Phase transition detection in the community detection for hypergraph network via tensor method

Network community detection is a crucial task in studying the structure of complex networks. However, traditional community detection methods primarily rely on the low-order structure of networks, overlooking high-order connection patterns, thus failing to fully capture the multi-scale structure of complex networks. The community detection problem can be regarded as a graph partitioning problem, aimed at partitioning the network into substructures with dense internal connections and sparse external connections by identifying the inter-cluster edge between communities. This paper investigates the general model of 3-uniform hypergraph subnetworks connected by random hyperedges. By leveraging hypergraph Laplacian spectral theory and inequality bounding, we demonstrate the existence of a critical phase transition when employing spectral clustering for community detection, contingent upon the probability of random inter-cluster hyperedge connections. Specifically, as the phase transition is approached, the community detection performance transitions from nearly perfect detectability to lower detectability. We derive upper and lower bounds for the phase transition and prove its exact location when the sizes of the two subnetworks are equal. Using simulated and real-world networks, we illustrate how to utilize empirical estimation of these bounds to validate the reliability of detected communities.