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The edge-averaging process on graphs with random initial opinions.

Dor Elboim1, Yuval Peres2, Ron Peretz3

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Summary
This summary is machine-generated.

This study shows that averaging opinions on random graph edges converges much faster than previously known, especially for disordered initial opinions. The convergence time is significantly reduced, offering faster consensus in networks.

Keywords:
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Area of Science:

  • Network Science
  • Distributed Systems
  • Graph Theory

Background:

  • Nodes in networks (e.g., sensor, social) have initial opinions.
  • Estimating the average opinion using local operations is a key goal.
  • The edge-averaging process is a natural algorithm for this task.

Purpose of the Study:

  • To analyze the convergence rate of the edge-averaging process.
  • To determine the time complexity for reaching approximate consensus.
  • To investigate convergence behavior on both finite and infinite graphs.

Main Methods:

  • Utilized random graph theory and stochastic processes.
  • Analyzed the edge-averaging process with independent Poisson clocks.
  • Derived convergence time bounds for disordered initial opinions.

Main Results:

  • For finite graphs with disordered initial opinions, consensus time is O(n^2), which is sharp.
  • For infinite graphs, convergence to the mean occurs in O(log n) time for opinions in L^2.
  • Almost sure convergence is established for infinite graphs when opinions are in L^p with p > 2.

Conclusions:

  • The edge-averaging algorithm exhibits significantly faster convergence than previously established.
  • Disordered initial opinions dramatically accelerate the consensus-reaching process.
  • The findings provide theoretical guarantees for opinion dynamics in large-scale networks.