Abstract

The mixing time of random walks on a graph has found broad applications across both theoretical and practical aspects of computer science, with the application effects depending on the behavior of mixing time. It is extensively believed that real-world networks, especially social networks, are fast mixing with their mixing time at most $O(\log N)$ where $N$ is the number of vertices. However, the behavior of mixing time in the real-life networks has not been examined carefully, and exactly analytical research for mixing time in models mimicking real networks is still lacking. In this paper, we first experimentally evaluate the mixing time of various real-world networks with scale-free small-world properties and show that their mixing time is much higher than anticipated. To better understand the behavior of the mixing time for real-world networks, we then analytically study the mixing time of the Apollonian network, which is simultaneously scale-free and small-world. To this end, we derive the recursive relations for all eigenvalues, especially the second largest eigenvalue modulus of the transition matrix, based on which we deduce a lower bound for the mixing time of the Apollonian network, which approximately scales sublinearly with $N$. Our results indicate that real-world networks are not always fast mixing, which has potential implications in the design of algorithms related to mixing time.

中文翻译：

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现实世界中的网络并非总是快速混合

摘要

图上随机游动的混合时间在计算机科学的理论和实践方面都得到了广泛的应用，其应用效果取决于混合时间的行为。广泛认为，现实世界的网络，尤其是社交网络，正在以其混合时间最多$ O（\ log N）$进行快速混合，其中$ N $是顶点数。但是，尚未仔细检查现实网络中混合时间的行为，并且仍然缺乏针对模拟真实网络的模型中的混合时间的精确分析研究。在本文中，我们首先通过实验评估各种具有无标度小世界属性的现实世界网络的混合时间，并表明它们的混合时间比预期的要长得多。为了更好地了解现实网络中混合时间的行为，然后，我们分析性地研究了Apollonian网络的混合时间，该时间同时是无标度和小世界的。为此，我们推导了所有特征值的递归关系，尤其是过渡矩阵的第二大特征值模量，在此基础上，我们推导出了Apollonian网络混合时间的下界，该界约用$ N $线性地缩放。我们的结果表明，现实世界中的网络并非总是快速混合，这在与混合时间相关的算法设计中具有潜在的含义。以此为基础，我们推导出了Apollonian网络的混合时间的下界，它与$ N $近似成线性关系。我们的结果表明，现实世界中的网络并非总是快速混合，这在与混合时间相关的算法设计中具有潜在的含义。以此为基础，我们推导出了Apollonian网络的混合时间的下界，它与$ N $近似成线性关系。我们的结果表明，现实世界中的网络并非总是快速混合，这在与混合时间相关的算法设计中具有潜在的含义。