The phenomenon of opinion disagreement has been empirically observed and reported in the literature, which is affected by various factors, such as the structure of social networks. An important discovery in network science is that most real-life networks, including social networks, are scale-free and sparse. In this paper, we study noisy opinion dynamics in sparse scale-free social networks to uncover the influence of power-law topology on opinion disagreement. We adopt the popular discrete-time DeGroot model for opinion dynamics in a graph, where nodes' opinions are subject to white noise. We first study opinion disagreement in many realistic and model networks with a scale-free topology, which approaches a constant, indicating that a scale-free structure is resistant to noise in the opinion dynamics. Moreover, existing algorithms for estimating opinion disagreement are computationally impractical for large-scale networks due to their high computational complexity. To solve this challenge, we introduce a sublinear-time algorithm to approximate this quantity with a theoretically guaranteed error. This algorithm efficiently simulates truncated random walks starting from a subset of nodes while preserving accurate estimation. Extensive experiments demonstrate its efficiency, accuracy, and scalability.

翻译：意见分歧现象已在文献中得到实证观察和报告，其受到社交网络结构等多种因素的影响。网络科学的一项重要发现是，包括社交网络在内的大多数现实网络具有无标度和稀疏特性。本文研究稀疏无标度社交网络中的嘈杂意见动态，以揭示幂律拓扑结构对意见分歧的影响。我们采用图论中经典的离散时间DeGroot模型描述意见动态，其中节点的意见受白噪声干扰。首先，我们研究多种具有无标度拓扑的现实网络和模型网络中的意见分歧，发现其趋于常数，表明无标度结构对意见动态中的噪声具有鲁棒性。此外，现有估计意见分歧的算法由于计算复杂度高，在大规模网络中难以实际应用。为解决这一挑战，我们提出一种次线性时间算法来近似估计该量值，并具有理论保证的误差界。该算法通过高效模拟从节点子集出发的截断随机游走，同时保持估计的准确性。大量实验验证了其高效性、准确性和可扩展性。