Estimating True Beliefs in Opinion Dynamics with Social Pressure

TL;DR

This work addresses recovering true beliefs from publicly declared opinions under social pressure in a networked Interacting Pólya Urn model, where each agent has a fixed inherent belief $\phi_i$ and bias $\gamma_i$. It introduces a maximum-likelihood estimator for heterogeneous biases on arbitrary weighted graphs and a low-dimensional inherent-belief estimator based on history and neighborhood declarations. The authors prove almost-sure consistency of these estimators, even in consensus regimes, and derive convergence-rate bounds that depend on network structure, including consensus rates tied to the largest eigenvalue of normalized adjacency matrices $\mathbf{J}_{\mathbf{\bone}}$ or $\mathbf{J}_{\mathbf{\zero}}$. Practically, the results enable accurate inference of latent beliefs from noisy, pressure-driven expressions, with quantified performance as the network evolves toward consensus or remains non-consensus.

Abstract

Social networks often exert social pressure, causing individuals to adapt their expressed opinions to conform to their peers. An agent in such systems can be modeled as having a (true and unchanging) inherent belief while broadcasting a declared opinion at each time step based on her inherent belief and the past declared opinions of her neighbors. An important question in this setting is parameter estimation: how to disentangle the effects of social pressure to estimate inherent beliefs from declared opinions. This is useful for forecasting when agents' declared opinions are influenced by social pressure while real-world behavior only depends on their inherent beliefs. To address this, Jadbabaie et al. formulated the Interacting Pólya Urn model of opinion dynamics under social pressure and studied it on complete-graph social networks using an aggregate estimator, and found that their estimator converges to the inherent beliefs unless majority pressure pushes the network to consensus. In this work, we studythis model on arbitrary networks, providing an estimator which converges to the inherent beliefs even in consensus situations. Finally, we bound the convergence rate of our estimator in both consensus and non-consensus scenarios; to get the bound for consensus scenarios (which converge slower than non-consensus) we additionally found how quickly the system converges to consensus.

Figures (4)

• Figure 1: Example network used in numerical simulations. Blue nodes have inherent belief $\phi_i = 1$ (i.e. $\gamma_i > 1$) and red have inherent belief $\phi_i = 0$ (i.e. $\gamma_i < 1$). All edges have weight $1$. As per opiniondynamicsCDC, this network approaches a consensus of $\bzero$.
• Figure 2: Simulation for the network from \ref{['fig::5-node-graph']}, comparing estimators for inherent beliefs. 'MLE' is the estimator \ref{['eq::steve']} (solid) and 'Eq' is the estimator \ref{['eq::estimator_eq_point_inherent']} (dashed). $1000$ instances of the network were run; each line corresponds to the average prediction of the estimator at the given time over these instances. Note that for certain agents, the average of 'Eq' converges faster than 'MLE'; however, when given an agent in general agreement with its neighbors, such as Agent 3, 'Eq' can converge extremely slowly (dashed green line).
• Figure 3: Empirical plot (over $1000$ experiments), for each agent $i$ of the network in \ref{['fig::5-node-graph']}, of the probability $\delta_i(t^*)$ that the MLE inherent belief estimator \ref{['eq::steve']} is wrong for some $t \geq t^*$. Each experiment was run for $100000$ steps; the curves are cut off at the first $t^*$ such that \ref{['eq::steve']} was always correct for all $t^* \leq t \leq 100000$ (i.e. $\delta = 0$ empirically).
• Figure 4: Plot illustrating example evolution of $V(\bbeta(t))$ and $\beta_i(t)$ for each agent $i$ on the network in \ref{['fig::5-node-graph']}.

Theorems & Definitions (58)

• definition 1
• remark 1
• definition 2
• definition 3: Estimator for Bias Parameter
• definition 4: Inherent Belief Estimator
• lemma 1
• proof
• proposition 1
• proof
• proposition 2