On the Limits of Information Spread by Memory-less Agents

TL;DR

It is demonstrated that any memory-less protocol with constant sample size requires at least an almost-linear number of rounds to converge, which sheds light on the convergence time of the “Minority” dynamics introduced by Becchetti et al., whose chaotic behavior is yet to be fully understood despite its apparent simplicity.

Abstract

We address the self-stabilizing bit-dissemination problem, designed to capture the challenges of spreading information and reaching consensus among entities with minimal cognitive and communication capacities. Specifically, a group of $n$ agents is required to adopt the correct opinion, initially held by a single informed individual, choosing from two possible opinions. In order to make decisions, agents are restricted to observing the opinions of a few randomly sampled agents, and lack the ability to communicate further and to identify the informed individual. Additionally, agents cannot retain any information from one round to the next. According to a recent publication by Becchetti et al. in SODA (2024), a logarithmic convergence time without memory is achievable in the parallel setting (where agents are updated simultaneously), as long as the number of samples is at least $Ω(\sqrt{n \log n})$. However, determining the minimal sample size for an efficient protocol to exist remains a challenging open question. As a preliminary step towards an answer, we establish the first lower bound for this problem in the parallel setting. Specifically, we demonstrate that it is impossible for any memory-less protocol with constant sample size, to converge with high probability in less than an almost-linear number of rounds. This lower bound holds even when agents are aware of both the exact value of $n$ and their own opinion, and encompasses various simple existing dynamics designed to achieve consensus. Beyond the bit-dissemination problem, our result sheds light on the convergence time of the ``minority'' dynamics, the counterpart of the well-known majority rule, whose chaotic behavior is yet to be fully understood despite the apparent simplicity of the algorithm.

Figures (4)

• Figure 1: Sketch of the proof of \ref{['lem:main']}. (a) By assumption (ii), with high probability, $Y_t$ cannot jump from below $a_1 \,n - t$ to above $a_2 \, n -t$ in a single step, let alone $a_2 \, n$. (b) In \ref{['claim:one_step_domination']}, we use assumption (i) and the properties of the Doob's decomposition to show that, if $a_1 \, n - t \leq Y_t \leq M_t$, then $Y_t$ cannot jump above $M_t$ in a single step. (c) In \ref{['claim:trapping_martingale']}, we use the Azuma-Hoeffding inequality to show that $M_t$ remains in the interval $[a_2\,n+T,a_3\,n-T]$ for at least $T$ rounds w.h.p. Overall, (a) (b) and (c) implies that $Y_t$ must remain below $a_3 \, n - T$ for at least $T$ rounds w.h.p., yielding the desired conclusion.
• Figure 2: Illustration of the arguments for Case 1. We consider a configuration in which the correct opinion is $1$. Constant $a_1$ is fixed arbitrarily in the interval $(r_\infty^{(k_0-1)},1)$ Then, $a_2$ is chosen according to \ref{['lemma:no_jump_to_consensus']} to ensure that $X_t$ cannot jump from below $a_1 \, n$ to above $a_2 \, n$. Finally, $a_3$ is set anywhere in the interval $(a_2,1]$. By assumption, $F_n < 0$ on $[a_1,a_3]$, and we can eventually apply \ref{['lem:main']}.
• Figure 3: Illustration of the arguments for Case 2. We consider a configuration in which the correct opinion is $0$. Constants $a_1, a_2$ and $a_3$ are chosen arbitrarily in the interval $(r_\infty^{(k_0-1)},1)$. By assumption, $F_n > 0$ on $[a_1,a_3]$. Moreover, once $a_2$ and $a_3$ are fixed, we give a lower-bound on $F_n$ on the interval $[r_n^{(k_0)},1]$, by letting $r_n^{(k_0)}$ be sufficiently close to $1$, in order to ensure that $X_t$ cannot jump from above $a_3 \, n$ to below $a_2 \, n$. Eventually, we are able to apply \ref{['cor:main_lemma_reversed']}.
• Figure 4: Depiction of the dual process behind the proof of \ref{['thm:secondary']}. The color of the circle in row $t$, column $i$, corresponds to the opinion of Agent $i$ in round $t$: it is black if $X_t^{(i)} = 1$, and white otherwise. An arrow is drawn from $(i,t+1)$ to $(j,t)$ if $S_t^{(i)} = j$, i.e., if Agent $i$ observes Agent $j$ in round $t$ (and thus adopts their opinion in round $t+1$). Red circles depict the locations of $n$ coalescing random walks going backward in time, and initially present at every location. Random walks at a location $i > 1$ make a move using the same randomness as the samples, while the source acts like a sink. If all random walks have coalesced in less than $T$ rounds, it implies that the opinion of each agent in round $T$ comes from the source, and thus that the dynamics has reached consensus on the correct opinion.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Theorem 6
• proof