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Granular DeGroot Dynamics -- a Model for Robust Naive Learning in Social Networks

Gideon Amir, Itai Arieli, Galit Ashkenazi-Golan, Ron Peretz

TL;DR

The paper studies opinion exchange on networks under DeGroot-like learning, where each agent receives a noisy signal about a true world state . To address fragility to stubborn agents and distorted monitoring, it proposes the raction{1}{m}-DeGroot updating rule, which restricts opinions to a finite grid and rounds at both the aggregation and update steps. The authors prove that on infinite graphs with bounded degree and sub-exponential growth, for any target accuracy  and failure probability  there exists an  such that the dynamics achieves $(,)$-learning, robust to finitely many adversarial agents and moderate distortions; they further extend these results to families of graphs and illustrate that increasing  enlarges the radius of influence of adversaries while improving accuracy, with asymptotic behavior recovering standard DeGroot as  . A general Lyapunov-energy framework underpins the convergence and robustness results, and the work also discusses a status quo-biased variant that fits the same robust criteria. Overall, the paper contributes a bounded-rationality mechanism that preserves informative learning while limiting adversarial influence in large-scale social networks, and it connects non-Bayesian learning with robust collective dynamics in a unified framework.

Abstract

We study a model of opinion exchange in social networks where a state of the world is realized and every agent receives a zero-mean noisy signal of the realized state. It is known from [Golub and Jackson 2010] that under DeGroot dynamics [DeGroot 1974] agents reach a consensus that is close to the state of the world when the network is large. The DeGroot dynamics, however, is highly non-robust and the presence of a single ``stubborn agent'' that does not adhere to the updating rule can sway the public consensus to any other value. We introduce a variant of DeGroot dynamics that we call \emph{ $\frac{1}{m}$-DeGroot}. $\frac{1}{m}$-DeGroot dynamics approximates standard DeGroot dynamics to the nearest rational number with $m$ as its denominator and like the DeGroot dynamics it is Markovian and stationary. We show that in contrast to standard DeGroot dynamics, $\frac{1}{m}$-DeGroot dynamics is highly robust both to the presence of stubborn agents and to certain types of misspecifications.

Granular DeGroot Dynamics -- a Model for Robust Naive Learning in Social Networks

TL;DR

The paper studies opinion exchange on networks under DeGroot-like learning, where each agent receives a noisy signal about a true world state . To address fragility to stubborn agents and distorted monitoring, it proposes the raction{1}{m}-DeGroot updating rule, which restricts opinions to a finite grid and rounds at both the aggregation and update steps. The authors prove that on infinite graphs with bounded degree and sub-exponential growth, for any target accuracy  and failure probability  there exists an  such that the dynamics achieves -learning, robust to finitely many adversarial agents and moderate distortions; they further extend these results to families of graphs and illustrate that increasing  enlarges the radius of influence of adversaries while improving accuracy, with asymptotic behavior recovering standard DeGroot as  . A general Lyapunov-energy framework underpins the convergence and robustness results, and the work also discusses a status quo-biased variant that fits the same robust criteria. Overall, the paper contributes a bounded-rationality mechanism that preserves informative learning while limiting adversarial influence in large-scale social networks, and it connects non-Bayesian learning with robust collective dynamics in a unified framework.

Abstract

We study a model of opinion exchange in social networks where a state of the world is realized and every agent receives a zero-mean noisy signal of the realized state. It is known from [Golub and Jackson 2010] that under DeGroot dynamics [DeGroot 1974] agents reach a consensus that is close to the state of the world when the network is large. The DeGroot dynamics, however, is highly non-robust and the presence of a single ``stubborn agent'' that does not adhere to the updating rule can sway the public consensus to any other value. We introduce a variant of DeGroot dynamics that we call \emph{ -DeGroot}. -DeGroot dynamics approximates standard DeGroot dynamics to the nearest rational number with as its denominator and like the DeGroot dynamics it is Markovian and stationary. We show that in contrast to standard DeGroot dynamics, -DeGroot dynamics is highly robust both to the presence of stubborn agents and to certain types of misspecifications.

Paper Structure

This paper contains 25 sections, 21 theorems, 101 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related literature
  3. Model and Preliminaries
  4. The Fragility of DeGroot dynamics
  5. Main Results
  6. Outline of the Technical Contribution
  7. Reduction from adversarial to stubborn agents
  8. Convergence of opinions
  9. Approximate learning
  10. Distorted monitoring
  11. General Framework
  12. List of notations
  13. Generalized results
  14. Main Results Fit into General Framework
  15. Status quo-biased DeGroot dynamics
  16. ...and 10 more sections

Key Result

Theorem 2.1

Let $G$ be a connected infinite graph of bounded degree. For every agent $i\in V$ it holds that $\lim_{t\rightarrow\infty}A_{i,t}=\mu$ almost surely.

Figures (3)

  • Figure 1: Simulation of two variants of the DeGroot dynamics. The underlying graph is the $100\times 100$ grid (with edges in all eight directions: N, NE, E,…). Initial opinions drawn i.i.d. uniform $[0,1]$ except for a single stubborn agent with opinion 0. The three images (left to right) capture the opinions of all agents at times $0$, $10,000$, and $100,0000$.
  • Figure 2: An illustration of Ginosar--Holzman's energy on the line graph $\mathbb Z$.
  • Figure 3: A sufficient condition for Ginosar--Holzman's energy not to increase is that $A_{i,t+1}$ is always closer to $g_D$ compared to $A_{i,t-1}$.

Theorems & Definitions (41)

  • Theorem 2.1
  • Lemma 2.1.1
  • Lemma 2.1.2
  • Definition 3.0.1
  • Definition 3.0.2
  • Theorem 3.1
  • Definition 3.1.1
  • Theorem 3.2
  • Theorem 4.1
  • Definition 5.0.1
  • ...and 31 more