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Opinion fluctuations and disagreement in social networks
Daron Acemoglu, Giacomo Como, Fabio Fagnani, Asuman Ozdaglar
TL;DR
The paper asks how persistent disagreement can remain despite communication and changing opinions. It studies a continuous-time stochastic gossip model with regular and stubborn agents, showing that competing stubborn opinions generate ergodic, non-degenerate fluctuations rather than consensus. In highly fluid large networks, most regular agents additionally exhibit approximately equal stationary first and second moments, although this does not imply approximate consensus.
Problem
Existing communication and learning models typically lead to consensus on strongly connected networks, leaving persistent disagreement unexplained.
Method
The paper analyzes an inhomogeneous continuous-time stochastic gossip process in which regular agents update from neighbors while stubborn agents retain fixed opinions, using Markov-chain characterizations of the dynamics.
Results
Competing stubborn opinions prevent consensus among regular agents, whose beliefs fluctuate ergodically and converge in law to a non-degenerate stationary random vector.
Takeaways & Limitations
In highly fluid large-scale networks, most regular agents have approximately equal stationary belief means and variances, a condition termed homogeneous influence.
Takeaways & Limitations
Homogeneous influence concerns marginal first and second moments and does not imply approximate consensus or any particular joint dependence among agents’ stationary beliefs.
Abstract
from arXiv · showhide
We study a tractable opinion dynamics model that generates long-run disagreements and persistent opinion fluctuations. Our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents: regular agents, who update their beliefs according to information that they receive from their social neighbors; and stubborn agents, who never update their opinions. When the society contains stubborn agents with different opinions, the belief dynamics never lead to a consensus (among the regular agents). Instead, beliefs in the society fail to converge almost surely, the belief profile keeps on fluctuating in an ergodic fashion, and it converges in law to a non-degenerate random vector. The structure of the network and the location of the stubborn agents within it shape the opinion dynamics. The expected belief vector evolves according to an ordinary differential equation coinciding with the Kolmogorov backward equation of a continuous-time Markov chain with absorbing states corresponding to the stubborn agents and converges to a harmonic vector, with every regular agent's value being the weighted average of its neighbors' values, and boundary conditions corresponding to the stubborn agents'. Expected cross-products of the agents' beliefs allow for a similar characterization in terms of coupled Markov chains on the network. We prove that, in large-scale societies which are highly fluid, meaning that the product of the mixing time of the Markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large, a condition of \emph{homogeneous influence} emerges, whereby the stationary beliefs' marginal distributions of most of the regular agents have approximately equal first and second moments.
1. Introduction
The paper proposes a stochastic gossip model with stubborn and regular agents to explain persistent disagreement and opinion fluctuations. It establishes stationary fluctuation results and characterizes how network structure and stubborn-agent placement shape influence, including homogeneous influence in highly fluid networks.
- Motivation: Consensus models often cannot explain persistent disagreement, motivating a model where communication and opinion changes coexist with enduring differences.The introduction contrasts typical learning models that lead to consensus on strongly connected networks with the persistence of disagreement despite communication.
- Model: The model uses continuous-time stochastic gossip: stubborn agents keep fixed opinions, while regular agents update toward weighted averages involving neighbors’ beliefs.Agents activate according to Poisson processes, meet social neighbors, and regular agents update their beliefs; the resulting opinions form a Markov process.
- Main results: Competing stubborn opinions prevent consensus among regular agents: their beliefs fail to converge almost surely, fluctuate ergodically, and converge in law to a non-degenerate stationary vector.This result applies under general conditions and provides the paper’s main mechanism for persistent disagreement and opinion fluctuations.
- Network effects: The expected belief vector follows a Markov-chain backward equation with stubborn agents as absorbing states and converges to a harmonic vector determined by network structure.The harmonic characterization also supports explicit solutions for social networks with particular structures or symmetries.
- Large-scale networks: In highly fluid large networks, most agents exhibit homogeneous influence: their stationary belief means and variances concentrate around approximately common values.Highly fluid means that the product of the stubborn-link fraction and the associated Markov-chain mixing time is small; the analysis also uses affine systems, time reversal, and applied-probability estimates.
2. Belief evolution model
The model describes continuous-time stochastic belief updates in a directed social network containing regular agents, who adapt their opinions, and stubborn agents, who do not. Its dual Markov-chain formulation explains persistent fluctuations and stationary disagreement, while network structure and stubborn-agent placement determine long-run influence.
- Model definition: The social network is a directed graph in which links indicate influence toward the updating agent; links to stubborn agents are incoming only, while regular-agent links may be uni- or bi-directional.The model uses a finite agent population and associates meeting rates with directed links.
- Model definition: Regular agents update their beliefs after independent Poisson-clock meetings with social neighbors, using a convex combination weighted by a trust parameter.Stubborn agents never change their opinions; each regular agent may be influenced by the stubborn agents reachable from it.
- Long-run behavior: Under the assumption that every regular agent is influenced by some stubborn agent, regular beliefs do not converge almost surely but fluctuate ergodically and converge in distribution to a non-degenerate stationary vector.The single-agent example also shows that the belief process fails to converge almost surely, while its time-reversed process converges rapidly to a stationary belief.
- Dual representation: A backward-time dual process traces each regular agent’s belief through a continuous-time Markov chain until it reaches an absorbing stubborn agent.For a fixed horizon, coalescing chains identify the opinion sources, and Assumption 1 ensures finite-time absorption with probability one.
- Dual representation: The stationary belief of a regular agent is the opinion of the stubborn agent reached by its dual chain, while stubborn agents retain their fixed opinions.The full belief vector converges in distribution to a stationary random vector represented through these absorption states.
