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.

翻译：在多数投票动力学中，社交网络中的一组 $n$ 名代理人被询问他们在未来选举中两个候选选项之间的偏好候选人。在每个时间步，进行新一轮投票，每位代理人根据其网络邻居的多数意见调整自己的投票。经过 $T$ 个时间步后，获得多数投票的候选人成为选举中的领先竞争者。通常，很难预测经过大量时间步后谁将成为领先候选人。我们从局部认证的角度研究预测特定时间步后领先候选人的问题，称之为选举预测。我们证明，在次指数增长的图中，选举预测允许一个大小为 $\mathcal{O}(\log n)$ 的证明标记方案。我们还为有界度图找到了非平凡的上界，其中证书的大小在 $n$ 中呈次线性。此外，我们探索了无限制情况下的下界，表明在任意 $n$ 节点图上，选举预测的局部可检查证明具有 $\Omega(n)$ 比特的证书。最后，我们证明即使在恒定增长的图中，我们的上界也是紧的。