This talk presents recent advances on community detection in geometric random graphs under the Soft Geometric Block Model (SGBM). The SGBM extends the classical Stochastic Block Model by incorporating spatial structure: vertices are randomly embedded in a compact metric space, and edge probabilities depend both on community labels and geometric distance. In such settings, classical spectral clustering methods that rely on the largest or smallest eigenvalues often fail, since those parts of the spectrum are dominated by geometric effects rather than community information. We introduce a higher-order spectral clustering algorithm for the dense regime with a fixed number k≥2 of equal-sized communities. Instead of selecting extreme eigenvalues, the algorithm targets the k−1 eigenvalues closest to a theoretically predicted value determined by the difference between intra- and inter-community connection probabilities. The associated eigenvectors provide an embedding into Rk−1, after which k-means clustering is applied. We prove spectral separation of these informative eigenvalues, characterize the limiting spectral distribution of the adjacency matrix, and establish weak consistency of the algorithm, together with strong consistency after a simple local refinement step. A key ingredient is a non-standard application of the Davis–Kahan theorem to control eigenspace perturbations when eigenvalues are not simple. Finally, we present a conjecture concerning spectral seriation for community recovery in one-dimensional geometric graphs. In this setting, we hypothesize that appropriate spectral ordering methods can recover the latent geometric arrangement of vertices and thereby reveal community structure. This conjecture suggests a deeper connection between geometry, spectral ordering, and clustering, and points toward new directions for understanding community detection in low-dimensional geometric networks.
Dr. Hariprasad Manjunath is an Assistant Professor at the School of Mathematics and Natural Sciences, Chanakya University, Bengaluru. He received his Ph.D. in Computational and Data Science from the Indian Institute of Science (IISc), Bengaluru, in 2022, where his research focused on fast algorithms for eigenvalues of periodic matrices. His research interests include spectral graph theory, matrix analysis, numerical linear algebra, and geometric random graphs. He has previously served as a Postdoctoral Researcher at INRIA, France, and as an Assistant Professor at IIIT Dharwad.