from only the times when nodes became infected, without observing transmission paths?
• RQ2 Is it possible to jointly estimate both the network topology and the transmission probabilities between nodes in a way that guarantees optimality?
• RQ3 How does the proposed convex optimization approach compare in accuracy and scalability to existing heuristic or threshold-based network inference methods?
• RQ4 To what extent can the method recover the true network structure and edge weights when applied to real-world diffusion data with unknown transmission dynamics?

Key findings

• The method achieves a break-even point of approximately 0.95 on both synthetic and real-world email and collaboration networks, indicating near-perfect precision-recall performance in network recovery.
• Edge transmission probability estimation error is less than 0.03 on synthetic and real datasets, demonstrating high accuracy in parameter recovery.
• The approach recovers the underlying network structure with high fidelity even when only a small number of cascades are observed, as shown on a 593-node email network.
• On a real recommendation network of 275 users and 1,522 edges, the method achieves a break-even point of 0.74, significantly outperforming NetInf (0.55) in precision-recall.
• The algorithm scales efficiently, inferring optimal networks of up to thousands of nodes in minutes, and processed a 275-node recommendation network in under 20 seconds.
• The method guarantees global optimality due to its convex formulation, unlike prior approximation-based approaches such as NetInf.