Empirical networks possess considerable heterogeneity of node connections, resulting in a small portion of nodes playing crucial roles in network structure and function. Yet, how to characterize nodes' influence and identify vital nodes is by far still unclear in the study of networks with higher-order interactions. In this paper, we introduce a multi-order graph obtained by incorporating the higher-order bipartite graph and the classical pairwise graph, and propose a Higher-order Augmented Random Walk (HoRW) model through random walking on it. This representation preserves as much information about the higher-interacting network as possible. The results indicate that the proposed method effectively addresses the localization problem of certain classical centralities. In contrast to random walks along pairwise interactions only, performing more walks along higher-order interactions assists in not only identifying the most important nodes but also distinguishing nodes that ranked in the middle and bottom. Our method outperforms classical centralities in identifying vital nodes and can scale to various tasks in networks, including information spread maximization and network dismantling problems. The proposed higher-order representation and the random walk model provide novel insights and potent tools for studying higher-order mechanisms and functionality.

翻译：经验网络具有显著的节点连接异质性，导致少量节点在网络结构与功能中扮演关键角色。然而，在具有高阶相互作用的网络研究中，如何刻画节点影响力并识别关键节点至今仍不明确。本文通过融合高阶二分图与经典成对图构建多阶图，并在此基础上提出高阶增广随机游走模型（Higher-order Augmented Random Walk, HoRW）。该表示方法尽可能完整地保留了高阶交互网络的信息。结果表明，所提方法有效解决了某些经典中心性指标的定位问题。与仅沿成对交互进行的随机游走相比，沿高阶交互进行更多游走不仅有助于识别最重要节点，还能对排名居中和靠后的节点进行区分。本方法在识别关键节点方面优于经典中心性指标，并可扩展至网络中的各类任务，包括信息传播最大化与网络拆解问题。所提出的高阶表示方法与随机游走模型为研究高阶机制与功能提供了全新视角与有力工具。