Local Certification of Majority Dynamics

TL;DR

This paper investigates the local certification problem of predicting the outcome of majority voting dynamics in social networks after multiple steps, providing bounds on proof sizes in various graph classes.

Contribution

It introduces a proof labeling scheme for Election-Prediction with sub-logarithmic size in graphs with sub-exponential growth and establishes tight bounds for general graphs.

Findings

Proof labeling scheme of size O(log n) for graphs with sub-exponential growth

Sub-linear certificate size for bounded degree graphs

Lower bound of Ω(n) bits for arbitrary graphs

Abstract

In majority voting dynamics, a group of $n$ agents in a social network are asked for their preferred candidate in a future election between two possible choices. At each time step, a new poll is taken, and each agent adjusts their vote according to the majority opinion of their network neighbors. After $T$ time steps, the candidate with the majority of votes is the leading contender in the election. In general, it is very hard to predict who will be the leading candidate after a large number of time-steps. We study, from the perspective of local certification, the problem of predicting the leading candidate after a certain number of time-steps, which we call Election-Prediction. We show that in graphs with sub-exponential growth Election-Prediction admits a proof labeling scheme of size $\mathcal{O}(\log n)$. We also find non-trivial upper bounds for graphs with a bounded degree, in which the size of the certificates are sub-linear in $n$. Furthermore, we explore lower bounds for the unrestricted case, showing that locally checkable proofs for Election-Prediction on arbitrary $n$-node graphs have certificates on $\Omega(n)$ bits. Finally, we show that our upper bounds are tight even for graphs of constant growth.