[Paper Review] Learning Networks from Random Walk-Based Node Similarities

Jeremy G. Hoskins, Cameron Musco+2 more|arXiv (Cornell University)|Jan 23, 2018

Complex Network Analysis Techniques47 references3 citations

TL;DR

This paper proposes a method to reconstruct social network graphs from a small, noisy subset of effective resistance measurements—random walk-based node similarities—using convex and non-convex optimization. It demonstrates that even 10–25% of noisy effective resistances enable recovery of over 20% of true edges and accurate approximation of all pairwise similarities, revealing major privacy risks in releasing such data.

ABSTRACT

Digital presence in the world of online social media entails significant privacy risks. In this work we consider a privacy threat to a social network in which an attacker has access to a subset of random walk-based node similarities, such as effective resistances (i.e., commute times) or personalized PageRank scores. Using these similarities, the attacker's goal is to infer as much information as possible about the underlying network, including any remaining unknown pairwise node similarities and edges. For the effective resistance metric, we show that with just a small subset of measurements, the attacker can learn a large fraction of edges in a social network, even when the measurements are noisy. We also show that it is possible to learn a graph which accurately matches the underlying network on all other effective resistances. This second observation is interesting from a data mining perspective, since it can be expensive to accurately compute all effective resistances. As an alternative, our graphs learned from just a subset of approximate effective resistances can be used as surrogates in a wide range of applications that use effective resistances to probe graph structure, including for graph clustering, node centrality evaluation, and anomaly detection. We obtain our results by formalizing the graph learning objective mathematically, using two optimization problems. One formulation is convex and can be solved provably in polynomial time. The other is not, but we solve it efficiently with projected gradient and coordinate descent. We demonstrate the effectiveness of these methods on a number of social networks obtained from Facebook. We also discuss how our methods can be generalized to other random walk-based similarities, such as personalized PageRank. Our code is available at https://github.com/cnmusco/graph-similarity-learning.

Motivation & Objective

• To investigate how much structural information about a network can be inferred from a small, noisy subset of random walk-based node similarities, particularly effective resistances.
• To develop efficient optimization methods for learning a graph that matches the original on all pairwise effective resistances using only partial measurements.
• To evaluate the privacy implications of releasing effective resistances or similar metrics, given their potential to expose sensitive network structure.
• To enable scalable graph mining by using learned surrogate graphs when computing all pairwise similarities is infeasible.

Proposed method

• Formalize graph learning as an optimization problem minimizing the least squares error between observed and predicted effective resistances.
• Propose a convex formulation using semidefinite programming to provably recover the graph under ideal conditions.
• Develop a non-convex, non-convex optimization approach using projected gradient and coordinate descent for scalable, efficient learning on large networks.
• Use block coordinate descent with random batches of 5,000 constraints to improve convergence and scalability on real-world graphs.
• Apply the least squares formulation to real Facebook ego networks with varying noise levels and sampling fractions.
• Normalize objective and error metrics to enable fair comparison across networks of different sizes.

Experimental results

Research questions

• RQ1 Can a small subset of noisy effective resistance measurements reveal a significant fraction of the true network structure, including edges and global topology?
• RQ2 How well can a learned graph approximate all pairwise effective resistances in the original network when only a fraction of them are observed?
• RQ3 To what extent does the quality of graph reconstruction depend on the number of sampled constraints and noise level in the measurements?
• RQ4 Can the proposed optimization methods scale to real-world social networks while maintaining high accuracy in edge recovery and similarity matching?
• RQ5 What are the privacy implications of releasing effective resistances or similar random walk-based similarities in online social networks?

Key findings

• Using only 10% of effective resistance measurements, the method recovered over 50% of true edges in small Facebook ego networks (FB Small A and FB Small C).
• For larger Facebook networks, the method recovered over 20% of true edges even with just 10% of the constraints, improving with lower noise and higher sampling fractions.
• Generalization error—measuring how well the learned graph matches the original on all effective resistances—was consistently low, indicating accurate global structure recovery.
• The least squares formulation outperformed the convex semidefinite program in practice, especially for small constraint sets, due to better scalability and convergence.
• The heaviest edges in the learned graphs aligned closely with true edges in the original network, as visually confirmed on the Grid graph, where the 224 heaviest edges matched the grid structure exactly.
• In some cases, using fewer constraints improved edge recovery, suggesting that optimization convergence may be more effective with sparser, higher-quality measurement sets.