[Paper Review] Message-Passing Methods for Complex Contagions

James P. Gleeson, Mason A. Porter|arXiv (Cornell University)|Mar 23, 2017

Complex Network Analysis Techniques4 citations

TL;DR

This paper develops message-passing methods to analytically predict global cascades in complex contagion dynamics on networks, using the Watts threshold model as a case study. It derives a cascade condition based on the spectral radius of a matrix derived from the nonbacktracking and degree matrices, enabling accurate prediction of whether a small initial seed can trigger a large-scale cascade, especially on real-world networks where configuration-model approximations fail.

ABSTRACT

Message-passing methods provide a powerful approach for calculating the expected size of cascades either on random networks (e.g., drawn from a configuration-model ensemble or its generalizations) asymptotically as the number $N$ of nodes becomes infinite or on specific finite-size networks. We review the message-passing approach and show how to derive it for configuration-model networks using the methods of (Dhar et al., 1997) and (Gleeson, 2008). Using this approach, we explain for such networks how to determine an analytical expression for a "cascade condition", which determines whether a global cascade will occur. We extend this approach to the message-passing methods for specific finite-size networks (Shrestha and Moore, 2014; Lokhov et al., 2015), and we derive a generalized cascade condition. Throughout this chapter, we illustrate these ideas using the Watts threshold model.

Motivation & Objective

• To develop analytical methods for predicting the size and occurrence of global cascades in complex contagion dynamics on networks.
• To extend message-passing techniques from infinite random networks to finite, real-world networks with complex topology.
• To derive a generalized cascade condition that accounts for network structure beyond degree distribution, especially in networks with strong clustering or short loops.
• To improve upon naive mean-field approximations by incorporating message-passing principles that respect the local tree-likeness of networks.
• To validate the method on real-world networks where configuration-model predictions fail, using the nonbacktracking matrix and its spectral properties.

Proposed method

• Derives message-passing equations for the Watts threshold model on configuration-model networks using the formalism of [1, 2, 3], accounting for node degrees and thresholds.
• Introduces a cascade condition based on the spectral radius of the matrix product DB, where D is a degree-weighted diagonal matrix and B is a nonbacktracking matrix.
• Applies the method to infinite networks to derive a critical threshold θ_crit that separates global cascade and no-cascade regimes.
• Adapts the approach to finite networks by solving a system of message-passing equations that track the probability of activation along paths.
• Uses the largest eigenvalue of the DB matrix as a predictor of cascade onset, replacing the simpler configuration-model threshold.
• Validates predictions against simulations on real-world networks, showing improved accuracy over degree-only models.

Experimental results

Research questions

• RQ1 Under what conditions does a small initial seed lead to a global cascade in complex contagion dynamics?
• RQ2 How can message-passing methods be adapted to predict cascade thresholds on finite, real-world networks with non-Poisson degree distributions?
• RQ3 Why do configuration-model predictions fail for certain real-world networks, and how can this be corrected?
• RQ4 What role does the nonbacktracking matrix play in determining the critical threshold for cascade propagation?
• RQ5 How does the spectral radius of the DB matrix relate to the onset of global cascades in complex contagions?

Key findings

• The cascade condition based on the spectral radius of the DB matrix accurately predicts global cascades on real-world networks, outperforming configuration-model approximations.
• For the 3-regular random graph, the configuration-model prediction and the spectral method yield identical critical thresholds, confirming accuracy in tree-like networks.
• On Facebook Caltech and Oklahoma networks, the two methods produce nearly identical results, indicating strong agreement in networks with low clustering.
• For Gowalla, PGP, and power grid networks, the configuration-model prediction significantly underestimates the true critical threshold, highlighting structural limitations of the mean-field approach.
• The spectral radius of the DB matrix provides a robust predictor of cascade onset, even in networks with high clustering or short loops where tree-like assumptions break down.
• The method reveals that network structure beyond degree distribution—particularly nonbacktracking connectivity—plays a crucial role in enabling or suppressing global cascades.