Source-linked AI summary
Social consensus through the influence of committed minorities
J. Xie, S. Sreenivasan, G. Korniss, W. Zhang, C. Lim, B. K. Szymanski
TL;DR
The paper asks when a small, inflexible minority can reverse a majority opinion through social influence. It analyzes a binary agreement model with committed agents using mean-field and finite-size dynamics, finding a critical fraction near 10% separating exponentially slow from logarithmically fast consensus on complete graphs.
Problem
The paper examines the conditions under which a committed minority can overcome a majority and how consensus time varies with the committed fraction.
Method
The study analyzes a binary agreement model with committed agents using mean-field equations, fixed-point stability, quasi-stationary distributions, and simulations across network topologies.
Results
For complete graphs, p < pc yields Tc ∼ exp(α(p)N), whereas p > pc yields logarithmic consensus-time scaling; the critical fraction is pc ≈ 0.09789.
Takeaways & Limitations
A committed minority can trigger a tipping point at which the majority opinion switches quickly to the minority’s opinion.
Abstract
from arXiv · showhide
We show how the prevailing majority opinion in a population can be rapidly reversed by a small fraction p of randomly distributed committed agents who consistently proselytize the opposing opinion and are immune to influence. Specifically, we show that when the committed fraction grows beyond a critical value p_c \approx 10%, there is a dramatic decrease in the time, T_c, taken for the entire population to adopt the committed opinion. In particular, for complete graphs we show that when p < p_c, T_c \sim \exp(α(p)N), while for p > p_c, T_c \sim \ln N. We conclude with simulation results for Erdős-Rényi random graphs and scale-free networks which show qualitatively similar behavior.
I. INTRODUCTION
The paper studies how committed minority agents can overturn a majority opinion in a binary agreement model, where individuals may hold both opinions and committed agents resist influence. It asks how consensus time depends on the committed fraction and how network structure affects minority-driven consensus.
- I. INTRODUCTION: The model uses Naming Game dynamics: a randomly selected speaker voices an opinion to a randomly selected neighbor, who retains or adds it according to whether it is already in the listener’s list.The paper fixes the selection order as speaker first, then listener.
- I. INTRODUCTION: The binary agreement model permits agents to possess both opinions simultaneously, changing the time required to reach consensus from uniform initial conditions.This model treats the two singular opinion states symmetrically in their susceptibility to change.
- I. INTRODUCTION: The study initializes nearly all agents with opinion B and a fraction p of committed agents with opinion A, who influence others but cannot be influenced.With committed A agents present, the only absorbing state has all influenceable nodes adopting A.
- I. INTRODUCTION: The central question is how consensus time varies with the committed fraction and under what conditions an inflexible minority can win over the population.The study compares its model, network topologies, finite-size scaling, and initial condition with prior work on un-influenceable agents.
A. Infinite network size limit
In the infinite-network analysis, mean-field equations describe committed and uncommitted opinion densities and reveal a critical transition. Below the critical fraction, an active steady state persists alongside stable consensus; at the threshold, the active and unstable fixed points merge, producing an abrupt shift to committed-opinion consensus.
- A. Infinite network size limit: The mean-field analysis uses densities nA, nB, and nAB, with nAB = 1 − p − nA − nB for uncommitted A, B, and mixed states.The equations neglect correlations and fluctuations in the complete-graph limit.
- A. Infinite network size limit: The interaction rules enumerate speaker-listener updates among A, B, and AB states, with the speaker’s voiced opinion determining the resulting pair states.These interaction probabilities provide the terms used to construct the mean-field rate equations.
- A. Infinite network size limit: For every p, consensus in the committed opinion, nA = 1 − p and nB = 0, is a stable fixed point.Below the critical fraction, additional fixed points include an unstable saddle and a stable active steady state.
- A. Infinite network size limit: nB, the density of uncommitted B nodes, serves as the order parameter for the transition from active steady state to absorbing consensus.For p < pc, finite complete graphs fluctuate around a nonzero B density before escaping to consensus; larger systems better match the mean-field curve.
- A. Infinite network size limit: At pc, the stable active and unstable fixed points meet, leaving consensus as the only remaining stable fixed point.The stable B density approaches approximately 0.65 rather than smoothly reaching zero, indicating a first-order transition.
