Structure-based influential nodes identification is a long-term challenge in the study of complex networks. While global centrality-based approaches are generally considered to be more accurate and reliable, the requirements of complete network information and high computational complexity are hard to meet, limiting their applications in many practical scenarios. In addition, recent studies have highlighted the effect of cyclic structures introducing redundant paths in network connectivity and exaggerating the importance of traditional centrality measures. In this work, we develop a new centrality metric, called Multi-Spanning Tree-based Degree Centrality (MSTDC), to quantify node importance with linear complexity by leveraging redundant ties. MSTDC is calculated using the aggregation of degrees of a small number of spanning trees constructed with a few randomly selected root nodes. Experiments on synthetic and empirical networks reveal that MSTDC obtains superior performance than other benchmark network centralities in identifying influential nodes from the perspective of both maintaining network connectivity and maximizing spreading capacity. In addition, we find that MSTDC is extraordinarily effective in networks with high clustering coefficients. Our study provides novel insights into the role of redundant ties in network structural and functional analyses.