SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1285-1314, June 2024.
Abstract. The voter model is a classical interacting particle system, modeling how global consensus is formed by local imitation. We analyze the time to consensus for a particular family of voter models when the underlying structure is a scale-free inhomogeneous random graph in the high edge–density regime, where this graph features a giant component. In this regime, we verify that the polynomial orders of consensus agree with those of their mean-field approximation in [A. Moinet, A. Barrat, and R. Pastor-Satorras, Phys. Rev. E, 98 (2018), 022303]. This “discursive” family of models has a symmetrized interaction to better model discussions and is indexed by a temperature parameter that, for certain parameters of the power law tail of the network’s degree distribution, is seen to produce two distinct phases of consensus speed. Our proofs rely on the well-known duality to coalescing random walks and a novel bound on the mixing time of these walks using the known fast mixing of the Erdős–Rényi giant subgraph. Unlike in the subcritical case [J. Fernley and M. Ortgiese, Random Structures Algorithms, 62 (2023), pp. 376–429], which requires tail exponent of the limiting degree distribution [math] as well as low edge density, in the giant component case, we also address the “ultrasmall world” power law exponents [math].

中文翻译：

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超临界无标度网络上的话语选民模型

SIAM 离散数学杂志，第 38 卷，第 2 期，第 1285-1314 页，2024 年 6 月
。摘要。选民模型是一个经典的相互作用粒子系统，模拟如何通过局部模仿形成全局共识。当底层结构是高边缘密度状态下的无标度非均匀随机图（该图具有巨大组件）时，我们分析了特定系列选民模型达成共识的时间。在这种情况下，我们验证了一致的多项式阶数与其在[A. Moinet、A. Barrat 和 R. Pastor-Satorras，物理学。修订版 E，98（2018），022303]。这种“散漫”模型系列具有对称交互作用，可以更好地进行模型讨论，并由温度参数索引，对于网络度分布的幂律尾部的某些参数，可以看出该温度参数会产生两个不同的共识速度阶段。我们的证明依赖于众所周知的合并随机游走的对偶性，以及使用已知的 Erdős-Rényi 巨型子图的快速混合来限制这些游走的混合时间。与亚临界情况不同[J. Fernley and M. Ortgiese, Random Structures Algorithms, 62 (2023), pp. 376–429]，这需要极限度分布的尾部指数[数学]以及低边缘密度，在巨型组件的情况下，我们还解决了“超小世界”幂律指数[数学]。