Spectral densities approximations of incidence-based locally treelike hypergraph matrices via the cavity method

Network science has significantly advanced our understanding of complex systems by representing them as graphs. Vertices correspond to system components, and edges capture pairwise interactions. However, many real-world systems (e.g., chemical reactions, brain networks, scientific collaborations) involve higher-order interactions that graphs fail to capture fully. Hypergraphs offer a more suitable framework, allowing interactions among multiple components, with each hyperedge connecting an arbitrary number of vertices. While significant progress has been made in studying the spectral properties of the matrix representation of a hypergraph, less attention has been given to efficiently computing its spectral density. Existing approaches primarily rely on direct diagonalization, which scales cubically with the number of vertices and is thus computationally prohibitive for large hypergraphs. In this work, we develop an efficient method for computing the spectral density of the signless Laplacian, adjacency, and Laplacian matrices of weighted hypergraphs using the cavity method. The cavity method is based on the incidence matrix, the most common way to represent hypergraphs. We further refine this approach to derive a more efficient method for unweighted hypergraphs that requires only the degree and order sequences. Finally, we validate the effectiveness of our processes demonstrating their computational efficiency and accuracy.