Binary-State Dynamics on Complex Networks: Pair Approximation and Beyond

Phys. Rev. X 3, 021004 – Published 29 April 2013
James P. Gleeson


A wide class of binary-state dynamics on networks—including, for example, the voter model, the Bass diffusion model, and threshold models—can be described in terms of transition rates (spin-flip probabilities) that depend on the number of nearest neighbors in each of the two possible states. High-accuracy approximations for the emergent dynamics of such models on uncorrelated, infinite networks are given by recently developed compartmental models or approximate master equations (AMEs). Pair approximations (PAs) and mean-field theories can be systematically derived from the AME. We show that PA and AME solutions can coincide under certain circumstances, and numerical simulations confirm that PA is highly accurate in these cases. For monotone dynamics (where transitions out of one nodal state are impossible, e.g., susceptible-infected disease spread or Bass diffusion), PA and the AME give identical results for the fraction of nodes in the infected (active) state for all time, provided that the rate of infection depends linearly on the number of infected neighbors. In the more general nonmonotone case, we derive a condition—that proves to be equivalent to a detailed balance condition on the dynamics—for PA and AME solutions to coincide in the limit t. This equivalence permits bifurcation analysis, yielding explicit expressions for the critical (ferromagnetic or paramagnetic transition) point of such dynamics, that is closely analogous to the critical temperature of the Ising spin model. Finally, the AME for threshold models of propagation is shown to reduce to just two differential equations and to give excellent agreement with numerical simulations. As part of this work, the Octave or Matlab code for implementing and solving the differential-equation systems is made available for download.


  • Received 8 October 2012
  • Published 29 April 2013

Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

James P. Gleeson*

  • MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland

  • *

Popular Summary

Should I open a Facebook account or not? Which of the two presidential candidates should I vote for in an American federal election? Will I get infected or not in a disease epidemic? Under what conditions does my choice or opinion become the popular one? The answer to each of these questions depends not only on the individual in question, but also crucially on their social or physical contacts with other individuals. Formulating this observation mathematically in terms of both local-contact-based binary decision-making processes and the concept of social network structures has led to many simple paradigmatic models, such as the voter model and the susceptible-infected model, that scientists study in order to understand how behaviors, opinions, and infectious diseases spread among human populations.

While these models are simple, analytical methods for tackling them are few and often not accurate, or achieve high accuracy at the cost of computational complexity, because dealing with interactions among many entities in a large system is known to be difficult, in general. In this paper, we present a low-complexity approximation approach, called pair approximation, and demonstrate that for certain classes of local decision rules, it achieves accuracy that is equivalent to that of a recently developed high-accuracy, high-complexity approach.

This new low-complexity approach should find broad utility in the theoretical studies of social phenomena at the population level. To facilitate the spread of its use, we have made a computational code available by download that implements the approach on the Octave or Matlab software platforms.

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