Consensus times for monotone aggregation dynamics

Abstract

We study an asynchronous consensus dynamics on $N$ agents: at each step a uniformly chosen agent replaces its state by $f(Y_1,\dots,Y_r)$, where $f$ is a fixed monotone aggregation rule and $Y_1,\dots,Y_r$ are the states of $r$ agents sampled uniformly with replacement. Let $T$ be the first time at which all agents agree. Let $f:\{0,1\}^r\to\{0,1\}$ be monotone and non-constant. The expected consensus time is governed by the two \emph{endpoint degrees} $D_0(f)=\#\{i:f(e_i)=1\}$ and $D_1(f)=\#\{i:f(\mathbf{1}-e_i)=0\}$, where $e_i$ is the $i$th standard basis vector and $\mathbf{1}$ the all-ones vector. If $D_0(f)\ne 1$ and $D_1(f)\ne 1$, then $E[T]=\Theta\bigl(N(1+\log d_f)\bigr)$, where $d_f$ counts the agents that must change state before the nearest attracting consensus is reached, so that $E[T]=O(N\log N)$ uniformly over initial states. If $f$ is a dictator, $E[T]$ is given by a voter-model formula and equals $\Theta\bigl(N^2 Ent(p_0)\bigr)$ with absolute constants, uniformly over the initial state, where $Ent$ is the binary entropy and $p_0$ the initial fraction of agents in state $1$. Otherwise exactly one of $D_0(f),D_1(f)$ equals $1$ and the other equals $0$. The worst-case expected consensus time is then $\Theta(N^{2-1/m})$, where $m\ge 2$ is the least number of coordinates that force the value $1$ in the {\em residual rule} of $f$, defined in the paper (of its dual, when $D_1(f)=1$).