  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.
