A Note on the Majority Dynamics in Inhomogeneous Random Graphs

Yilun Shang

doi:10.1007/s00025-021-01436-z

Abstract

In this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph K_n independently with probability p_n(e). Each vertex is independently assigned an initial state +1 (with probability p_+) or -1 (with probability 1-p_+), updated at each time step following the majority of its neighbors’ states. Under some regularity and density conditions of the edge probability sequence, if p_+ is smaller than a threshold, then G will display a unanimous state -1 asymptotically almost surely, meaning that the probability of reaching consensus tends to one as nrightarrow infty . The consensus reaching process has a clear difference in terms of the initial state assignment probability: In a dense random graph p_+ can be near a half, while in a sparse random graph p_+ has to be vanishing. The size of a dynamic monopoly in G is also discussed.

Highlights

Majority dynamics is a discrete-time deterministic process over a graph G with n vertices V = {1, 2, . . . , n}, where each vertex i ∈ V holds a state Ct(i) ∈ {−1, +1} at time step t ≥ 0
Majority dynamics (1.1) is the noiseless special case of the well-known majority-vote model in statistical physics [1,2,3], where each voter has a probability q choosing the minority of its neighbors and probability 1 − q the majority
Threshold and phase transition with respect to noise and other order parameters are the focus of these studies mainly using mean field approximation. Another closely related model is majority bootstrap percolation [4,5], where each vertex can have one of two colors, red or blue, and blue vertices update their color according to the majority rule while red vertices invariably retain their color

Summary

Introduction

Majority dynamics is a discrete-time deterministic process over a graph G with n vertices V = {1, 2, . N}, where each vertex i ∈ V holds a state Ct(i) ∈ {−1, +1} at time step t ≥ 0. The state configuration of G at t can be represented as a mapping Ct : V → {−1, +1}. Given an initial configuration C0, the process evolves following

Results

Main Results

Proofs