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Message-Passing Methods for Complex Contagions

James P. Gleeson, Mason A. Porter

TL;DR

This work develops and compares analytical methods for predicting global cascades in complex contagions, focusing on the Watts threshold model. Starting from naive mean-field, it introduces a direction-aware message-passing framework on configuration-model networks that yields accurate predictions for the steady-state active fraction $rho_infty$ and a cascade condition in the infinitesimal-seed limit. It further extends to networks with degree-degree correlations through a matrix-based criterion and to finite-size networks via edge-based messages and a nonbacktracking structure, arriving at a spectral-radius condition $\rho(D B) > 1$. The results provide rigorous, scalable tools for assessing cascade likelihood and thresholds in real-world networks, with implications for understanding diffusion, influence, and contagion on complex topologies.

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.

Message-Passing Methods for Complex Contagions

TL;DR

This work develops and compares analytical methods for predicting global cascades in complex contagions, focusing on the Watts threshold model. Starting from naive mean-field, it introduces a direction-aware message-passing framework on configuration-model networks that yields accurate predictions for the steady-state active fraction and a cascade condition in the infinitesimal-seed limit. It further extends to networks with degree-degree correlations through a matrix-based criterion and to finite-size networks via edge-based messages and a nonbacktracking structure, arriving at a spectral-radius condition . The results provide rigorous, scalable tools for assessing cascade likelihood and thresholds in real-world networks, with implications for understanding diffusion, influence, and contagion on complex topologies.

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 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.

Paper Structure

This paper contains 10 sections, 24 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Configuration-model networks
  3. Naive mean-field approximation
  4. Message-passing for configuration-model networks
  5. The criticality condition (i.e., "cascade condition")
  6. Networks with degree--degree correlations
  7. Matrix criticality condition
  8. Message-passing for finite-size networks
  9. Criticality condition for finite-size networks
  10. Conclusions

Figures (3)

  • Figure 1: The expected steady-state fraction $\rho_\infty$ of active nodes for cascades in the Watts threshold model (WTM), where every node has the same threshold $r=0.18$ (so a node becomes active when its fraction of active neighbors is at least as large as $18\%$). The networks are Erdős--Rényi random graphs ($G(N,m)$, where $m$ is the total number of edges) with mean degree $z$ (so they have a Poisson degree distribution $p_k = z^k e^{-z}/k!$), and the initial seed fraction is $\rho_0=10^{-3}$. The numerical simulation results, shown by the black squares, are a mean over 100 realizations on networks with $N=10^5$ nodes. The blue dashed curve shows the result of the naive mean-field approximation given by Eqs. (\ref{['GP4']}) and (\ref{['GP4a']}), and the red solid curve is for the message-passing approach of Eqs. (\ref{['GP8']}) and (\ref{['GP10']}).
  • Figure 2: Schematic for the method described in Sec. \ref{['sec:config']}. We suppose that the contagion spreads upward from level $n-1$ to level $n$ and beyond. The assumption of infinite network size allows us to consider the limit of an infinite number of levels, terminating with the "top" (or "root") node of the tree approximation.
  • Figure 3: Schematic for the message-passing approach of Sec. \ref{['sec:GP4']}.