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Analysis of complex contagions in random multiplex networks (1204.0491v3)

Published 2 Apr 2012 in physics.soc-ph and cs.SI

Abstract: We study the diffusion of influence in random multiplex networks where links can be of $r$ different types, and for a given content (e.g., rumor, product, political view), each link type is associated with a content dependent parameter $c_i$ in $[0,\infty]$ that measures the relative bias type-$i$ links have in spreading this content. In this setting, we propose a linear threshold model of contagion where nodes switch state if their "perceived" proportion of active neighbors exceeds a threshold \tau. Namely, a node connected to $m_i$ active neighbors and $k_i-m_i$ inactive neighbors via type-$i$ links will turn active if $\sum{c_i m_i}/\sum{c_i k_i}$ exceeds its threshold \tau. Under this model, we obtain the condition, probability and expected size of global spreading events. Our results extend the existing work on complex contagions in several directions by i) providing solutions for coupled random networks whose vertices are neither identical nor disjoint, (ii) highlighting the effect of content on the dynamics of complex contagions, and (iii) showing that content-dependent propagation over a multiplex network leads to a subtle relation between the giant vulnerable component of the graph and the global cascade condition that is not seen in the existing models in the literature.

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Authors (2)
  1. Osman Yagan (61 papers)
  2. Virgil Gligor (15 papers)
Citations (160)
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Summary

Analysis of Complex Contagions in Random Multiplex Networks

This paper presents an in-depth examination of influence diffusion within random multiplex networks, where links can manifest as one of multiple types and each type holds a content-dependent bias for spreading information. The authors propose a linear threshold model aimed at determining contagion behavior based on nodes turning active when the perceived proportion of their active neighbors surpasses a threshold. The model encompasses complex contagions and investigates conditions for the emergence of global spreading events.

Main Contributions

The paper makes several significant extensions over existing work in the field of complex contagions:

  1. Multiplex Network Framework: The paper advances prior models by accommodating coupled random networks where vertices are neither identical nor disjoint, offering a more realistic representation of various interconnected networks.
  2. Content-Dependent Dynamics: The introduction of content parameters quantifies the effect of different link types on information propagation, leading to nuanced contagion dynamics not captured by simpler models.
  3. Giant Vulnerable Component Relation: A subtle relationship between the vulnerable components of graphs and the conditions for global cascades is demonstrated, providing insights distinct from prevailing models which typically do not consider link-type bias.

Methodology

The authors utilize generating functions and a self-consistency approach to derive several key results regarding cascade dynamics:

  • Threshold and Probability of Global Cascades: By employing neighborhood influence response functions as a measuring framework, the authors derive sufficient conditions for global cascades. These conditions hinge on the existence of a giant in-component among vulnerable nodes.
  • Expected Cascade Size: The analytical model predicts the final size of global influence diffusion, employing recursive techniques to capture dynamics of active states within the branching processes inherent in such networks.

Analytical and Simulation Insights

The findings revealed by simulations corroborate the analytical predictions, showcasing:

  • Content Impact: Different contents with varied biases significantly alter spreading dynamics, underscoring the critical role that content-dependent parameters play in contagion processes.
  • Cascade Windows: The cascade windows, denoting ranges of connectivity and node thresholds that support global spreading events, were heavily influenced by content factors.
  • Effect of Clustering: While the model assumes localized tree-like network structures, the inclusion of clustering in simulations reveals dual effects: decreased cascade sizes at low connectivity and increased sizes past certain connectivity thresholds.

Implications and Future Directions

This paper illuminates practical and theoretical implications for understanding cascade behavior in complex networks, particularly relevant for social media influence, information dissemination, and infrastructure robustness. The work suggests several intriguing avenues for future research:

  • Extending analysis to more clustered and degree-correlated networks.
  • Developing rigorous models beyond mean-field approximations.
  • Applying the framework to real-world networks for empirical validation of theoretical predictions.

These directions promise to deepen insights into the control and management of cascading processes across various domains. The quantification of content-dependent link biases in particular, opens potential for precision-targeted strategies in both marketing and network security contexts.