双曲图上的选民模型
The voter model on the hyperbolic graph
- CRiSM, Department of Statistics, University of Warwick（华威大学统计系CRiSM）
- Department of Mathematics, Aarhus University（奥胡斯大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究双曲随机图巨连通分支上的选民模型，确定了不同参数下的共识时间阶及相变，通过关联相遇集与乘积链电网目标顶点完成证明。
AI中文摘要：
我们研究双曲随机图的巨连通分支上的选民模型，该模型属于空间无标度网络，处于稀疏且线性巨分量 regime α∈(1/2,1)。当顶点数n→∞时，我们发现淬火平均共识时间的阶为n^(2−1/α)，且概率任意接近1。该结论可推广至每个顶点以速率q(v)=d(v)^φ改变观点的选民模型，其中我们还确定了所有φ≥0时的共识时间阶，这些阶分为3个区域，在φ=2−2α处存在相变。对于上界，我们的主要证明思路是将相遇集与乘积链电网中适当高度的某个固定目标顶点关联起来，以严格论证Durrett提出的论点。
英文摘要：
We consider the voter model on the giant component of a hyperbolic random graph, which is a spatial scale-free network, in the sparse and linear-giant regime $α\in(1/2,1)$. We find that the quenched expected consensus time has order $n^{2-1/α}$, as the number of vertices $n\to\infty$, with probability arbitrarily close to one. This is generalised to the voter model where each vertex changes its opinion at rates $q(v)={\rm d}(v)^φ$, where we also establish the consensus time orders for all $φ\geq 0$. These orders have 3 regimes, with a phase transition at $φ=2-2α$. For the upper bounds, our main proof idea is to connect the meeting set to some fixed target vertex of appropriate height in the product chain electrical network, to make rigorous an argument due to Durrett.
