Node-level out-of-distribution (OOD) detection on graphs has received significant attention from the machine learning community. However, previous approaches are evaluated using unrealistic benchmarks that consider only randomly selected OOD nodes, failing to reflect the interactions among nodes. In this paper, we introduce a new challenging task to model the interactions of OOD nodes in a graph, termed spreading OOD detection, where a newly emerged OOD node spreads its property to neighboring nodes. We curate realistic benchmarks by employing the epidemic spreading models that simulate the spreading of OOD nodes on the graph. We also showcase a Spreading COVID-19" dataset to demonstrate the applicability of spreading OOD detection in real-world scenarios. Furthermore, to effectively detect spreading OOD samples under the proposed benchmark setup, we present a new approach called energy distribution-based detector (EDBD), which includes a novel energy-aggregation scheme. EDBD is designed to mitigate undesired mixing of OOD scores between in-distribution (ID) and OOD nodes. Our extensive experimental results demonstrate the superiority of our approach over state-of-the-art methods in both spreading OOD detection and conventional node-level OOD detection tasks across seven benchmark datasets. The source code is available at https://github.com/daehoum1/edbd.