随机图上Gossip观点动力学中的集中性
Concentration in Gossip Opinion Dynamics over Random Graphs
AI总结：
研究随机图上Gossip观点动力学中的集中不等式，通过引入期望图上的Gossip模型近似原始过程，推导了高概率界，并利用矩阵扰动和马尔可夫链集中不等式分析网络结构对最终观点的影响，验证了理论发现。
摘要：
我们研究随机图上Gossip观点动力学中的集中不等式。在该模型中，网络由具有独立边的随机图模型生成，智能体通过网络随机进行两两交互。在此过程中，常规智能体对邻居的观点取平均后更新，而固执智能体不改变观点。为近似原始过程，我们引入了期望图上的Gossip模型，该期望图通过对随机图模型生成的所有可能网络取平均得到。利用集中不等式，我们推导了随机图上的期望最终观点向量与期望图上的期望最终观点向量之间距离的高概率界。借助矩阵扰动结果，我们展示了这种集中性如何帮助研究网络结构对期望最终观点的影响，分两种情况：（i）当固执智能体的影响较大时，期望最终观点发生极化，且接近固执智能体的观点；（ii）当固执智能体的影响较小时，期望最终观点彼此接近。借助马尔可夫链的集中不等式，我们获得了时间平均观点与期望图上的期望最终观点之间距离的高概率界。在仿真中，我们验证了理论发现，并研究了具有社区结构的随机块模型上的Gossip模型。
英文摘要：
We study concentration inequalities in gossip opinion dynamics over random graphs. In the model, a network is generated from a random graph model with independent edges, and agents interact pairwise randomly over the network. During the process, regular agents average neighbors' opinions and then update, whereas stubborn agents do not change opinions. To approximate the original process, we introduce a gossip model over an expected graph, obtained by averaging all possible networks generated from the random graph model. Using concentration inequalities, we derive high-probability bounds for the distance between the expected final opinion vectors over the random graph and over the expected graph. Leveraging matrix perturbation results, we show how such concentration can help study the effect of network structure on the expected final opinions in two cases: (i) When the influence of stubborn agents is large, the expected final opinions polarize and are close to stubborn agents' opinions. (ii) When the influence of stubborn agents is small, the expected final opinions are close to each other. With the help of concentration inequalities for Markov chains, we obtain high-probability bounds for the distance between time-averaged opinions and the expected final opinions over the expected graph. In simulation, we validate the theoretical findings, and study a gossip model over a stochastic block model that has community structure.
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