3. Convergence in distribution and ergodicity of the beliefs
With competing stubborn agents, regular agents do not converge to consensus: their beliefs fluctuate indefinitely while converging in distribution to a stationary random vector. The analysis uses an iterated affine-system representation and a time-reversed process to establish convergence and ergodic behavior.
- Competing stubborn opinions prevent consensus among regular agents, whose beliefs fail to converge almost surely and fluctuate ergodically.
- The actual belief process is analyzed through an iterated affine function system and a time-reversed process with matching marginal distributions.The time-reversed process converges almost surely even though the actual process generally does not.
- The belief vector converges in distribution to a stationary random vector whose law remains invariant over time.
- Ergodic averages of continuous functions of the beliefs converge almost surely to expectations under the stationary distribution.
- For every regular agent influenced by at least two stubborn agents with different beliefs, the stationary belief is non-degenerate.Such an agent therefore continues fluctuating rather than stabilizing at a deterministic limit.
4. Expected beliefs and belief crossproducts
Expected beliefs and cross-products are characterized through single and coupled continuous-time Markov chains on the social network. This yields harmonic boundary-value descriptions for means and corresponding coupled-chain characterizations for second moments.
- The coupled chain evolves independently off the diagonal, can coalesce after meeting, and leaves a component fixed after it reaches a stubborn agent.
- Expected beliefs and expected cross-products satisfy the same linear differential system as transition probabilities of coupled Markov chains.
- Hitting probabilities of the stubborn set characterize stationary expected beliefs and belief cross-products.
- Stationary expected beliefs are the unique solution of a harmonic system with stubborn agents supplying boundary values.For regular agents, each expected belief is determined by the weighted neighbor relation encoded by the generator.
- Stationary cross-products satisfy a coupled harmonic system, with factorization when at least one coordinate is stubborn.
- Asymptotic empirical averages of beliefs and cross-products equal their stationary expectations, independently of initial regular-agent beliefs.
5. Explicit computations of stationary expected beliefs and variances
Explicit examples show how network topology and stubborn-agent placement determine stationary expected beliefs and variances. Tree, star, barbell, cycle, and toroidal networks yield distinct spatial patterns, including interpolation, polarization, and topology-dependent influence.
- Tree topology: Tree networks linearly interpolate between the two stubborn agents along their connecting path, with attached components inheriting the value of the path vertex.For a line graph, expected beliefs interpolate linearly, while variances form a parabola that peaks centrally and vanish at stubborn agents.
- Stationary variances: Regular agents’ beliefs fluctuate ergodically around values determined by their relative distances from the two stubborn agents, with fluctuation amplitude largest at central nodes.Agents closer to one stubborn agent experience stronger influence from it, while equidistant central nodes have maximal fluctuation amplitude.
- Star topology: In a star graph, placing one stubborn agent at the center makes every regular agent share its belief; placing both stubborn agents off-center yields their arithmetic average.These contrasting outcomes demonstrate the importance of stubborn-agent location within the same topology.
- Barbell topology: In a barbell network, expected beliefs polarize as the population grows: agents in each complete-graph half approach the opinion of that half’s stubborn agent.The corresponding figure shows expected average beliefs concentrating around the respective stubborn opinions in the two halves.
- Cayley graphs: For cycles and d-dimensional tori, the analysis uses Abelian Cayley-graph structure, random-walk hitting times, Green functions, and eigenvectors to characterize stationary beliefs.The one-dimensional torus is a cycle, while higher-dimensional cases provide the stated toroidal networks.
6. Homogeneous influence in highly fluid social networks
Highly fluid social networks make stubborn agents’ influence approximately homogeneous across most regular agents, while network geometry and stubborn-agent placement determine whether this occurs.
- Network fluidity: The analysis characterizes stationary expected beliefs and variances through the underlying social network’s geometry and stubborn-agent set.The framework uses Markov-chain hitting probabilities, invariant measures, and mixing times to study these moments.
- Network fluidity: The social-network transition structure can be represented by an irreducible, aperiodic stochastic matrix, including the lazy random walk on a connected graph.For canonical undirected-graph networks, the invariant measure is degree-based and mixing can be bounded using conductance.
- Homogeneous influence: Theorem 4 shows that stationary expected beliefs and, when θav = 1, variances of most regular agents concentrate around the moments of a weighted-mean belief Z.The resulting homogeneous influence concerns marginal first and second moments rather than the full joint distribution.
- Homogeneous influence: Highly fluid networks are those where mixing sufficiently offsets the relative linkage to stubborn agents, allowing most regular agents’ stationary moments to approach common values.The relevant mechanism is loss of memory of the starting point before the Markov chain hits the stubborn set.
- Examples: Barbell-like networks are never highly fluid when at least one stubborn agent exists, and their expected stationary beliefs can polarize instead of becoming homogeneous.This contrasts network structures with fast mixing and weak relative linkage to stubborn agents against bottlenecked structures.
- Examples: Erdős–Rényi graphs in the connected regime provide an example where increasing population size produces the concentration predicted by homogeneous influence.The cited regime has p = cn^-1 log n, mixing time O(log n), and degree bounds proportional to log n.
7. Conclusion
The paper identifies stubborn agents as a mechanism producing persistent disagreement and opinion fluctuations, and characterizes these phenomena using Markov-chain duality. In highly fluid networks, stationary expected beliefs and variances become nearly constant across agents.
- Stubborn agents generate persistent fluctuations and disagreement because regular agents’ beliefs do not converge almost surely and instead fluctuate ergodically.
- A duality argument characterizes expected stationary beliefs through hitting probabilities of a Markov chain on the social network.
- Hitting probabilities of coupled Markov chains characterize correlations between the stationary beliefs of pairs of regular agents.
- In highly fluid networks, stationary expected beliefs and variances are almost constant, producing homogeneous influence among most agents.