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The edge-averaging process on graphs with random initial opinions

Dor Elboim, Yuval Peres, Ron Peretz

TL;DR

This work analyzes the edge-averaging process on graphs with random initial opinions, proving that on finite graphs with i.i.d. initial data bounded by 1, the expected time to near-consensus scales as $\mathcal{O}(\log^2 n)$ and is tight on cycles. It introduces a fragmentation representation $m_t$ that expresses $f_t(o)$ as a weighted sum of initial opinions and uses a quadratic dispersion $Q_t=\sum_v m_t(v)^2$ to derive concentration and convergence bounds. For infinite graphs, the authors establish $L^p$-convergence to the mean for any $p\ge1$, with rates $\|f_t(o)-\mu\|_p$ decaying like $t_*^{(1-p)/(2p)}$, and prove almost-sure convergence when $p>4$. The results bridge global and local convergence analyses, tie the dynamics to fragmentation bounds, and provide sharp lower bounds, offering a detailed picture of fast consensus under typical random initial conditions. These insights have implications for distributed averaging in networks, sensor fusion, and social learning, where initial heterogeneity is common and rapid convergence to the true average is desirable.

Abstract

In several settings (e.g., sensor networks and social networks), nodes of a graph are equipped with initial opinions, and the goal is to estimate the average of these opinions using local operations. A natural algorithm to achieve this is the edge-averaging process, where edges are repeatedly selected at random (according to independent Poisson clocks) and the opinions on the nodes of each selected edge are replaced by their average. The effectiveness of this algorithm is determined by its convergence rate. It is known that on a finite graph of $n$ nodes, the opinions reach approximate consensus in polynomial time. We prove that the convergence is much faster when the initial opinions are disordered (independent identically distributed): the time to reach approximate consensus is $O (\log^2n)$, and this bound is sharp. For infinite graphs, we show that for every $p\geq 1$, if the initial opinions are in $L^p$, then the opinion at each vertex converges to the mean in $L^p$, and if $p>4$, then almost sure convergence holds as well.

The edge-averaging process on graphs with random initial opinions

TL;DR

This work analyzes the edge-averaging process on graphs with random initial opinions, proving that on finite graphs with i.i.d. initial data bounded by 1, the expected time to near-consensus scales as and is tight on cycles. It introduces a fragmentation representation that expresses as a weighted sum of initial opinions and uses a quadratic dispersion to derive concentration and convergence bounds. For infinite graphs, the authors establish -convergence to the mean for any , with rates decaying like , and prove almost-sure convergence when . The results bridge global and local convergence analyses, tie the dynamics to fragmentation bounds, and provide sharp lower bounds, offering a detailed picture of fast consensus under typical random initial conditions. These insights have implications for distributed averaging in networks, sensor fusion, and social learning, where initial heterogeneity is common and rapid convergence to the true average is desirable.

Abstract

In several settings (e.g., sensor networks and social networks), nodes of a graph are equipped with initial opinions, and the goal is to estimate the average of these opinions using local operations. A natural algorithm to achieve this is the edge-averaging process, where edges are repeatedly selected at random (according to independent Poisson clocks) and the opinions on the nodes of each selected edge are replaced by their average. The effectiveness of this algorithm is determined by its convergence rate. It is known that on a finite graph of nodes, the opinions reach approximate consensus in polynomial time. We prove that the convergence is much faster when the initial opinions are disordered (independent identically distributed): the time to reach approximate consensus is , and this bound is sharp. For infinite graphs, we show that for every , if the initial opinions are in , then the opinion at each vertex converges to the mean in , and if , then almost sure convergence holds as well.
Paper Structure (12 sections, 13 theorems, 83 equations, 2 figures)

This paper contains 12 sections, 13 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Global convergence
  4. Local convergence
  5. Other related work
  6. Fragmentation
  7. Proofs: bounded initial opinions
  8. Proofs: initial opinions in $L^p$
  9. Lower bounds
  10. Existence and uniqueness of the averaging process
  11. Open problems and future directions
  12. Acknowledgements

Key Result

Theorem 1.1

Consider the edge-averaging process on a graph $G=(V,E)$ with $|V|=n$. Suppose that the initial opinions $\{f_0(v)\}_{v\in V}$ are i.i.d. random variables that satisfy $|f_0(v)|\leq 1$ a.s. Then for every $\varepsilon >0$, we have where $C>0$ is a universal constant.

Figures (2)

  • Figure 1: Simulation of the edge-averaging process on the $100\times 100$ discrete torus with uniform random (top) and deterministic (bottom) binary initial opinions. Snapshots taken at times (left to right) $0$, $7$, $60$, and $600$.
  • Figure 2: A simulation of the edge-averaging process on the $100\times 100$ discrete torus. The graphs show the oscillation of the opinions $\mathrm{osc}(f_t)$ as a function of time for random (blue) and deterministic (orange) initial opinions.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Claim 2.3
  • proof
  • ...and 18 more