Random walks have been proven to be useful for constructing various algorithms to gain information on networks. Algorithm node2vec employs biased random walks to realize embeddings of nodes into low-dimensional spaces, which can then be used for tasks such as multi-label classification and link prediction. The performance of the node2vec algorithm in these applications is considered to depend on properties of random walks that the algorithm uses. In the present study, we theoretically and numerically analyse random walks used by the node2vec. Those random walks are second-order Markov chains. We exploit the mapping of its transition rule to a transition probability matrix among directed edges to analyse the stationary probability, relaxation times in terms of the spectral gap of the transition probability matrix, and coalescence time. In particular, we show that node2vec random walk accelerates diffusion when walkers are designed to avoid both backtracking and visiting a neighbour of the previously visited node but do not avoid them completely.

中文翻译：

---

网络上的node2vec随机游走分析

随机游走已被证明对于构建各种算法来获取网络信息非常有用。算法node2vec采用有偏随机游走来实现节点嵌入到低维空间中，然后可用于多标签分类和链接预测等任务。在这些应用中，node2vec 算法的性能被认为取决于该算法使用的随机游走的属性。在本研究中，我们从理论上和数值上分析了 node2vec 使用的随机游走。这些随机游走是二阶马尔可夫链。我们利用其转移规则到有向边之间的转移概率矩阵的映射来分析平稳概率、转移概率矩阵的谱间隙方面的弛豫时间以及合并时间。特别是，我们表明，当步行者被设计为避免回溯和访问先前访问过的节点的邻居但不完全避免它们时，node2vec 随机游走会加速扩散。