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The widespread availability of geotagged data combined with modern map services allows for the accurate attachment of data to spatial networks. Applying statistical analysis, such as hotspot detection, over spatial networks is very important for precise quantification and patterns analysis, which empowers effective decision-making in various important applications. Existing hotspot detection algorithms on spatial networks either lack sufficient statistical evidence on detected hotspots, such as clustering, or they provide statistical evidence at a prohibitive computational overhead. In this paper, we propose efficient algorithms for detecting hotspots based on the network local K-function for predefined and unknown hotspot radii. The K-function is a widely adopted statistical approach for network pattern analysis that enables the understanding of the density and distribution of activities and events happening within the spatial network. However, its practical application has been limited due to the inefficiency of state-of-the-art algorithms, particularly for large-sized networks. Extensive experimental evaluation using real and synthetic datasets shows that our algorithms are up to 28 times faster than the state-of-the-art algorithms in computing hotspots with a predefined radius and up to more than four orders of magnitude faster in identifying hotspots without a predefined radius. Additionally, to address dynamic changes in the spatial network, we propose an incremental hotspot detection approach that efficiently updates hotspot computations by leveraging prior results as new events are added.