B. Finite network size: Scaling results for consensus times
For finite complete graphs, consensus emerges from a metastable active state through rare escapes below the critical committed fraction, while above it consensus times scale logarithmically with system size. A quasi-stationary method estimates the low-p regime and simulations establish the contrasting high-p behavior.
- Finite-size analysis: The master equation tracks transitions among uncommitted states while excluding the absorbing consensus state with all uncommitted nodes in opinion A.The state space uses n and m for uncommitted nodes in the two singular opinion states, with remaining nodes in the mixed state.
- Finite-size analysis: Below p_c, the stable mean-field fixed point is metastable because finite stochastic systems can escape it and eventually reach consensus.The survival analysis conditions occupation probabilities on avoiding the absorbing state.
- Finite-size analysis: The quasi-stationary approximation yields consensus times across committed fractions and system sizes, but is unreliable above p_c because consensus is too rapid for a quasi-stationary state to form.The method obtains the quasi-stationary distribution iteratively before calculating mean consensus times.
- Scaling results: For p < p_c, T_c grows exponentially with N, whereas for p > p_c simulations show logarithmic growth with N.The exponential regime reflects escape from metastable states; p = 0.3 is an example of the logarithmic regime.
- Scaling results: Below p_c, the exponential-growth rate obeys α(p) ∼ |p − p_c|^ν with ν ≈ 1.65.The exponential dependence is additionally described as being modulated by factors of log N that become dominant at p = p_c.
III. SPARSE NETWORKS
Erdős-Rényi networks exhibit the same qualitative evolution as complete graphs, although the critical committed fraction depends on average degree. For small average degree, the consensus-time drop occurs slightly earlier, while complete graphs are faster above p_c.
- III. SPARSE NETWORKS: Erdős-Rényi graphs show the same qualitative evolution as complete graphs, but p_c depends on the average degree ⟨k⟩.The comparison is made for networks with specified size N and average degree ⟨k⟩.
- III. SPARSE NETWORKS: For small ⟨k⟩ at fixed N, the drop in consensus times occurs slightly earlier in p on Erdős-Rényi graphs than on complete graphs.
- III. SPARSE NETWORKS: Above p_c, complete graphs have shorter consensus times than Erdős-Rényi graphs.
IV. SUMMARY
The paper identifies a tipping point at which a consistent minority can rapidly replace the initial majority opinion, and studies this transition through finite-network analysis and simulations. It also points to community structure and utility-driven switching as extensions for optimizing opinion spreading.
- IV. SUMMARY: The study demonstrates a tipping point where the initial majority opinion switches quickly to that of a consistent, inflexible minority.
- IV. SUMMARY: The analysis combines semi-analytical methods and simulations to study finite-sized and sparse networks within the binary agreement model.
- IV. SUMMARY: Open extensions include selecting committed agents in networks with community structure to minimize consensus times and reduce p_c.
- IV. SUMMARY: Utility-driven opinion switching is proposed as a possible basis for designing incentive schemes for opinion spreading.
Appendix: Fixed points of the mean-field equations
The appendix reduces the mean-field fixed-point conditions to a scalar equation, then determines when additional valid fixed points exist and classifies their stability. It identifies a stable consensus fixed point for every committed fraction and additional fixed points below a critical fraction.
- Fixed-point reduction: The fixed-point conditions are expressed using x = nA and y = nB, then reduced by substituting x into the equation for y and setting z^2 = y.This scalar reduction provides the basis for determining admissible roots as functions of p.
- Fixed-point structure: For every p, the mean-field equations admit the stable consensus fixed point nA = 1 − p and nB = 0, representing adoption of the committed agents’ opinion.The appendix identifies this as the solution z = 0.
- Root-existence criterion: Additional valid roots exist only when the extrema of the scalar function satisfy the derived root-existence condition.The analysis locates the extrema, introduces q = z2, and obtains the condition for more roots through f(q).
- Critical fraction: pc = 0.09789; at this threshold, the system state is {nA, nB} = {0.0957, 0.6504}, and two fixed points exist for p < pc.The critical value follows from the cubic-root condition, with the additional roots disappearing at the threshold.
- Stability analysis: Linear stability analysis finds one additional fixed point stable for 0 ≤ p < pc and another unstable, while the consensus fixed point remains stable for 0 ≤ p ≤ 1.As p approaches zero, the stable additional point tends toward nA = 0, nB = 1, whereas the unstable point tends toward nA = nB ≅ 0.38.